List of space groups 230 space groups in three dimensions , given by a number index, and a full name in Hermann–Mauguin notation , and a short name . The long names are given with spaces for readability. The groups each have a point group of the unit cell.Symbols In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type . Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group .Bravais lattices in three dimensions:P primitiveI body centered F face centered A centered on A faces onlyB centered on B faces onlyC centered on C faces onlyR rhombohedralm within the point groups can be replaced by a glide plane , labeled as a , b , or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell . The d glide is often called the diamond glide plane as it features in the diamond structure .,, or glide translation along half the lattice vector of this face glide translation along with half a face diagonal glide planes with translation along a quarter of a face diagonal. two glides with the same glide plane and translation along two half-lattice vectors. gyration point can be replaced by a screw axis denoted by a number, n , where the angle of rotation is . The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° rotation followed by a translation of ½ of the lattice vector. 31 is a 120° rotation followed by a translation of ⅓ of the lattice vector.1 , 31 , 32 , 41 , 42 , 43 , 61 , 62 , 63 , 64 , and 65 .Schoenflies notationspace group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is somtimes supplemented with a symbol of the form which specifies the bravais lattice . Here is the lattice system , and is the centering type.Fedorov symbols , h , or a . The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups. Symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. All the other space groups are asymmorphic. Example for point group 4/mmm : the symmorphic space groups are P4/mmm and I4/mmm ; hemisymmorphic space groups should contain axial combination 422, these are P4/mcc, P4/nbm, P4/nnc, and I4/mcm.Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold 1 1 P1 P 1 1s 2 P P 2s
List of Monoclinic List of Orthorhombic Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold Fibrifold 16 222 rowspan=9 P222 P 2 2 2 9s 17 222 P2221 P 2 2 21 4a - 18 222 P21 21 2 P 21 21 2 7a - 19 222 P21 21 21 P 21 21 21 8a - 20 222 C2221 C 2 2 21 5a - 21 222 C222 C 2 2 2 10s - 22 222 F222 F 2 2 2 12s - 23 222 I222 I 2 2 2 11s - 24 222 I21 21 21 I 21 21 21 6a - 25 mm2 rowspan=22 Pmm2 P m m 2 13s 26 mm2 Pmc21 P m c 21 9a , - 27 mm2 Pcc2 P c c 2 5h - 28 mm2 Pma2 P m a 2 6h , - 29 mm2 Pca21 P c a 21 11a - 30 mm2 Pnc2 P n c 2 7h , - 31 mm2 Pmn21 P m n 21 10a , - 32 mm2 Pba2 P b a 2 9h - 33 mm2 Pna21 P n a 21 12a , - 34 mm2 Pnn2 P n n 2 8h - 35 mm2 Cmm2 C m m 2 14s - 36 mm2 Cmc21 C m c 21 13a , - 37 mm2 Ccc2 C c c 2 10h - 38 mm2 Amm2 A m m 2 15s , - 39 mm2 Aem2 A b m 2 11h , - 40 mm2 Ama2 A m a 2 12h , - 41 mm2 Aea2 A b a 2 13h , - 42 mm2 Fmm2 F m m 2 17s - 43 mm2 Fdd2 F dd2 16h - 44 mm2 Imm2 I m m 2 16s - 45 mm2 Iba2 I b a 2 15h - 46 mm2 Ima2 I m a 2 14h , - 47 rowspan=28 rowspan=28 Pmmm P 2/m 2/m 2/m 18s 48 Pnnn P 2/n 2/n 2/n 19h - - 49 Pccm P 2/c 2/c 2/m 17h - - 50 Pban P 2/b 2/a 2/n 18h - - 51 Pmma P 21 /m 2/m 2/a 14a , - - 52 Pnna P 2/n 21 /n 2/a 17a , - - 53 Pmna P 2/m 2/n 21 /a 15a , - - 54 Pcca P 21 /c 2/c 2/a 16a , - - 55 Pbam P 21 /b 21 /a 2/m 22a - - 56 Pccn P 21 /c 21 /c 2/n 27a - - 57 Pbcm P 2/b 21 /c 21 /m 23a , - - 58 Pnnm P 21 /n 21 /n 2/m 25a - - 59 Pmmn P 21 /m 21 /m 2/n 24a - - 60 Pbcn P 21 /b 2/c 21 /n 26a , - - 61 Pbca P 21 /b 21 /c 21 /a 29a - - 62 Pnma P 21 /n 21 /m 21 /a 28a , - - 63 Cmcm C 2/m 2/c 21 /m 18a , - - 64 Cmca C 2/m 2/c 21 /a 19a , - - 65 Cmmm C 2/m 2/m 2/m 19s - - 66 Cccm C 2/c 2/c 2/m 20h - - 67 Cmme C 2/m 2/m 2/e 21h - - 68 Ccce C 2/c 2/c 2/e 22h - - 69 Fmmm F 2/m 2/m 2/m 21s - - 70 Fddd F 2/d 2/d 2/d 24h - - 71 Immm I 2/m 2/m 2/m 20s - - 72 Ibam I 2/b 2/a 2/m 23h - - 73 Ibca I 2/b 2/c 2/a 21a - - 74 Imma I 2/m 2/m 2/a 20a - -
List of Tetragonal Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold 75 4 rowspan=6 P4 P 4 22s 76 4 P41 P 41 30a - 77 4 P42 P 42 33a - 78 4 P43 P 43 31a - 79 4 I4 I 4 23s - 80 4 I41 I 41 32a - 81 rowspan=2 P P 26s 82 I I 27s - 83 4/m rowspan=6 P4/m P 4/m 28s 84 4/m P42 /m P 42 /m 41a - 85 4/m P4/n P 4/n 29h - 86 4/m P42 /n P 42 /n 42a - 87 4/m I4/m I 4/m 29s - 88 4/m I41 /a I 41 /a 40a - 89 422 rowspan=10 P422 P 4 2 2 30s 90 422 P421 2 P421 2 43a - 91 422 P41 22 P 41 2 2 44a - 92 422 P41 21 2 P 41 21 2 48a - 93 422 P42 22 P 42 2 2 47a - 94 422 P42 21 2 P 42 21 2 50a - 95 422 P43 22 P 43 2 2 45a - 96 422 P43 21 2 P 43 21 2 49a - 97 422 I422 I 4 2 2 31s - 98 422 I41 22 I 41 2 2 46a - 99 4mm rowspan=12 P4mm P 4 m m 24s 100 4mm P4bm P 4 b m 26h - 101 4mm P42 cm P 42 c m 37a - 102 4mm P42 nm P 42 n m 38a - 103 4mm P4cc P 4 c c 25h - 104 4mm P4nc P 4 n c 27h - 105 4mm P42 mc P 42 m c 36a - 106 4mm P42 bc P 42 b c 39a - 107 4mm I4mm I 4 m m 25s - 108 4mm I4cm I 4 c m 28h - 109 4mm I41 md I 41 m d 34a - 110 4mm I41 cd I 41 c d 35a - 111 2m rowspan=12 P2m P 2 m 32s 112 2m P2c P 2 c 30h - 113 2m P21 m P 21 m 52a - 114 2m P21 c P 21 c 53a - 115 2m Pm2 P m 2 33s - 116 2m Pc2 P c 2 31h - 117 2m Pb2 P b 2 32h - 118 2m Pn2 P n 2 33h - 119 2m Im2 I m 2 35s - 120 2m Ic2 I c 2 34h - 121 2m I2m I 2 m 34s - 122 2m I2d I 2 d 51a - 123 4/m 2/m 2/m rowspan=20 P4/mmm P 4/m 2/m 2/m 36s 124 4/m 2/m 2/m P4/mcc P 4/m 2/c 2/c 35h - 125 4/m 2/m 2/m P4/nbm P 4/n 2/b 2/m 36h - 126 4/m 2/m 2/m P4/nnc P 4/n 2/n 2/c 37h - 127 4/m 2/m 2/m P4/mbm P 4/m 21 /b 2/m 54a - 128 4/m 2/m 2/m P4/mnc P 4/m 21 /n 2/c 56a - 129 4/m 2/m 2/m P4/nmm P 4/n 21 /m 2/m 55a - 130 4/m 2/m 2/m P4/ncc P 4/n 21 /c 2/c 57a - 131 4/m 2/m 2/m P42 /mmc P 42 /m 2/m 2/c 60a - 132 4/m 2/m 2/m P42 /mcm P 42 /m 2/c 2/m 61a - 133 4/m 2/m 2/m P42 /nbc P 42 /n 2/b 2/c 63a - 134 4/m 2/m 2/m P42 /nnm P 42 /n 2/n 2/m 62a - 135 4/m 2/m 2/m P42 /mbc P 42 /m 21 /b 2/c 66a - 136 4/m 2/m 2/m P42 /mnm P 42 /m 21 /n 2/m 65a - 137 4/m 2/m 2/m P42 /nmc P 42 /n 21 /m 2/c 67a - 138 4/m 2/m 2/m P42 /ncm P 42 /n 21 /c 2/m 65a - 139 4/m 2/m 2/m I4/mmm I 4/m 2/m 2/m 37s - 140 4/m 2/m 2/m I4/mcm I 4/m 2/c 2/m 38h - 141 4/m 2/m 2/m I41 /amd I 41 /a 2/m 2/d 59a - 142 4/m 2/m 2/m I41 /acd I 41 /a 2/c 2/d 58a -
List of Trigonal Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold 143 3 rowspan=4 P3 P 3 38s 144 3 P31 P 31 68a - 145 3 P32 P 32 69a - 146 3 R3 R 3 39s - 147 rowspan=2 P P 51s 148 R R 52s - 149 32 rowspan=7 P312 P 3 1 2 45s 150 32 P321 P 3 2 1 44s - 151 32 P31 12 P 31 1 2 72a - 152 32 P31 21 P 31 2 1 70a - 153 32 P32 12 P 32 1 2 73a - 154 32 P32 21 P 32 2 1 71a - 155 32 R32 R 3 2 46s - 156 3m rowspan=6 P3m1 P 3 m 1 40s 157 3m P31m P 3 1 m 41s - 158 3m P3c1 P 3 c 1 39h - 159 3m P31c P 3 1 c 40h - 160 3m R3m R 3 m 42s - 161 3m R3c R 3 c 41h - 162 2/m rowspan=6 P1m P 1 2/m 56s 163 2/m P1c P 1 2/c 46h - 164 2/m Pm1 P 2/m 1 55s - 165 2/m Pc1 P 2/c 1 45h - 166 2/m Rm R 2/m 57s - 167 2/m Rc R 2/c 47h -
List of Hexagonal Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold 168 6 rowspan=6 P6 P 6 49s 169 6 P61 P 61 74a - 170 6 P65 P 65 75a - 171 6 P62 P 62 76a - 172 6 P64 P 64 77a - 173 6 P63 P 63 78a - 174 P P 43s 175 6/m rowspan=2 P6/m P 6/m 53s 176 6/m P63 /m P 63 /m 81a - 177 622 rowspan=6 P622 P 6 2 2 54s 178 622 P61 22 P 61 2 2 82a - 179 622 P65 22 P 65 2 2 83a - 180 622 P62 22 P 62 2 2 84a - 181 622 P64 22 P 64 2 2 85a - 182 622 P63 22 P 63 2 2 86a - 183 6mm rowspan=4 P6mm P 6 m m 50s 184 6mm P6cc P 6 c c 44h - 185 6mm P63 cm P 63 c m 80a - 186 6mm P63 mc P 63 m c 79a - 187 m2 rowspan=4 Pm2 P m 2 48s 188 m2 Pc2 P c 2 43h - 189 m2 P2m P 2 m 47s - 190 m2 P2c P 2 c 42h - 191 6/m 2/m 2/m rowspan=4 P6/mmm P 6/m 2/m 2/m 58s 192 6/m 2/m 2/m P6/mcc P 6/m 2/c 2/c 48h - 193 6/m 2/m 2/m P63 /mcm P 63 /m 2/c 2/m 87a - 194 6/m 2/m 2/m P63 /mmc P 63 /m 2/m 2/c 88a -
List of Cubic Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Conway Fibrifold Fibrifold 195 23 rowspan=5 P23 P 2 3 59s 196 23 F23 F 2 3 61s - 197 23 I23 I 2 3 60s - 198 23 P21 3 P 21 3 89a - 199 23 I21 3 I 21 3 90a - 200 2/m rowspan=7 Pm P 2/m 62s 201 2/m Pn P 2/n 49h - 202 2/m Fm F 2/m 64s - 203 2/m Fd F 2/d 50h - 204 2/m Im I 2/m 63s - 205 2/m Pa P 21 /a 91a - 206 2/m Ia I 21 /a 92a - 207 432 rowspan=8 P432 P 4 3 2 68s 208 432 P42 32 P 42 3 2 98a - 209 432 F432 F 4 3 2 70s - 210 432 F41 32 F 41 3 2 97a - 211 432 I432 I 4 3 2 69s - 212 432 P43 32 P 43 3 2 94a - 213 432 P41 32 P 41 3 2 95a - 214 432 I41 32 I 41 3 2 96a - 215 3m rowspan=6 P3m P 3 m 65s 216 3m F3m F 3 m 67s - 217 3m I3m I 3 m 66s - 218 3m P3n P 3 n 51h - 219 3m F3c F 3 c 52h - 220 3m I3d I 3 d 93a - 221 4/m 2/m rowspan=10 Pmm P 4/m 2/m 71s 222 4/m 2/m Pnn P 4/n 2/n 53h - 223 4/m 2/m Pmn P 42 /m 2/n 102a - 224 4/m 2/m Pnm P 42 /n 2/m 103a - 225 4/m 2/m Fmm F 4/m 2/m 73s - 226 4/m 2/m Fmc F 4/m 2/c 54h - 227 4/m 2/m Fdm F 41 /d 2/m 100a - 228 4/m 2/m Fdc F 41 /d 2/c 101a - 229 4/m 2/m Imm I 4/m 2/m 72s - 230 4/m 2/m Iad I 41 /a 2/d 99a -
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