List of small groups


The following list in mathematics contains the finite groups of small order up to group isomorphism.

Counts

For the number of nonisomorphic groups of order is
For labeled groups, see.

Glossary

Each group is named by their Small Groups library index as Goi, where o is the order of the group, and i is the index of the group within that order.
Common group names:
The notations Zn and Dihn have the advantage that point groups in three dimensions Cn and Dn do not have the same notation. There are more isometry groups than these two, of the same abstract group type.
The notation denotes the direct product of the two groups; Gn denotes the direct product of a group with itself n times. G ⋊ H denotes a semidirect product where H acts on G; this may also depend on the choice of action of H on G
Abelian and simple groups are noted. The equality sign denotes isomorphism.
The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.
In the lists of subgroups, the trivial group and the group itself are not listed. Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses.

List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups.
The numbers of nonisomorphic abelian groups of orders are
For labeled Abelian groups, see.
OrderIDGoiGroupNontrivial proper SubgroupsCycle
graph		Properties
11G11Z1 = S1 = A2–Trivial. Cyclic. Alternating. Symmetric. Elementary.
22G21Z2 = S2 = Dih1–Simple. Symmetric. Cyclic. Elementary.
33G31Z3 = A3–Simple. Alternating. Cyclic. Elementary.
44G41Z4 = Dic1Z2Cyclic.
45G42Z22 = K4 = Dih2Z2 Elementary. Product.
56G51Z5–Simple. Cyclic. Elementary.
68G62Z6 = Z3 × Z2Z3, Z2Cyclic. Product.
79G71Z7–Simple. Cyclic. Elementary.
810G81Z4, Z2Cyclic.
811G82Z4 × Z2Z22, Z4, Z2 Product.
814G85Z23Z22, Z2 Product. Elementary.
915G91Z9Z3Cyclic.
916G92Z32Z3 Elementary. Product.
1018G102Z10 = Z5 × Z2Z5, Z2Cyclic. Product.
1119G111Z11–Simple. Cyclic. Elementary.
1221G122Z12 = Z4 × Z3Z6, Z4, Z3, Z2Cyclic. Product.
1224G125Z6 × Z2 = Z3 × Z22Z6, Z3, Z2, Z22Product.
1325G131Z13–Simple. Cyclic. Elementary.
1427G142Z14 = Z7 × Z2Z7, Z2Cyclic. Product.
1528G151Z15 = Z5 × Z3Z5, Z3Cyclic. Product.
1629G161Z16Z8, Z4, Z2Cyclic.
1630G162Z42Z2, Z4, Z22, Product.
1633G165Z8 × Z2Z2, Z4, Z22, Z8, Product.
1638G1610Z4 × Z22Z2, Z4, Z22, Z23, Product.
1642G1614Z24 = K42Z2, Z22, Z23 Product. Elementary.
1743G171Z17–Simple. Cyclic. Elementary.
1845G182Z18 = Z9 × Z2Z9, Z6, Z3, Z2Cyclic. Product.
1848G185Z6 × Z3 = Z32 × Z2Z6, Z3, Z2Product.
1949G191Z19–Simple. Cyclic. Elementary.
2051G202Z20 = Z5 × Z4Z10, Z5, Z4, Z2Cyclic. Product.
2054G205Z10 × Z2 = Z5 × Z22Z5, Z2Product.
2156G212Z21 = Z7 × Z3Z7, Z3Cyclic. Product.
2258G222Z22 = Z11 × Z2Z11, Z2Cyclic. Product.
2359G231Z23–Simple. Cyclic. Elementary.
2461G242Z24 = Z8 × Z3Z12, Z8, Z6, Z4, Z3, Z2Cyclic. Product.
2468G249Z12 × Z2 = Z6 × Z4
= Z4 × Z3 × Z2		Z12, Z6, Z4, Z3, Z2Product.
2474G2415Z6 × Z22 = Z3 × Z23Z6, Z3, Z2Product.
2575G251Z25Z5Cyclic.
2576G252Z52Z5Product. Elementary.
2678G262Z26 = Z13 × Z2Z13, Z2Cyclic. Product.
2779G271Z27Z9, Z3Cyclic.
2780G272Z9 × Z3Z9, Z3Product.
2783G275Z33Z3Product. Elementary.
2885G282Z28 = Z7 × Z4Z14, Z7, Z4, Z2Cyclic. Product.
2887G284Z14 × Z2 = Z7 × Z22Z14, Z7, Z4, Z2Product.
2988G291Z29–Simple. Cyclic. Elementary.
3092G304Z30 = Z15 × Z2 = Z10 × Z3
= Z6 × Z5 = Z5 × Z3 × Z2		Z15, Z10, Z6, Z5, Z3, Z2Cyclic. Product.
3193G311Z31–Simple. Cyclic. Elementary.

List of small non-abelian groups

The numbers of non-abelian groups, by order, are counted by.
However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are
OrderIDGoiGroupNontrivial proper SubgroupsCycle
graph		Properties
67G61Dih3 = S3 = D6Z3, Z2 Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group
812G83Dih4 = D8Z4, Z22, Z2 Dihedral group. Extraspecial group. Nilpotent.
813G84Q8 = Dic2 = <2,2,2>Z4, Z2Quaternion group, Hamiltonian group. all subgroups are normal without the group being abelian. The smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G. Extraspecial group Binary dihedral group. Nilpotent.
1017G101Dih5 = D10Z5, Z2 Dihedral group, Frobenius group
1220G121Q12 = Dic3 = <3,2,2>
= Z3 ⋊ Z4		Z2, Z3, Z4, Z6Binary dihedral group
1222G123A4 = K4 ⋊ Z3
= ⋊ Z3		Z22, Z3, Z2 Alternating group. No subgroups of order 6, although 6 divides its order. Frobenius group
1223G124Dih6 = D12 = Dih3 × Z2Z6, Dih3, Z22, Z3, Z2 Dihedral group, product
1426G141Dih7 = D14Z7, Z2 Dihedral group, Frobenius group
1631G163G4,4 = K4 ⋊ Z4
⋊ Z2		E8, Z4 × Z2, Z4, K4, Z2 Has the same number of elements of every order as the Pauli group. Nilpotent.
1632G164Z4 ⋊ Z4The squares of elements do not form a subgroup. Has the same number of elements of every order as Q8 × Z2. Nilpotent.
1634G166Z8 ⋊ Z2Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q8 × Z2 are also modular. Nilpotent.
1635G167Dih8 = D16Z8, Dih4, Z22, Z4, Z2 Dihedral group. Nilpotent.
1636G168QD16The order 16 quasidihedral group. Nilpotent.
1637G169Q16 = Dic4 = <4,2,2>generalized quaternion group, binary dihedral group. Nilpotent.
1639G1611Dih4 × Z2Dih4,, Z23, Z22, Z4, Z2 Product. Nilpotent.
1640G1612Q8 × Z2Hamiltonian, product. Nilpotent.
1641G1613 ⋊ Z2The Pauli group generated by the Pauli matrices. Nilpotent.
1844G181Dih9 = D18Dihedral group, Frobenius group
1846G183S3 × Z3Product
1847G184 ⋊ Z2Frobenius group
2050G201Q20 = Dic5 = <5,2,2>Binary dihedral group
2052G203Z5 ⋊ Z4Frobenius group
2053G204Dih10 = Dih5 × Z2 = D20Dihedral group, product
2155G211Z7 ⋊ Z3Z7, Z3 Smallest non-abelian group of odd order. Frobenius group
2257G221Dih11 = D22Z11, Z2 Dihedral group, Frobenius group
2460G241Z3 ⋊ Z8Central extension of S3
2462G243SL = 2T = Q8 ⋊ Z3Binary tetrahedral group
2463G244Q24 = Dic6 = <6,2,2> = Z3 ⋊ Q8Binary dihedral
2464G245Z4 × S3Product
2465G246Dih12Dihedral group
2466G247Dic3 × Z2 = Z2 × Product
2467G248 ⋊ Z2 = Z3 ⋊ Dih4Double cover of dihedral group
2469G2410Dih4 × Z3Product. Nilpotent.
2470G2411Q8 × Z3Product. Nilpotent.
2471G2412S4Symmetric group. Has no normal Sylow subgroups.
2472G2413A4 × Z2Product
2473G2414D12× Z2Product
2677G261Dih13Dihedral group, Frobenius group
2781G273Z32 ⋊ Z3All non-trivial elements have order 3. Extraspecial group. Nilpotent.
2782G274Z9 ⋊ Z3Extraspecial group. Nilpotent.
2884G281Z7 ⋊ Z4Binary dihedral group
2886G283Dih14Dihedral group, product
3089G301Z5 × S3Product
3090G302Z3 × Dih5Product
3091G303Dih15Dihedral group, Frobenius group

Classifying groups of small order

Small groups of prime power order pn are given as follows:
Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p complement include:
The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:
It contains explicit descriptions of the available groups in computer readable format.
The smallest order for which the SmallGroups library does not have information is 2048.