Table of Small Finite Groups

Order 1: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC1.svg 1 = S1 = A2 Y Y Y

Order 2: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC2.svg ℤ2 = S2 Y Y Y

Order 3: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC3.svg ℤ3 = A3 Y Y Y

Order 4: 2 Groups

Abelian Cyclic Simple
GroupDiagramMiniC4.svg ℤ4 Y Y
GroupDiagramMiniD4.svg ℤ22 = K4 Y

Order 5: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC5.svg ℤ5 Y Y Y

Order 6: 2 Groups

Abelian Cyclic Simple
GroupDiagramMiniC6.svg ℤ6 = 3 × 2 Y Y
GroupDiagramMiniD6.svg Dih3 = S3 

Order 7: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC7.svg ℤ7 Y Y Y

Order 8: 5 Groups

Abelian Cyclic Simple
GroupDiagramMiniC8.svg ℤ8 Y Y
GroupDiagramMiniC2C4.svg ℤ4 × 2 Y
GroupDiagramMiniC2x3.svg ℤ23 Y
GroupDiagramMiniD8.svg Dih4
GroupDiagramMiniQ8.svg Dic2 = Q8

Order 9: 2 Groups

Abelian Cyclic Simple
GroupDiagramMiniC9.svg ℤ9 Y Y
GroupDiagramMiniC3x2.svg ℤ3 × 3 Y

Order 10: 2 Groups

Abelian Cyclic Simple
GroupDiagramMiniC10.svg ℤ10 = ℤ5 × 2 Y Y
GroupDiagramMiniD10.svg Dih5

Order 11: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC11.svg ℤ11 Y Y Y

Order 12: 5 Groups

Abelian Cyclic Simple
GroupDiagramMiniC12.svg ℤ12 = ℤ4 × 3 Y Y
GroupDiagramMiniC2C6.svg ℤ6 × 2 = ℤ3 × 22 Y
GroupDiagramMiniX12.svg Dic3
GroupDiagramMiniA4.svg A4
GroupDiagramMiniD12.svg Dih6 = Dih3 × 2

Order 13: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC13.svg ℤ13 Y Y Y

Order 14: 2 Groups

Abelian Cyclic Simple
GroupDiagramMiniC14.svg ℤ14 = ℤ7 × 2 Y Y
GroupDiagramMiniD14.svg Dih7

Order 15: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC15.svg ℤ15 = ℤ5 × 3 Y Y

Order 16: 14 Groups

Abelian Cyclic Simple
GroupDiagramMiniC16.svg ℤ16 Y Y
GroupDiagramC2C8.svg ℤ8 × 2 Y
GroupDiagramMiniC4x2.svg ℤ42 Y
GroupDiagramMiniC2x2C4.svg ℤ4 × 22 Y
GroupDiagramMiniC2x4.svg ℤ24 Y
GroupDiagramMiniD16.svg Dih8
GroupDiagramMiniQ16.svg Dic4
GroupDiagramMiniC2D8.svg Dih4 × 2
GroupDiagramMiniC2Q8.svg Dic2 × 2
GroupDiagramMiniG44.svg K4 ⋊ 4
GroupDiagramMinix3.svg ℤ4 ⋊ 4
GroupDiagramMOD16.svg ℤ8 ⋊ 2
GroupDiagramMiniC2x2C4.svg (4 × 2) ⋊ 2
GroupDiagramMiniQH16.svg QD16

Order 17: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC17.svg ℤ17 Y Y Y

Order 18: 5 Groups

Abelian Cyclic Simple
GroupDiagramMiniC18.svg ℤ18 = 9 × 2 Y Y
GroupDiagramMiniC3C6.png ℤ6 × 3 = 32 × 2 Y
GroupDiagramMiniD18.png Dih9
GroupDiagramMiniC3D6.png S3 × 3
GroupDiagramMiniG18-4.png (3 × 3) ⋊ 2

Order 19: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC19.svg ℤ19 Y Y Y

Order 20: 5 Groups

Abelian Cyclic Simple
GroupDiagramMiniC20.svg ℤ20 = 5 × 4 Y Y
GroupDiagramMiniC2C10.png ℤ10 × 2 = 5 × 3 × 2 Y
GroupDiagramMiniD20.png Dih10 = Dih5 × 2
GroupDiagramMiniQ20.png Dic5
GroupDiagramMiniC5semiprodC4.png ℤ5 ⋊ 4

Order 21: 2 Groups

Abelian Cyclic Simple
GroupDiagramMiniC21.svg ℤ20 = 7 × 3 Y Y
Frob21 cycle graph.svg ℤ7 ⋊ 3

Order 22: 2 Groups

Abelian Cyclic Simple
GroupDiagramMiniC22.svg ℤ22 = 11 × 2 Y Y
 Dih11

Order 23: 1 Group

Abelian Cyclic Simple
GroupDiagramMiniC23.svg ℤ23 Y Y Y

Order 24: 14 Groups

Abelian Cyclic Simple
GroupDiagramMiniC24.svg ℤ24 = 8 × 3 Y Y
12 × 2 = 6 × 4 Y
6 × 22 = 3 × 23 Y
Dih12
Dih6 × 2
Dih4 × 3
Symmetric group 4; cycle graph.svg S4
S3 × 4
GroupDiagramMiniA4xC2.png A4 × 2
GroupDiagramMiniQ24.png Dic6
Dic3 × 2
Dic2 × 3
3 ⋊ 8
3 ⋊ Dih4
SL(2,3); Cycle graph.svg SL(2,3) = 2T

Order 25: 2 Groups

Abelian Cyclic Simple
25 Y Y
52 Y

Order 26: 2 Groups

Abelian Cyclic Simple
26 = 13 × 2 Y Y
Dih13

Order 27: 5 Groups

Abelian Cyclic Simple
27 Y Y
9 × 3 Y
33 Y
32 ⋊ 3
9 ⋊ 3

Order 28: 3 Groups

Abelian Cyclic Simple
28 = 7 × 4 Y Y
14 × 2 = 7 × 22 Y
Dih14
Z7 ⋊ Z4

Order 29: 1 Group

Abelian Cyclic Simple
29 Y Y Y

Order 30: 4 Groups

Abelian Cyclic Simple
30 = 15 × 2 = 10 × 3 = 6 × 5 Y Y
Dih15
Dih5 × 3
S3 × 5

Order 31: 1 Group

(Order p)

Abelian Cyclic Simple
31 Y Y Y

Order 32: 51 Groups

I’m not going to list these all here, but seven of them are abelian.

(All cycle graphs stolen from Wikipedia)

Notable Groups

A5: Smallest simple group that isn’t cyclic. Order 60.

22: Smallest non-cyclic group.

Dih3: Smallest non-abelian group.

A4: First example where n divides |G| but there is no element of order n. (n = 6)

Z7 ⋊ Z3: Smallest non-abelian group of odd size.

Dih3: Smallest group with a normal subgroup that isn’t isomorphic to one of its subgroups.

General Patterns

1 group of order p
2 groups of order p2
5 groups of order p3
15 groups of order p4, for p > 2
2 groups of order pq for q-1 divisible by p
1 group of order pq for q-1 not divisible by p

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