Finite Group DatabasebetaFinite Group DatabasebetaOrder 72.
35 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×D3 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3⋊Dic3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #67 | — | 3 | 2 | no | maximal |
| 4 | C2×C32 | C22 | 1 | 1 | yes | characteristic |
| 5 | C3×D3 | — | 2 | 2 | no | — |
| 6 | C22×C3 | — | 3 | 2 | no | — |
| 7 | C2×D3 | D3 | 1 | 2 | yes | complemented |
| 8 | Dic3 | — | 3 | 2 | no | — |
| 9 | Dic3 | — | 6 | 1 | no | — |
| 10 | C32 | D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | D4 | — | 9 | 1 | no | Sylow |
| 12 | C2×C3 | — | 2 | 1 | no | — |
| 13 | C2×C3 | — | 6 | 2 | no | — |
| 14 | C2×C3 | C2×D3 | 1 | 2 | yes | — |
| 15 | D3 | — | 2 | 2 | no | — |
| 16 | C22 | — | 3 | 2 | no | — |
| 17 | C4 | — | 9 | 1 | no | — |
| 18 | C3 | — | 2 | 1 | no | — |
| 19 | C3 | #67 | 1 | 2 | yes | complemented |
| 20 | C2 | — | 6 | 2 | no | — |
| 21 | C2 | D32 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32 |
| commutator, Fitting, socle, cosocle | C2×C32 | C22 |
| hypocenter | C32 | D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×D3 | C2 | semidirect |
| 2. C3⋊Dic3 | C2 | semidirect |
| 7. C2×D3 | D3 | semidirect |
| 10. C32 | D4 | semidirect |
| 19. C3 | #67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D3 | 2 |
| C3⋊Dic3 | 2 |
| #67 | 3 |