Finite Group DatabasebetaFinite Group DatabasebetaOrder 72.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×D3 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×A4 | — | 3 | 1 | no | maximal |
| 4 | C3×D3 | — | 4 | 1 | no | maximal |
| 5 | C22×C3 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×D3 | — | 3 | 1 | no | — |
| 7 | C2×D3 | — | 3 | 1 | no | — |
| 8 | A4 | — | 2 | 1 | no | — |
| 9 | A4 | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 10 | C32 | — | 4 | 1 | no | Sylow |
| 11 | C23 | — | 3 | 1 | no | Sylow |
| 12 | C2×C3 | — | 3 | 1 | no | — |
| 13 | C2×C3 | — | 12 | 1 | no | — |
| 14 | D3 | — | 3 | 1 | no | — |
| 15 | D3 | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 16 | C22 | — | 9 | 1 | no | — |
| 17 | C22 | — | 9 | 1 | no | — |
| 18 | C22 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C3 | — | 4 | 1 | no | — |
| 20 | C3 | — | 8 | 1 | no | — |
| 21 | C3 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2 | — | 3 | 1 | no | — |
| 23 | C2 | — | 3 | 1 | no | — |
| 24 | C2 | — | 9 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C3 | C2×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×A4 | C2 | semidirect |
| 2. C22×D3 | C3 | semidirect |
| 5. C22×C3 | C2×C3 | semidirect |
| 9. A4 | D3 | direct (N × H) |
| 15. D3 | A4 | direct (N × H) |
| 18. C22 | C3×D3 | semidirect |
| 21. C3 | C2×A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×A4 | 2 |
| C22×D3 | 3 |
| C2×A4 | 3 |
| C3×D3 | 4 |