Finite Group DatabasebetaFinite Group DatabasebetaOrder 72.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | S4 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | C3×D4 | — | 3 | 1 | no | maximal |
| 4 | C3×D3 | — | 4 | 1 | no | maximal |
| 5 | C22×C3 | — | 3 | 1 | no | — |
| 6 | C22×C3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | A4 | — | 2 | 1 | no | — |
| 8 | A4 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3×C4 | — | 3 | 1 | no | — |
| 10 | C32 | — | 4 | 1 | no | Sylow |
| 11 | D4 | — | 3 | 1 | no | Sylow |
| 12 | C2×C3 | — | 3 | 1 | no | — |
| 13 | C2×C3 | — | 6 | 1 | no | — |
| 14 | D3 | — | 4 | 1 | no | — |
| 15 | C22 | — | 3 | 1 | no | — |
| 16 | C22 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | C4 | — | 3 | 1 | no | — |
| 18 | C3 | — | 4 | 1 | no | — |
| 19 | C3 | — | 8 | 1 | no | — |
| 20 | C3 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 21 | C2 | — | 3 | 1 | no | — |
| 22 | C2 | — | 6 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | S4 |
| commutator, cosocle, hypocenter | A4 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×A4 | C2 | semidirect |
| 2. S4 | C3 | direct (N × H) |
| 6. C22×C3 | D3 | semidirect |
| 8. A4 | C2×C3 | semidirect |
| 16. C22 | C3×D3 | semidirect |
| 20. C3 | S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×A4 | 2 |
| S4 | 3 |
| C3×D4 | 3 |
| C3×D3 | 4 |