Finite Group DatabasebetaFinite Group DatabasebetaOrder 72.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #67 | — | 3 | 1 | no | maximal |
| 5 | C2×C9 | — | 2 | 1 | no | — |
| 6 | C2×C9 | C22 | 1 | 1 | yes | characteristic |
| 7 | D9 | — | 2 | 1 | no | — |
| 8 | C22×C3 | D3 | 1 | 1 | yes | characteristic |
| 9 | C2×D3 | — | 3 | 1 | no | — |
| 10 | Dic3 | — | 3 | 1 | no | — |
| 11 | C9 | D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | D4 | — | 9 | 1 | no | Sylow |
| 13 | C2×C3 | — | 2 | 1 | no | — |
| 14 | C2×C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 15 | D3 | — | 6 | 1 | no | — |
| 16 | C22 | — | 9 | 1 | no | — |
| 17 | C22 | D9 | 1 | 1 | yes | characteristic, complemented |
| 18 | C4 | — | 9 | 1 | no | — |
| 19 | C3 | #67 | 1 | 1 | yes | characteristic |
| 20 | C2 | — | 2 | 1 | no | — |
| 21 | C2 | — | 18 | 1 | no | — |
| 22 | C2 | C2×D9 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D9 |
| commutator, cosocle | C2×C9 | C22 |
| Frattini, socle | C2×C3 | C2×D3 |
| Fitting | C22×C9 | C2 |
| hypercenter | C22 | D9 |
| hypocenter | C9 | D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9 | C2 | semidirect |
| 2. C2×D9 | C2 | semidirect |
| 3. Dic9 | C2 | semidirect |
| 11. C9 | D4 | semidirect |
| 17. C22 | D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9 | 2 |
| C2×D9 | 2 |
| Dic9 | 2 |
| #67 | 3 |