Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3⋊(C3⋊C16) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C32×C8 | C4 | 1 | 1 | yes | characteristic |
| 3 | C3⋊C16 | — | 6 | 2 | no | — |
| 4 | C32×C4 | C8 | 1 | 1 | yes | characteristic |
| 5 | C32 | — | 9 | 1 | no | maximal, Sylow |
| 6 | C3×C8 | — | 2 | 2 | no | — |
| 7 | C2×C32 | C16 | 1 | 1 | yes | characteristic |
| 8 | C16 | — | 9 | 1 | no | — |
| 9 | C3×C4 | — | 2 | 2 | no | — |
| 10 | C32 | C32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C8 | #157 | 1 | 1 | yes | characteristic |
| 12 | C2×C3 | — | 2 | 2 | no | — |
| 13 | C4 | #622 | 1 | 1 | yes | characteristic |
| 14 | C3 | — | 2 | 2 | no | — |
| 15 | C2 | #3708 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | #157 |
| commutator, hypocenter | C32 | C32 |
| Fitting | C32×C8 | C4 |
| socle, prime radical | C2×C32 | C16 |
| cosocle | C3⋊(C3⋊C16) | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. C32 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊(C3⋊C16) | 2 |
| C32 | 9 |