Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
205 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D32 |
| commutator, socle, cosocle | C22×C32 | C23 |
| Fitting | C2×C32×C4 | C22 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | #121 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×D12 | C2 | semidirect |
| 2. C2×#626 | C2 | semidirect |
| 3. C2×#626 | C2 | semidirect |
| 4. C2×D3×Dic3 | C2 | semidirect |
| 5. #3752 | C2 | semidirect |
| 6. C3×Dic3⋊C4 | C2 | semidirect |
| 7. (C2×D3)⋊Dic3 | C2 | semidirect |
| 27. C2×D12 | D3 | semidirect |
| 35. Dic3⋊C4 | D3 | semidirect |
| 45. C2×C3×D3 | D4 | semidirect |
| 49. C3×Dic3 | D4 | semidirect |
| 112. C2×D3 | #67 | semidirect |
| 122. Dic3 | D12 | semidirect |
| 123. C32 | #121 | semidirect |
| 166. C3 | #895 | semidirect |
| 167. C3 | #939 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D12 | 2 |
| C2×#626 | 2 |
| C2×D3×Dic3 | 2 |
| #3752 | 2 |
| C3×Dic3⋊C4 | 2 |
| (C2×D3)⋊Dic3 | 2 |
| #939 | 3 |
| #895 | 3 |