Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
347 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D32 |
| commutator, cosocle | C2×C32 | C24 |
| Frattini | C2 | C22×D32 |
| Fitting | C2×C32×C4 | C22 |
| socle | C22×C32 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C2×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#634 | C2 | semidirect |
| 2. C2×C3×D12 | C2 | semidirect |
| 3. C2×C3×C4×D3 | C2 | semidirect |
| 4. C2×#625 | C2 | semidirect |
| 5. C2×D3×Dic3 | C2 | semidirect |
| 6. #3795 | C2 | direct (N × H) |
| 13. #634 | C22 | semidirect |
| 15. C3×D12 | C22 | semidirect |
| 16. C3×C4×D3 | C22 | semidirect |
| 17. #625 | C22 | semidirect |
| 18. D3×Dic3 | C22 | semidirect |
| 26. C2×D12 | D3 | semidirect |
| 28. C2×C4×D3 | D3 | semidirect |
| 58. D12 | C2×D3 | semidirect |
| 60. C4×D3 | C2×D3 | semidirect |
| 64. C2×C32 | #41 | semidirect |
| 93. C32 | C2×#41 | semidirect |
| 108. C2×C3 | #235 | semidirect |
| 109. C2×C3 | #237 | semidirect |
| 123. C3 | C2×#235 | semidirect |
| 124. C3 | C2×#237 | semidirect |
| 127. C2 | #3795 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#634 | 2 |
| C2×C3×D12 | 2 |
| C2×C3×C4×D3 | 2 |
| C2×#625 | 2 |
| C2×D3×Dic3 | 2 |
| #3795 | 2 |
| C2×#237 | 3 |
| C2×#235 | 3 |