Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
355 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 355 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 | C23×C32 | C22 |
| socle | C22×C32 | C23 |
| hypercenter | C23 | D32 |
| hypocenter | C32 | C2×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#638 | C2 | semidirect |
| 2. C2×C3×#67 | C2 | semidirect |
| 3. C22×C3×Dic3 | C2 | semidirect |
| 4. C2×#627 | C2 | semidirect |
| 5. C2×#626 | C2 | semidirect |
| 6. C2×#624 | C2 | semidirect |
| 7. #3804 | C2 | direct (N × H) |
| 8. C2×D3×Dic3 | C2 | semidirect |
| 14. #638 | C22 | semidirect |
| 16. C3×#67 | C22 | semidirect |
| 20. C2×C3×Dic3 | C22 | semidirect |
| 21. #627 | C22 | semidirect |
| 22. #626 | C22 | semidirect |
| 23. #624 | C22 | semidirect |
| 24. D3×Dic3 | C22 | semidirect |
| 31. C2×#67 | D3 | semidirect |
| 33. C22×Dic3 | D3 | semidirect |
| 71. #67 | C2×D3 | semidirect |
| 79. C2×Dic3 | C2×D3 | semidirect |
| 87. C2×C32 | #41 | semidirect |
| 127. C32 | C2×#41 | semidirect |
| 148. C2×C3 | #235 | semidirect |
| 149. C2×C3 | #237 | semidirect |
| 168. C3 | C2×#235 | semidirect |
| 169. C3 | C2×#237 | semidirect |
| 173. C2 | #3804 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#638 | 2 |
| C2×C3×#67 | 2 |
| C22×C3×Dic3 | 2 |
| C2×#627 | 2 |
| C2×#626 | 2 |
| C2×#624 | 2 |
| #3804 | 2 |
| C2×D3×Dic3 | 2 |
| C2×#237 | 3 |
| C2×#235 | 3 |