Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
596 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#405 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #405 | D3 |
| hypocenter | C3 | #3155 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#908 | C2 | semidirect |
| 2. C2×#904 | C2 | semidirect |
| 3. C22×D24 | C2 | semidirect |
| 4. #5525 | C2 | semidirect |
| 5. C3×#405 | C2 | semidirect |
| 6. #4712 | C2 | semidirect |
| 7. #4696 | C2 | semidirect |
| 8. #4695 | C2 | semidirect |
| 16. C2×Dic12 | C22 | semidirect |
| 21. C2×D24 | C22 | semidirect |
| 22. C2×D24 | C22 | semidirect |
| 23. C2×#204 | C22 | semidirect |
| 24. C2×#204 | C22 | semidirect |
| 27. #875 | C22 | semidirect |
| 28. #874 | C22 | semidirect |
| 29. C3×#97 | C22 | semidirect |
| 30. C3×C4×C8 | C22 | semidirect |
| 38. #405 | D3 | semidirect |
| 52. C3×M16 | D4 | semidirect |
| 53. C2×C3×C8 | D4 | semidirect |
| 87. #97 | C2×D3 | semidirect |
| 88. C4×C8 | C2×D3 | semidirect |
| 133. M16 | D12 | semidirect |
| 134. C2×C8 | D12 | semidirect |
| 197. C3 | #3155 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#908 | 2 |
| C2×#904 | 2 |
| C22×D24 | 2 |
| #5525 | 2 |
| C3×#405 | 2 |
| #4712 | 2 |
| #4696 | 2 |
| #4695 | 2 |
| #3155 | 3 |