Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D24 |
| commutator, cosocle | C3×C8 | C24 |
| Frattini | C8 | C23×D3 |
| Fitting | C3×#504 | C2 |
| socle | C2×C3 | C22×D8 |
| hypercenter | #504 | D3 |
| hypocenter | C3 | #3416 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5753 | C2 | semidirect |
| 2. #5752 | C2 | semidirect |
| 3. C3×#504 | C2 | semidirect |
| 4. #4909 | C2 | semidirect |
| 5. #4908 | C2 | semidirect |
| 6. #4904 | C2 | semidirect |
| 7. C2×#800 | C2 | semidirect |
| 14. C2×Dic12 | C22 | semidirect |
| 16. #904 | C22 | semidirect |
| 18. C3×M32 | C22 | semidirect |
| 19. C2×C3×C16 | C22 | semidirect |
| 20. Dic24 | C22 | semidirect |
| 21. #800 | C22 | semidirect |
| 22. #800 | C22 | semidirect |
| 23. #800 | C22 | semidirect |
| 24. D48 | C22 | semidirect |
| 31. #504 | D3 | semidirect |
| 67. M32 | C2×D3 | semidirect |
| 68. C2×C16 | C2×D3 | semidirect |
| 144. C3 | #3416 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5753 | 2 |
| #5752 | 2 |
| C3×#504 | 2 |
| #4909 | 2 |
| #4908 | 2 |
| #4904 | 2 |
| C2×#800 | 2 |
| #3416 | 3 |