Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
456 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#384 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C23×D3 | C23 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2383 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#384 | C2 | semidirect |
| 5. #4978 | C2 | semidirect |
| 6. #4976 | C2 | semidirect |
| 7. #4937 | C2 | semidirect |
| 18. #876 | C4 | semidirect |
| 28. #384 | D3 | semidirect |
| 40. C3×C42 | C2×C4 | semidirect |
| 44. Dic3⋊C4 | C2×C4 | semidirect |
| 93. C42 | C4×D3 | semidirect |
| 140. C3 | #2383 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#876 | 2 |
| C3×#384 | 2 |
| #4990 | 2 |
| #4989 | 2 |
| #4978 | 2 |
| #4976 | 2 |
| #4937 | 2 |
| #2383 | 3 |