Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#384 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2823 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#876 | C2 | semidirect |
| 2. C3×#384 | C2 | semidirect |
| 3. #4994 | C2 | semidirect |
| 5. #4981 | C2 | semidirect |
| 22. #384 | D3 | semidirect |
| 31. C3×C42 | D4 | semidirect |
| 73. C42 | D12 | semidirect |
| 112. C3 | #2823 | 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 |
| #4994 | 2 |
| #4991 | 2 |
| #4981 | 2 |
| #2823 | 3 |