Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
932 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 932 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, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#380 | C2 |
| hypercenter | #380 | D3 |
| hypocenter | C3 | #2615 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#235 | C2 | semidirect |
| 2. C2×#930 | C2 | semidirect |
| 3. C2×#895 | C2 | semidirect |
| 4. C3×#380 | C2 | semidirect |
| 5. #4987 | C2 | semidirect |
| 6. #4960 | C2 | semidirect |
| 7. #4673 | C2 | semidirect |
| 8. #4671 | C2 | semidirect |
| 37. #380 | D3 | semidirect |
| 42. C22×C3×C4 | D4 | semidirect |
| 63. C2×C4×D3 | D4 | semidirect |
| 154. C22×C4 | D12 | semidirect |
| 264. C3 | #2615 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#235 | 2 |
| C2×#930 | 2 |
| C2×#895 | 2 |
| C3×#380 | 2 |
| #4987 | 2 |
| #4960 | 2 |
| #4673 | 2 |
| #4671 | 2 |
| #2615 | 3 |