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 | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#500 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #500 | D3 |
| hypocenter | C3 | #3169 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #500 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5725 | C2 | semidirect |
| 2. C3×#500 | C2 | semidirect |
| 3. #5088 | C2 | semidirect |
| 4. #4883 | C2 | semidirect |
| 5. #4872 | C2 | semidirect |
| 6. #4695 | C2 | semidirect |
| 7. C4×D3×C8 | C2 | semidirect |
| 26. #500 | D3 | semidirect |
| 87. C4×D3 | D8 | semidirect |
| 88. C3×C8 | #41 | semidirect |
| 145. C8 | #239 | semidirect |
| 165. C3 | #3169 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5725 | 2 |
| C3×#500 | 2 |
| #5088 | 2 |
| #4883 | 2 |
| #4872 | 2 |
| #4695 | 2 |
| C4×D3×C8 | 2 |
| #3169 | 3 |