Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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×#400 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #400 | D3 |
| hypocenter | C3 | #2572 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #400 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#400 | C2 | semidirect |
| 3. #4990 | C2 | semidirect |
| 4. #4973 | C2 | semidirect |
| 5. #4673 | C2 | semidirect |
| 6. #4666 | C2 | semidirect |
| 8. #4664 | C2 | semidirect |
| 9. #4651 | C2 | semidirect |
| 32. #400 | D3 | semidirect |
| 226. C3 | #2572 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#400 | 2 |
| #4991 | 2 |
| #4990 | 2 |
| #4973 | 2 |
| #4673 | 2 |
| #4666 | 2 |
| #4665 | 2 |
| #4664 | 2 |
| #4651 | 2 |
| #2572 | 3 |