Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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 | #2398 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #400 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#887 | C2 | semidirect |
| 3. C3×#400 | C2 | semidirect |
| 5. #4985 | C2 | semidirect |
| 6. #4675 | C2 | semidirect |
| 7. #4673 | C2 | semidirect |
| 9. #4666 | C2 | semidirect |
| 10. #4666 | C2 | semidirect |
| 11. #4648 | C2 | semidirect |
| 41. #400 | D3 | semidirect |
| 262. C3 | #2398 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#893 | 2 |
| C2×#887 | 2 |
| C3×#400 | 2 |
| #4991 | 2 |
| #4985 | 2 |
| #4675 | 2 |
| #4673 | 2 |
| #4672 | 2 |
| #4666 | 2 |
| #4648 | 2 |
| #2398 | 3 |