Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
400 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#424 | C2 |
| prime radical | C2×#235 | C22 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #424 | D3 |
| hypocenter | C3 | #2922 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #424 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#424 | C2 | semidirect |
| 4. Dic3×M16 | C2 | semidirect |
| 6. C2×C8×Dic3 | C2 | semidirect |
| 7. #4999 | C2 | semidirect |
| 8. #4840 | C2 | semidirect |
| 9. #4833 | C2 | semidirect |
| 35. #424 | D3 | semidirect |
| 182. C3 | #2922 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5522 | 2 |
| C3×#424 | 2 |
| #5129 | 2 |
| Dic3×M16 | 2 |
| #5110 | 2 |
| C2×C8×Dic3 | 2 |
| #4999 | 2 |
| #4840 | 2 |
| #4833 | 2 |
| #2922 | 3 |