Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
504 conjugacy classes of subgroups; only the 233 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 233 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 | C22×C3 | C23×C4 |
| Fitting | C2×C3×#117 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | C2×#117 | D3 |
| hypocenter | C3 | #2346 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #4984 | C2 | semidirect |
| 5. Dic3×C4⋊C4 | C2 | semidirect |
| 7. #4936 | C2 | semidirect |
| 8. #4932 | C2 | semidirect |
| 11. C3×#117 | C4 | semidirect |
| 56. #117 | Dic3 | semidirect |
| 182. C3 | #2346 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#117 | 2 |
| #5212 | 2 |
| C4×#217 | 2 |
| #4984 | 2 |
| Dic3×C4⋊C4 | 2 |
| #4951 | 2 |
| #4936 | 2 |
| #4932 | 2 |
| #2346 | 3 |