Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
432 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 139 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×#396 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2401 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #4986 | C2 | semidirect |
| 3. Dic3×C4⋊C4 | C2 | semidirect |
| 6. C3⋊#401 | C2 | semidirect |
| 7. #4648 | C2 | semidirect |
| 136. C3 | #2401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#396 | 2 |
| #4986 | 2 |
| Dic3×C4⋊C4 | 2 |
| #4953 | 2 |
| Dic3×C22⋊C4 | 2 |
| C3⋊#401 | 2 |
| #4648 | 2 |
| #2401 | 3 |