Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
676 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 173 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×C4×C4⋊C4 | C2 |
| prime radical | C2×#874 | C2 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2548 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#875 | C2 | semidirect |
| 3. C2×C4×Dic6 | C2 | semidirect |
| 4. C3×C4×C4⋊C4 | C2 | semidirect |
| 5. #4988 | C2 | semidirect |
| 6. #4939 | C2 | semidirect |
| 7. #4672 | C2 | semidirect |
| 31. C4×C4⋊C4 | D3 | semidirect |
| 48. C2×Dic6 | D4 | semidirect |
| 136. C3×C4 | #124 | semidirect |
| 184. C4 | #875 | semidirect |
| 186. C3 | #2548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#875 | 2 |
| C2×#874 | 2 |
| C2×C4×Dic6 | 2 |
| C3×C4×C4⋊C4 | 2 |
| #4988 | 2 |
| #4939 | 2 |
| #4672 | 2 |
| #2548 | 3 |