Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1224 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 237 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 | C24×C3×C4 | C2 |
| hypercenter | C24×C4 | D3 |
| hypocenter | C3 | #2847 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×C4 | C2 | semidirect |
| 2. C2×#953 | C2 | semidirect |
| 3. C2×#930 | C2 | semidirect |
| 4. #5213 | C2 | semidirect |
| 5. #5212 | C2 | semidirect |
| 12. C22×D12 | C22 | semidirect |
| 16. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 17. C2×C4⋊Dic3 | C22 | semidirect |
| 18. C3⋊#95 | C22 | semidirect |
| 19. C24×C4 | D3 | semidirect |
| 26. C24×C3 | D4 | semidirect |
| 33. C22×C3×C4 | D4 | semidirect |
| 81. C24 | D12 | semidirect |
| 92. C22×C4 | #67 | semidirect |
| 103. C22×C3 | #121 | semidirect |
| 111. C3×C4 | #120 | semidirect |
| 148. C22 | #930 | semidirect |
| 154. C4 | #953 | semidirect |
| 155. C3 | #2847 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C4 | 2 |
| C2×#953 | 2 |
| C2×#930 | 2 |
| #5213 | 2 |
| #5212 | 2 |
| #2847 | 3 |