Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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 | C2×C3×#120 | C2 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | #2847 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Dic3 | C2 | semidirect |
| 2. C2×C3×#120 | C2 | semidirect |
| 3. C2×#953 | C2 | semidirect |
| 4. C2×#939 | C2 | semidirect |
| 5. #5220 | C2 | semidirect |
| 6. #4958 | C2 | semidirect |
| 7. #4942 | C2 | semidirect |
| 12. C22×#67 | C22 | semidirect |
| 20. C2×#217 | C22 | semidirect |
| 22. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 23. C2×Dic3⋊C4 | C22 | semidirect |
| 24. C3⋊#95 | C22 | semidirect |
| 28. C2×#120 | D3 | semidirect |
| 34. C24×C3 | D4 | semidirect |
| 50. C22×Dic3 | D4 | semidirect |
| 107. C24 | #67 | semidirect |
| 139. C22×C3 | #121 | semidirect |
| 149. Dic3 | #120 | semidirect |
| 196. C22 | #939 | semidirect |
| 204. C3 | #2847 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Dic3 | 2 |
| C2×C3×#120 | 2 |
| C2×#953 | 2 |
| C2×#939 | 2 |
| #5220 | 2 |
| #4958 | 2 |
| #4942 | 2 |
| #2847 | 3 |