Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1348 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 213 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 | C23×C3×D4 | C2 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | #2433 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×D4 | C2 | semidirect |
| 2. C2×#953 | C2 | semidirect |
| 3. C2×#940 | C2 | semidirect |
| 4. C2×#930 | C2 | semidirect |
| 5. #5223 | C2 | semidirect |
| 6. #5221 | C2 | semidirect |
| 7. C4×#217 | C2 | semidirect |
| 16. C22×#67 | C22 | semidirect |
| 17. C22×D12 | C22 | semidirect |
| 20. C2×#217 | C22 | semidirect |
| 23. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 24. C2×C4⋊Dic3 | C22 | semidirect |
| 25. C2×C4×Dic3 | C22 | semidirect |
| 26. C3⋊#95 | C22 | semidirect |
| 27. C23×D4 | D3 | semidirect |
| 46. C2×C3×D4 | D4 | semidirect |
| 48. C22×C3×C4 | D4 | semidirect |
| 61. #217 | D4 | semidirect |
| 114. C2×Dic3 | C2×D4 | semidirect |
| 135. C2×D4 | #67 | semidirect |
| 139. C22×C4 | #67 | semidirect |
| 156. C22×C3 | #127 | semidirect |
| 166. C3×C4 | #120 | semidirect |
| 222. C22 | #940 | semidirect |
| 230. C4 | #953 | semidirect |
| 231. C3 | #2433 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D4 | 2 |
| C2×#953 | 2 |
| C2×#940 | 2 |
| C2×#930 | 2 |
| #5223 | 2 |
| #5221 | 2 |
| C4×#217 | 2 |
| #2433 | 3 |