Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
2528 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C22×C3×C22⋊C4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C22×C22⋊C4 | D3 |
| hypocenter | C3 | C22×#121 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#67 | C2 | semidirect |
| 2. C23×#67 | C2 | semidirect |
| 3. C23×D12 | C2 | semidirect |
| 4. C23×C4×D3 | C2 | semidirect |
| 5. C22×C3×C22⋊C4 | C2 | semidirect |
| 6. C22×(C2×D3)⋊C4 | C2 | semidirect |
| 7. C22×Dic3⋊C4 | C2 | semidirect |
| 8. C2×#884 | C2 | direct (N × H) |
| 17. C22×#67 | C22 | semidirect |
| 18. C22×#67 | C22 | semidirect |
| 22. C22×D12 | C22 | semidirect |
| 24. C22×C4×D3 | C22 | semidirect |
| 25. C2×C3×C22⋊C4 | C22 | semidirect |
| 26. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 27. C2×Dic3⋊C4 | C22 | semidirect |
| 28. #884 | C22 | direct (N × H) |
| 36. C22×C22⋊C4 | D3 | semidirect |
| 47. C23×D3 | D4 | semidirect |
| 56. C2×#67 | C23 | semidirect |
| 57. C2×#67 | C23 | semidirect |
| 59. C22×Dic3 | D4 | semidirect |
| 63. C2×D12 | C23 | semidirect |
| 65. C2×C4×D3 | C23 | semidirect |
| 66. C3×C22⋊C4 | C23 | semidirect |
| 67. (C2×D3)⋊C4 | C23 | semidirect |
| 68. Dic3⋊C4 | C23 | semidirect |
| 87. C2×C22⋊C4 | C2×D3 | semidirect |
| 104. C22×D3 | C2×D4 | semidirect |
| 114. C2×Dic3 | C2×D4 | semidirect |
| 150. C22⋊C4 | C22×D3 | semidirect |
| 155. C22×C3 | #121 | semidirect |
| 167. C2×D3 | C22×D4 | semidirect |
| 171. Dic3 | C22×D4 | semidirect |
| 204. C2×C3 | C2×#121 | semidirect |
| 222. C22 | #884 | direct (N × H) |
| 231. C3 | C22×#121 | semidirect |
| 235. C2 | C2×#884 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#67 | 2 |
| C23×D12 | 2 |
| C23×C4×D3 | 2 |
| C22×C3×C22⋊C4 | 2 |
| C22×(C2×D3)⋊C4 | 2 |
| C22×Dic3⋊C4 | 2 |
| C2×#884 | 2 |
| C22×#121 | 3 |