Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1224 conjugacy classes of subgroups; only the 368 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 368 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| socle | C23×C3 | C24 |
| prime radical | C2×C3×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×C4 | C2 | semidirect |
| 3. C2×C3×C4×D4 | C2 | semidirect |
| 4. C3×#386 | C2 | semidirect |
| 5. C3×#385 | C2 | semidirect |
| 6. C3×#379 | C2 | semidirect |
| 7. C3×C4×C22⋊C4 | C2 | semidirect |
| 8. C4×#120 | C3 | direct (N × H) |
| 14. C23×C3×C4 | C22 | semidirect |
| 15. C3×#120 | C4 | direct (N × H) |
| 17. C2×C3×C4⋊C4 | C22 | semidirect |
| 21. C2×C3×C22⋊C4 | C22 | semidirect |
| 22. C2×C3×C42 | C22 | semidirect |
| 23. C3×#95 | C22 | semidirect |
| 24. C24×C4 | C2×C3 | semidirect |
| 26. C2×C4×D4 | C2×C3 | semidirect |
| 27. #386 | C2×C3 | semidirect |
| 28. #385 | C2×C3 | semidirect |
| 29. #379 | C2×C3 | semidirect |
| 30. C4×C22⋊C4 | C2×C3 | semidirect |
| 33. C24×C3 | C2×C4 | semidirect |
| 38. C2×C3×D4 | C2×C4 | semidirect |
| 44. C22×C3×C4 | D4 | semidirect |
| 54. C3×C22⋊C4 | C2×C4 | semidirect |
| 61. C23×C4 | C22×C3 | semidirect |
| 62. #120 | C3×C4 | direct (N × H) |
| 64. C2×C4⋊C4 | C22×C3 | semidirect |
| 68. C2×C22⋊C4 | C22×C3 | semidirect |
| 69. C2×C42 | C22×C3 | semidirect |
| 70. #95 | C22×C3 | semidirect |
| 79. C23×C3 | C22×C4 | semidirect |
| 93. C2×C3×C4 | C22×C4 | semidirect |
| 98. C24 | C2×C3×C4 | semidirect |
| 103. C2×D4 | C2×C3×C4 | semidirect |
| 109. C22×C4 | C3×D4 | semidirect |
| 119. C22⋊C4 | C2×C3×C4 | semidirect |
| 129. C22×C3 | C4×D4 | semidirect |
| 137. C3×C4 | #120 | direct (N × H) |
| 146. C23 | C22×C3×C4 | semidirect |
| 160. C2×C4 | C22×C3×C4 | semidirect |
| 176. C22 | C3×C4×D4 | semidirect |
| 184. C4 | C3×#120 | direct (N × H) |
| 185. C3 | C4×#120 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C4 | 2 |
| C2×C3×#120 | 2 |
| C2×C3×C4×D4 | 2 |
| C3×#386 | 2 |
| C3×#385 | 2 |
| C3×#379 | 2 |
| C3×C4×C22⋊C4 | 2 |
| C4×#120 | 3 |