Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
4544 conjugacy classes of subgroups; only the 1112 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 1112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C26×C3 | C2 | semidirect |
| 2. C23×C3×D4 | C2 | semidirect |
| 3. C2×C3×#120 | C2 | direct (N × H) |
| 4. C22×C3×C22⋊C4 | C2 | semidirect |
| 5. C22×#120 | C3 | direct (N × H) |
| 8. C25×C3 | C22 | semidirect |
| 9. C25×C3 | C22 | semidirect |
| 11. C22×C3×D4 | C22 | semidirect |
| 12. C23×C3×C4 | C22 | semidirect |
| 13. C3×#120 | C22 | direct (N × H) |
| 14. C2×C3×C22⋊C4 | C22 | semidirect |
| 15. C26 | C2×C3 | semidirect |
| 16. C23×D4 | C2×C3 | semidirect |
| 17. C2×#120 | C2×C3 | direct (N × H) |
| 18. C22×C22⋊C4 | C2×C3 | semidirect |
| 23. C24×C3 | D4 | semidirect |
| 26. C24×C3 | C23 | semidirect |
| 27. C24×C3 | C23 | semidirect |
| 29. C2×C3×D4 | C23 | semidirect |
| 31. C22×C3×C4 | C23 | semidirect |
| 32. C3×C22⋊C4 | C23 | semidirect |
| 35. C25 | C22×C3 | semidirect |
| 36. C25 | C22×C3 | semidirect |
| 38. C22×D4 | C22×C3 | semidirect |
| 39. C23×C4 | C22×C3 | semidirect |
| 40. #120 | C22×C3 | direct (N × H) |
| 41. C2×C22⋊C4 | C22×C3 | semidirect |
| 49. C23×C3 | C2×D4 | semidirect |
| 52. C23×C3 | C24 | semidirect |
| 55. C2×C3×C4 | C24 | semidirect |
| 60. C24 | C3×D4 | semidirect |
| 63. C24 | C23×C3 | semidirect |
| 64. C24 | C23×C3 | semidirect |
| 66. C2×D4 | C23×C3 | semidirect |
| 68. C22×C4 | C23×C3 | semidirect |
| 69. C22⋊C4 | C23×C3 | semidirect |
| 75. C22×C3 | #120 | direct (N × H) |
| 77. C22×C3 | C22×D4 | semidirect |
| 87. C23 | C2×C3×D4 | semidirect |
| 90. C23 | C24×C3 | semidirect |
| 93. C2×C4 | C24×C3 | semidirect |
| 96. C2×C3 | C2×#120 | direct (N × H) |
| 103. C22 | C3×#120 | direct (N × H) |
| 105. C22 | C22×C3×D4 | semidirect |
| 108. C3 | C22×#120 | direct (N × H) |
| 111. C2 | C2×C3×#120 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×C3 | 2 |
| C23×C3×D4 | 2 |
| C2×C3×#120 | 2 |
| C22×C3×C22⋊C4 | 2 |
| C22×#120 | 3 |