Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1344 conjugacy classes of subgroups; only the 776 normal subgroups are shown.
This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 776 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#560 | C2 | direct (N × H) |
| 2. C2×C3×#127 | C2 | semidirect |
| 3. C2×C3×#126 | C2 | semidirect |
| 4. C2×C3×#123 | C2 | semidirect |
| 5. C2×C3×#121 | C2 | semidirect |
| 6. C2×C3×#120 | C2 | semidirect |
| 7. C2×#560 | C3 | direct (N × H) |
| 11. C22×C3×D4 | C22 | semidirect |
| 12. C23×C3×C4 | C22 | semidirect |
| 13. C3×#127 | C22 | semidirect |
| 14. C3×#126 | C22 | semidirect |
| 15. C3×#123 | C22 | semidirect |
| 16. C3×#121 | C22 | semidirect |
| 17. C3×#120 | C22 | semidirect |
| 18. C2×C3×C4⋊C4 | C22 | semidirect |
| 21. C2×C3×C22⋊C4 | C22 | semidirect |
| 22. C2×C3×C42 | C22 | semidirect |
| 23. #560 | C2×C3 | direct (N × H) |
| 24. C2×#127 | C2×C3 | semidirect |
| 25. C2×#126 | C2×C3 | semidirect |
| 26. C2×#123 | C2×C3 | semidirect |
| 27. C2×#121 | C2×C3 | semidirect |
| 28. C2×#120 | C2×C3 | semidirect |
| 39. C2×C3×D4 | C23 | semidirect |
| 43. C22×C3×C4 | C23 | semidirect |
| 44. C3×C4⋊C4 | C23 | semidirect |
| 47. C3×C22⋊C4 | C23 | semidirect |
| 48. C3×C42 | C23 | semidirect |
| 52. C22×D4 | C22×C3 | semidirect |
| 53. C23×C4 | C22×C3 | semidirect |
| 54. #127 | C22×C3 | semidirect |
| 55. #126 | C22×C3 | semidirect |
| 56. #123 | C22×C3 | semidirect |
| 57. #121 | C22×C3 | semidirect |
| 58. #120 | C22×C3 | semidirect |
| 59. C2×C4⋊C4 | C22×C3 | semidirect |
| 62. C2×C22⋊C4 | C22×C3 | semidirect |
| 63. C2×C42 | C22×C3 | semidirect |
| 92. C2×D4 | C23×C3 | semidirect |
| 96. C22×C4 | C23×C3 | semidirect |
| 97. C4⋊C4 | C23×C3 | semidirect |
| 100. C22⋊C4 | C23×C3 | semidirect |
| 101. C42 | C23×C3 | semidirect |
| 135. C2×C3 | #560 | direct (N × H) |
| 149. C3 | C2×#560 | direct (N × H) |
| 153. C2 | C3×#560 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#560 | 2 |
| C2×C3×#127 | 2 |
| C2×C3×#126 | 2 |
| C2×C3×#123 | 2 |
| C2×C3×#121 | 2 |
| C2×C3×#120 | 2 |
| C2×#560 | 3 |