Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
756 conjugacy classes of subgroups; only the 426 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 426 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#127 | C2 | semidirect |
| 2. C32×#126 | C2 | semidirect |
| 3. C32×#123 | C2 | semidirect |
| 4. C32×#121 | C2 | semidirect |
| 5. C32×#120 | C2 | semidirect |
| 6. C3×#560 | C3 | direct (N × H) |
| 10. C2×C32×D4 | C22 | semidirect |
| 11. C22×C32×C4 | C22 | semidirect |
| 12. C32×C4⋊C4 | C22 | semidirect |
| 15. C32×C22⋊C4 | C22 | semidirect |
| 16. C32×C42 | C22 | semidirect |
| 17. C3×#127 | C2×C3 | semidirect |
| 18. C3×#126 | C2×C3 | semidirect |
| 19. C3×#123 | C2×C3 | semidirect |
| 20. C3×#121 | C2×C3 | semidirect |
| 21. C3×#120 | C2×C3 | semidirect |
| 32. #560 | C32 | direct (N × H) |
| 36. C2×C3×D4 | C22×C3 | semidirect |
| 37. C22×C3×C4 | C22×C3 | semidirect |
| 38. C3×C4⋊C4 | C22×C3 | semidirect |
| 41. C3×C22⋊C4 | C22×C3 | semidirect |
| 42. C3×C42 | C22×C3 | semidirect |
| 51. #127 | C2×C32 | semidirect |
| 52. #126 | C2×C32 | semidirect |
| 53. #123 | C2×C32 | semidirect |
| 54. #121 | C2×C32 | semidirect |
| 55. #120 | C2×C32 | semidirect |
| 73. C2×D4 | C22×C32 | semidirect |
| 74. C22×C4 | C22×C32 | semidirect |
| 75. C4⋊C4 | C22×C32 | semidirect |
| 78. C22⋊C4 | C22×C32 | semidirect |
| 79. C42 | C22×C32 | semidirect |
| 88. C32 | #560 | direct (N × H) |
| 111. C3 | C3×#560 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#127 | 2 |
| C32×#126 | 2 |
| C32×#123 | 2 |
| C32×#121 | 2 |
| C32×#120 | 2 |
| C3×#560 | 3 |