Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
720 conjugacy classes of subgroups; only the 438 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 438 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×#125 | C2 | semidirect |
| 3. C32×#123 | C2 | semidirect |
| 4. C32×#121 | C2 | semidirect |
| 5. C32×#121 | C2 | semidirect |
| 6. C32×C4×D4 | C2 | semidirect |
| 7. C32×#117 | C2 | semidirect |
| 8. C3×#540 | C3 | direct (N × H) |
| 22. C3×#127 | C2×C3 | semidirect |
| 23. C3×#125 | C2×C3 | semidirect |
| 24. C3×#123 | C2×C3 | semidirect |
| 25. C3×#121 | C2×C3 | semidirect |
| 26. C3×#121 | C2×C3 | semidirect |
| 27. C3×C4×D4 | C2×C3 | semidirect |
| 28. C3×#117 | C2×C3 | semidirect |
| 41. #540 | C32 | direct (N × H) |
| 66. #127 | C2×C32 | semidirect |
| 67. #125 | C2×C32 | semidirect |
| 68. #123 | C2×C32 | semidirect |
| 69. #121 | C2×C32 | semidirect |
| 70. #121 | C2×C32 | semidirect |
| 71. C4×D4 | C2×C32 | semidirect |
| 72. #117 | C2×C32 | semidirect |
| 113. C32 | #540 | direct (N × H) |
| 141. C3 | C3×#540 | 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×#125 | 2 |
| C32×#123 | 2 |
| C32×#121 | 2 |
| C32×C4×D4 | 2 |
| C32×#117 | 2 |
| C3×#540 | 3 |