Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
744 conjugacy classes of subgroups; only the 474 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 474 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24×C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C32×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×2−1+4 | C2 | semidirect |
| 2. C32×#137 | C2 | semidirect |
| 3. C32×#135 | C2 | semidirect |
| 4. C2×C32×Q16 | C2 | semidirect |
| 5. C32×#131 | C2 | semidirect |
| 6. C3×#578 | C3 | direct (N × H) |
| 18. C3×2−1+4 | C2×C3 | semidirect |
| 19. C3×#137 | C2×C3 | semidirect |
| 20. C3×#135 | C2×C3 | semidirect |
| 21. C2×C3×Q16 | C2×C3 | semidirect |
| 22. C3×#131 | C2×C3 | semidirect |
| 34. #578 | C32 | direct (N × H) |
| 51. 2−1+4 | C2×C32 | semidirect |
| 52. #137 | C2×C32 | semidirect |
| 53. #135 | C2×C32 | semidirect |
| 54. C2×Q16 | C2×C32 | semidirect |
| 55. #131 | C2×C32 | semidirect |
| 86. C32 | #578 | direct (N × H) |
| 106. C3 | C3×#578 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×2−1+4 | 2 |
| C32×#137 | 2 |
| C32×#135 | 2 |
| C2×C32×Q16 | 2 |
| C32×#131 | 2 |
| C3×#578 | 3 |