Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
246 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 138 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 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C32 |
| prime radical | C32×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C2×C32×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C32×Q16 | C2 | semidirect |
| 3. C32×Q32 | C2 | semidirect |
| 4. C32×SD32 | C2 | semidirect |
| 5. C32×M32 | C2 | semidirect |
| 6. C3×#510 | C3 | direct (N × H) |
| 17. C2×C3×Q16 | C2×C3 | semidirect |
| 18. C3×Q32 | C2×C3 | semidirect |
| 19. C3×SD32 | C2×C3 | semidirect |
| 20. C3×M32 | C2×C3 | semidirect |
| 31. #510 | C32 | direct (N × H) |
| 48. C2×Q16 | C2×C32 | semidirect |
| 49. Q32 | C2×C32 | semidirect |
| 50. SD32 | C2×C32 | semidirect |
| 51. M32 | C2×C32 | semidirect |
| 80. C32 | #510 | direct (N × H) |
| 100. C3 | C3×#510 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#135 | 2 |
| C2×C32×Q16 | 2 |
| C32×Q32 | 2 |
| C32×SD32 | 2 |
| C32×M32 | 2 |
| C3×#510 | 3 |