Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
31452 conjugacy classes of subgroups; only the 1706 normal subgroups are shown.
This group has 31452 conjugacy classes of subgroups — too many to draw. Only its 1706 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2846 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C26×C4 | C2 | semidirect |
| 3. C2×C4×#120 | C2 | semidirect |
| 4. #14694 | C2 | semidirect |
| 5. #14693 | C2 | semidirect |
| 6. #14679 | C2 | semidirect |
| 7. C4×#379 | C2 | semidirect |
| 13. C22×C4×D4 | C22 | semidirect |
| 14. #2846 | C4 | direct (N × H) |
| 16. C2×#386 | C22 | semidirect |
| 17. C2×#385 | C22 | semidirect |
| 21. C2×C4×C22⋊C4 | C22 | semidirect |
| 22. C22⋊#95 | C22 | semidirect |
| 25. C26 | C2×C4 | semidirect |
| 32. C24×C4 | D4 | semidirect |
| 35. C24×C4 | C23 | semidirect |
| 38. C2×#120 | C2×C4 | semidirect |
| 40. C22×C4⋊C4 | C23 | semidirect |
| 44. C22×C22⋊C4 | C23 | semidirect |
| 45. C22×C42 | C23 | semidirect |
| 50. #379 | C2×C4 | semidirect |
| 52. C2×#95 | C23 | semidirect |
| 65. C22×D4 | C22×C4 | semidirect |
| 91. C2×C22⋊C4 | C22×C4 | semidirect |
| 107. C24 | C4×D4 | semidirect |
| 109. C24 | C23×C4 | semidirect |
| 129. C22×C4 | #120 | semidirect |
| 130. C22×C4 | C23×C4 | semidirect |
| 179. C22 | C4×#120 | semidirect |
| 187. C4 | #2846 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×C4 | 2 |
| C2×#2846 | 2 |
| C2×C4×#120 | 2 |
| #14694 | 2 |
| #14693 | 2 |
| #14679 | 2 |
| C4×#379 | 2 |