Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4432 conjugacy classes of subgroups.
The subgroup lattice has 404 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #37566 | C2 | semidirect |
| 2. #37565 | C2 | semidirect |
| 3. #37562 | C2 | semidirect |
| 5. #35345 | C2 | semidirect |
| 6. #35344 | C2 | semidirect |
| 7. #35235 | C2 | semidirect |
| 8. #35132 | C2 | semidirect |
| 9. #35120 | C2 | semidirect |
| 14. #2864 | C22 | semidirect |
| 15. #2864 | C22 | semidirect |
| 16. #2862 | C22 | semidirect |
| 17. #2859 | C22 | semidirect |
| 19. #2843 | C22 | semidirect |
| 22. #2666 | C22 | semidirect |
| 23. #2666 | C22 | semidirect |
| 29. #2612 | C22 | semidirect |
| 30. #2612 | C22 | semidirect |
| 33. #2606 | C22 | semidirect |
| 34. #2606 | C22 | semidirect |
| 35. #2605 | C22 | semidirect |
| 36. #2605 | C22 | semidirect |
| 37. #2570 | C22 | semidirect |
| 39. #2550 | C22 | semidirect |
| 45. #2331 | C22 | semidirect |
| 46. #2330 | C22 | semidirect |
| 83. C22×C42 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #37566 | 2 |
| #37565 | 2 |
| #37562 | 2 |
| #37477 | 2 |
| #35345 | 2 |
| #35344 | 2 |
| #35235 | 2 |
| #35132 | 2 |
| #35120 | 2 |
| #35065 | 2 |
| #34012 | 2 |
| C2×#2283 | 2 |