Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4448 conjugacy classes of subgroups.
The subgroup lattice has 375 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #37609 | C2 | semidirect |
| 2. #37566 | C2 | semidirect |
| 3. #37562 | C2 | semidirect |
| 5. #36691 | C2 | semidirect |
| 7. #34878 | C2 | semidirect |
| 8. #34644 | C2 | semidirect |
| 9. #34595 | C2 | semidirect |
| 13. #2845 | C22 | semidirect |
| 14. #2844 | C22 | semidirect |
| 15. #2843 | C22 | semidirect |
| 18. #2666 | C22 | semidirect |
| 19. #2666 | C22 | semidirect |
| 20. #2666 | C22 | semidirect |
| 25. #2612 | C22 | semidirect |
| 28. #2606 | C22 | semidirect |
| 29. #2605 | C22 | semidirect |
| 36. #2402 | C22 | semidirect |
| 37. #2402 | C22 | semidirect |
| 38. #2399 | C22 | semidirect |
| 39. #2399 | C22 | semidirect |
| 42. #2295 | C22 | semidirect |
| 75. C22×C42 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #37609 | 2 |
| #37566 | 2 |
| #37562 | 2 |
| #37478 | 2 |
| #36691 | 2 |
| #34958 | 2 |
| #34878 | 2 |
| #34644 | 2 |
| #34595 | 2 |
| #34012 | 2 |
| C2×#2283 | 2 |