Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5058 conjugacy classes of subgroups.
The subgroup lattice has 365 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. #36645 | C2 | semidirect |
| 2. #36640 | C2 | semidirect |
| 3. #35345 | C2 | semidirect |
| 4. #35344 | C2 | semidirect |
| 5. #35294 | C2 | semidirect |
| 6. #35104 | C2 | semidirect |
| 8. #34912 | C2 | semidirect |
| 10. #34074 | C2 | semidirect |
| 16. #2868 | C22 | semidirect |
| 17. #2864 | C22 | semidirect |
| 18. #2862 | C22 | semidirect |
| 19. #2859 | C22 | semidirect |
| 20. #2859 | C22 | semidirect |
| 21. #2859 | C22 | semidirect |
| 30. #2606 | C22 | semidirect |
| 31. #2605 | C22 | semidirect |
| 32. #2570 | C22 | semidirect |
| 33. #2550 | C22 | semidirect |
| 37. #2331 | C22 | semidirect |
| 38. #2330 | C22 | semidirect |
| 40. C4×C4⋊Q8 | C22 | semidirect |
| 41. C4×#125 | C22 | semidirect |
| 43. #2295 | C22 | semidirect |
| 80. C22×C42 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36645 | 2 |
| #36640 | 2 |
| #35345 | 2 |
| #35344 | 2 |
| #35294 | 2 |
| #35104 | 2 |
| #35033 | 2 |
| #34912 | 2 |
| #34634 | 2 |
| #34074 | 2 |
| #34012 | 2 |
| C2×C4×#117 | 2 |