Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5744 conjugacy classes of subgroups.
The subgroup lattice has 427 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #34978 | C2 | semidirect |
| 2. #34966 | C2 | semidirect |
| 4. #34948 | C2 | semidirect |
| 5. #34938 | C2 | semidirect |
| 6. #34912 | C2 | semidirect |
| 7. #34878 | C2 | semidirect |
| 9. #34552 | C2 | semidirect |
| 10. C4×#533 | C2 | semidirect |
| 16. #2612 | C22 | semidirect |
| 19. #2606 | C22 | semidirect |
| 20. #2605 | C22 | semidirect |
| 21. #2571 | C22 | semidirect |
| 22. #2570 | C22 | semidirect |
| 24. #2552 | C22 | semidirect |
| 25. #2551 | C22 | semidirect |
| 26. #2550 | C22 | semidirect |
| 34. #2402 | C22 | semidirect |
| 35. #2402 | C22 | semidirect |
| 36. #2399 | C22 | semidirect |
| 37. #2399 | C22 | semidirect |
| 40. C4×C4⋊Q8 | C22 | semidirect |
| 41. C4×C4⋊Q8 | C22 | semidirect |
| 42. C4×#125 | C22 | semidirect |
| 43. C4×#125 | C22 | semidirect |
| 48. #2295 | C22 | semidirect |
| 57. C42×Q8 | C22 | semidirect |
| 93. C22×C42 | Q8 | semidirect |
| 388. C2×C4 | #533 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #34978 | 2 |
| #34966 | 2 |
| #34958 | 2 |
| #34948 | 2 |
| #34938 | 2 |
| #34912 | 2 |
| #34878 | 2 |
| #34739 | 2 |
| #34552 | 2 |
| C4×#533 | 2 |
| #34012 | 2 |
| C2×C4×#117 | 2 |