Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6601 conjugacy classes of subgroups.
The subgroup lattice has 314 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C24 |
| commutator, socle | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| prime radical | #412 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60331 | C2 | semidirect |
| 3. #60308 | C2 | semidirect |
| 4. #60305 | C2 | semidirect |
| 5. #60280 | C2 | semidirect |
| 6. #60278 | C2 | semidirect |
| 7. #60204 | C2 | semidirect |
| 8. C2×C8×#41 | C2 | semidirect |
| 18. #2994 | C22 | semidirect |
| 19. #2991 | C22 | semidirect |
| 20. #2990 | C22 | semidirect |
| 21. #2986 | C22 | semidirect |
| 22. #2984 | C22 | semidirect |
| 23. #2979 | C22 | semidirect |
| 24. #2977 | C22 | semidirect |
| 25. #2974 | C22 | semidirect |
| 26. #2966 | C22 | semidirect |
| 27. #2965 | C22 | semidirect |
| 30. C8×#41 | C22 | semidirect |
| 31. C8×#41 | C22 | semidirect |
| 32. C8×#41 | C22 | semidirect |
| 33. C8×#41 | C22 | semidirect |
| 34. #2960 | C22 | semidirect |
| 35. C2×C8×Q8 | C22 | semidirect |
| 36. #2931 | C22 | semidirect |
| 37. #2929 | C22 | semidirect |
| 38. C2×C8×D4 | C22 | semidirect |
| 39. #2922 | C22 | semidirect |
| 40. #2920 | C22 | semidirect |
| 41. C2×#431 | C22 | semidirect |
| 42. C4×#131 | C22 | semidirect |
| 43. C22×C4×C8 | C22 | semidirect |
| 212. #41 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #412 | 2 |
| #60331 | 2 |
| #60308 | 2 |
| #60305 | 2 |
| #60280 | 2 |
| #60278 | 2 |
| #60204 | 2 |
| C2×C8×#41 | 2 |