Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6879 conjugacy classes of subgroups.
The subgroup lattice has 505 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C26 |
| commutator, socle | C22 | C25×C4 |
| prime radical | C2×#3480 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60219 | C2 | semidirect |
| 3. #60218 | C2 | semidirect |
| 4. #60214 | C2 | semidirect |
| 5. M16×#41 | C2 | semidirect |
| 6. C2×Q8×M16 | C2 | semidirect |
| 7. #60144 | C2 | semidirect |
| 8. #60125 | C2 | semidirect |
| 9. C4×#568 | C2 | semidirect |
| 26. #2971 | C22 | semidirect |
| 27. #2970 | C22 | semidirect |
| 28. #2969 | C22 | semidirect |
| 29. #2968 | C22 | semidirect |
| 30. #2967 | C22 | semidirect |
| 31. #2966 | C22 | semidirect |
| 32. #2965 | C22 | semidirect |
| 33. C8×#41 | C22 | semidirect |
| 36. Q8×M16 | C22 | semidirect |
| 37. Q8×M16 | C22 | semidirect |
| 38. C2×#446 | C22 | semidirect |
| 39. C2×C8×Q8 | C22 | semidirect |
| 40. #2935 | C22 | semidirect |
| 41. #2935 | C22 | semidirect |
| 44. #2932 | C22 | semidirect |
| 45. #2931 | C22 | semidirect |
| 46. #2925 | C22 | semidirect |
| 47. #2925 | C22 | semidirect |
| 52. #2875 | C22 | semidirect |
| 53. C4×#131 | C22 | semidirect |
| 200. C2×M16 | #41 | semidirect |
| 377. M16 | C2×#41 | semidirect |
| 380. C2×C8 | C2×#41 | semidirect |
| 432. Q8 | #568 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3480 | 2 |
| #60219 | 2 |
| #60218 | 2 |
| #60214 | 2 |
| M16×#41 | 2 |
| C2×Q8×M16 | 2 |
| #60144 | 2 |
| #60125 | 2 |
| C4×#568 | 2 |