Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1535 conjugacy classes of subgroups.
The subgroup lattice has 379 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C24 |
| commutator, socle | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33485 | C2 | semidirect |
| 3. #33484 | C2 | semidirect |
| 4. #33479 | C2 | semidirect |
| 5. C16×#41 | C2 | semidirect |
| 6. Q8×M32 | C2 | semidirect |
| 7. D4×M32 | C2 | semidirect |
| 8. #33418 | C2 | semidirect |
| 9. C2×C4×M32 | C2 | semidirect |
| 26. #2183 | C22 | semidirect |
| 27. Q8×C16 | C22 | semidirect |
| 28. #2169 | C22 | semidirect |
| 29. #2168 | C22 | semidirect |
| 30. D4×C16 | C22 | semidirect |
| 31. #2163 | C22 | semidirect |
| 32. #2163 | C22 | semidirect |
| 33. #2162 | C22 | semidirect |
| 34. #2162 | C22 | semidirect |
| 40. C4×M32 | C22 | semidirect |
| 41. C4×M32 | C22 | semidirect |
| 42. C2×#346 | C22 | semidirect |
| 43. C2×C4×C16 | C22 | semidirect |
| 184. M32 | #41 | direct (N × H) |
| 218. #41 | M32 | direct (N × H) |
| 295. C16 | C2×#41 | semidirect |
| 303. Q8 | C2×M32 | semidirect |
| 306. D4 | C2×M32 | semidirect |
| 325. C2×C4 | C2×M32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×#41 | 2 |
| #33485 | 2 |
| #33484 | 2 |
| #33479 | 2 |
| C16×#41 | 2 |
| Q8×M32 | 2 |
| D4×M32 | 2 |
| #33418 | 2 |
| C2×C4×M32 | 2 |