Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6308 conjugacy classes of subgroups.
The subgroup lattice has 387 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 | #62686 | 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. #60330 | C2 | semidirect |
| 3. #60306 | C2 | semidirect |
| 4. #60304 | C2 | semidirect |
| 5. #60279 | C2 | semidirect |
| 6. #60277 | C2 | semidirect |
| 7. #60204 | C2 | semidirect |
| 8. C2×C8×#41 | C2 | semidirect |
| 20. #2993 | C22 | semidirect |
| 21. #2992 | C22 | semidirect |
| 22. #2989 | C22 | semidirect |
| 23. #2985 | C22 | semidirect |
| 24. #2983 | C22 | semidirect |
| 25. #2981 | C22 | semidirect |
| 26. #2976 | C22 | semidirect |
| 27. #2975 | C22 | semidirect |
| 29. #2966 | C22 | semidirect |
| 30. #2966 | C22 | semidirect |
| 31. #2965 | C22 | semidirect |
| 32. #2965 | C22 | semidirect |
| 33. C8×#41 | C22 | semidirect |
| 34. C8×#41 | C22 | semidirect |
| 35. C8×#41 | C22 | semidirect |
| 36. C8×#41 | C22 | semidirect |
| 37. #2960 | C22 | semidirect |
| 38. C2×C8×Q8 | C22 | semidirect |
| 39. #2931 | C22 | semidirect |
| 40. #2929 | C22 | semidirect |
| 42. C2×C8×D4 | C22 | semidirect |
| 43. #2922 | C22 | semidirect |
| 44. #2920 | C22 | semidirect |
| 46. C2×#431 | C22 | semidirect |
| 47. C4×#131 | C22 | semidirect |
| 49. C22×C4×C8 | C22 | semidirect |
| 272. C22×C4 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62686 | 2 |
| #60330 | 2 |
| #60306 | 2 |
| #60304 | 2 |
| #60279 | 2 |
| #60277 | 2 |
| #60204 | 2 |
| C2×C8×#41 | 2 |