Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6366 conjugacy classes of subgroups.
The subgroup lattice has 392 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3428 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60668 | C2 | semidirect |
| 3. #60667 | C2 | semidirect |
| 4. #60666 | C2 | semidirect |
| 5. #60664 | C2 | semidirect |
| 6. #60660 | C2 | semidirect |
| 7. C2×#3164 | C2 | semidirect |
| 25. #3166 | C22 | semidirect |
| 26. #3165 | C22 | semidirect |
| 30. #3164 | C22 | semidirect |
| 31. #3164 | C22 | semidirect |
| 32. #3164 | C22 | semidirect |
| 33. #3164 | C22 | semidirect |
| 34. #3164 | C22 | semidirect |
| 36. #3163 | C22 | semidirect |
| 37. #3163 | C22 | semidirect |
| 38. #3162 | C22 | semidirect |
| 39. C2×#501 | C22 | semidirect |
| 40. #3160 | C22 | semidirect |
| 41. C2×#500 | C22 | semidirect |
| 42. C2×#499 | C22 | semidirect |
| 62. #568 | Q8 | semidirect |
| 63. C2×#131 | Q8 | semidirect |
| 170. #131 | C2×Q8 | semidirect |
| 171. #131 | C2×Q8 | semidirect |
| 172. #131 | C2×Q8 | semidirect |
| 173. C2×M16 | C2×Q8 | semidirect |
| 174. C2×M16 | C2×Q8 | semidirect |
| 178. C22×C8 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62854 | 2 |
| #60668 | 2 |
| #60667 | 2 |
| #60666 | 2 |
| #60664 | 2 |
| #60660 | 2 |
| C2×#3164 | 2 |
| #60109 | 2 |
| #60108 | 2 |
| C2×#2913 | 2 |
| #60058 | 2 |