Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
9138 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.
This group has 9138 conjugacy classes of subgroups — too many to draw. Only its 3490 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#126 | C2 | semidirect |
| 2. C2×#2823 | C2 | direct (N × H) |
| 3. C22×#398 | C2 | semidirect |
| 4. C22×#396 | C2 | semidirect |
| 6. C22×#384 | C2 | semidirect |
| 9. C22×#126 | C22 | semidirect |
| 17. #2823 | C22 | direct (N × H) |
| 18. C2×#398 | C22 | semidirect |
| 19. C2×#396 | C22 | semidirect |
| 21. C2×#384 | C22 | semidirect |
| 32. C2×#126 | C23 | semidirect |
| 45. C22×C42 | D4 | semidirect |
| 47. #398 | C23 | semidirect |
| 48. #396 | C23 | semidirect |
| 50. #384 | C23 | semidirect |
| 87. C2×C42 | C2×D4 | semidirect |
| 127. C42 | C22×D4 | semidirect |
| 160. C22 | #2823 | direct (N × H) |
| 172. C2 | C2×#2823 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#126 | 2 |
| C2×#2823 | 2 |
| C22×#398 | 2 |
| C22×#396 | 2 |
| C22×#394 | 2 |
| C22×#384 | 2 |