Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
10314 conjugacy classes of subgroups; only the 3906 normal subgroups are shown.
This group has 10314 conjugacy classes of subgroups — too many to draw. Only its 3906 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×#2860 | C2 | direct (N × H) |
| 4. C22×#389 | C2 | semidirect |
| 5. C22×#388 | C2 | semidirect |
| 6. C22×C43 | C2 | semidirect |
| 9. C22×#126 | C22 | semidirect |
| 17. #2860 | C22 | direct (N × H) |
| 19. C2×#389 | C22 | semidirect |
| 20. C2×#388 | C22 | semidirect |
| 22. C2×C43 | C22 | semidirect |
| 31. C2×#126 | C23 | semidirect |
| 42. C22×C42 | D4 | semidirect |
| 47. #389 | C23 | semidirect |
| 48. #388 | C23 | semidirect |
| 50. C43 | C23 | semidirect |
| 84. C2×C42 | C2×D4 | semidirect |
| 112. C22×C4 | #126 | semidirect |
| 126. C42 | C22×D4 | semidirect |
| 152. C2×C4 | C2×#126 | semidirect |
| 160. C22 | #2860 | direct (N × H) |
| 170. C4 | C22×#126 | semidirect |
| 172. C2 | C2×#2860 | 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×#2860 | 2 |
| C22×#390 | 2 |
| C22×#389 | 2 |
| C22×#388 | 2 |
| C22×C43 | 2 |