Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5514 conjugacy classes of subgroups; only the 1330 normal subgroups are shown.
This group has 5514 conjugacy classes of subgroups — too many to draw. Only its 1330 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#492 | C2 | semidirect |
| 2. C2×#2868 | C2 | semidirect |
| 4. #33300 | C2 | direct (N × H) |
| 5. C2×C42×C8 | C2 | semidirect |
| 11. C2×#492 | C22 | semidirect |
| 14. #2868 | C22 | semidirect |
| 19. C42×C8 | C22 | semidirect |
| 38. C2×C4×C8 | D4 | semidirect |
| 67. C2×C42 | SD16 | semidirect |
| 72. C4×C8 | C2×D4 | semidirect |
| 95. C2×C8 | #127 | semidirect |
| 99. C42 | C2×SD16 | semidirect |
| 115. C2×C4 | #492 | semidirect |
| 121. C8 | C2×#127 | semidirect |
| 131. C4 | C2×#492 | semidirect |
| 134. C2 | #33300 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#492 | 2 |
| C2×#2868 | 2 |
| C2×#2867 | 2 |
| #33300 | 2 |
| C2×C42×C8 | 2 |