Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
413 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2215 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #33617 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2264 | C2 | semidirect |
| 2. #33974 | C2 | semidirect |
| 4. C2×#2222 | C2 | semidirect |
| 5. #13702 | C2 | semidirect |
| 6. #13701 | C2 | semidirect |
| 7. #13636 | C2 | semidirect |
| 8. #13625 | C2 | semidirect |
| 9. C2×#1420 | C2 | semidirect |
| 17. C2×Q64 | C22 | semidirect |
| 18. C2×SD64 | C22 | semidirect |
| 33. #1425 | C22 | semidirect |
| 34. #1420 | C22 | semidirect |
| 35. #1420 | C22 | semidirect |
| 36. #1419 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2264 | 2 |
| #33974 | 2 |
| #33617 | 2 |
| C2×#2222 | 2 |
| #13702 | 2 |
| #13701 | 2 |
| #13636 | 2 |
| #13625 | 2 |
| C2×#1420 | 2 |