Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
442 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ 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×#2263 | C2 | semidirect |
| 2. C2×#2258 | C2 | semidirect |
| 3. #33610 | C2 | semidirect |
| 4. #33607 | C2 | semidirect |
| 5. #13699 | C2 | semidirect |
| 6. #13697 | C2 | semidirect |
| 7. #13696 | C2 | semidirect |
| 8. #13694 | C2 | semidirect |
| 9. #13634 | C2 | semidirect |
| 10. #13615 | C2 | semidirect |
| 11. C2×#1415 | C2 | semidirect |
| 22. C2×M64 | C22 | semidirect |
| 24. C22×C32 | C22 | semidirect |
| 27. #2216 | C22 | semidirect |
| 30. #2215 | C22 | semidirect |
| 35. #1424 | C22 | semidirect |
| 36. #1423 | C22 | semidirect |
| 37. #1416 | C22 | semidirect |
| 38. #1416 | C22 | semidirect |
| 41. #1415 | C22 | semidirect |
| 42. #1415 | C22 | semidirect |
| 76. M64 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2263 | 2 |
| C2×#2258 | 2 |
| #33610 | 2 |
| #33607 | 2 |
| #13699 | 2 |
| #13697 | 2 |
| #13696 | 2 |
| #13694 | 2 |
| #13634 | 2 |
| #13615 | 2 |
| C2×#1415 | 2 |