Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
470 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 122 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 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ 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. #33853 | C2 | semidirect |
| 3. D4×D16 | C2 | semidirect |
| 4. #13725 | C2 | semidirect |
| 5. #13697 | C2 | semidirect |
| 6. #13686 | C2 | semidirect |
| 7. #13685 | C2 | semidirect |
| 8. #13663 | C2 | semidirect |
| 9. #13661 | C2 | semidirect |
| 10. C4×SD64 | C2 | semidirect |
| 11. #13646 | C2 | semidirect |
| 12. C2×#508 | C22 | semidirect |
| 18. C2×SD64 | C22 | semidirect |
| 20. C2×D32 | C22 | semidirect |
| 21. C2×D32 | C22 | semidirect |
| 22. C2×M64 | C22 | semidirect |
| 24. #2246 | C22 | semidirect |
| 26. #2215 | C22 | semidirect |
| 31. C4×Q32 | C22 | semidirect |
| 34. #1424 | C22 | semidirect |
| 35. C4⋊C32 | C22 | semidirect |
| 36. #1416 | C22 | semidirect |
| 38. #1415 | C22 | semidirect |
| 39. #1415 | C22 | semidirect |
| 40. C22⋊C32 | C22 | semidirect |
| 41. C4×C32 | C22 | semidirect |
| 71. SD64 | D4 | semidirect |
| 117. Q32 | C2×D4 | semidirect |
| 144. C32 | C2×D4 | semidirect |
| 256. C4 | #2263 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2263 | 2 |
| #33853 | 2 |
| D4×D16 | 2 |
| #13725 | 2 |
| #13697 | 2 |
| #13686 | 2 |
| #13685 | 2 |
| #13663 | 2 |
| #13661 | 2 |
| C4×SD64 | 2 |
| #13646 | 2 |