Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
502 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 130 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. C22×D32 | C2 | semidirect |
| 2. D4×D16 | C2 | semidirect |
| 3. #13725 | C2 | semidirect |
| 4. #13694 | C2 | semidirect |
| 5. #13685 | C2 | semidirect |
| 6. #13661 | C2 | semidirect |
| 7. C4×D32 | C2 | semidirect |
| 8. D4×C32 | C2 | semidirect |
| 9. C22×D16 | C22 | semidirect |
| 13. C2×D32 | C22 | semidirect |
| 14. C2×D32 | C22 | semidirect |
| 15. C2×D32 | C22 | semidirect |
| 16. C22×C32 | C22 | semidirect |
| 17. #2246 | C22 | semidirect |
| 18. #2215 | C22 | semidirect |
| 21. C4×D16 | C22 | semidirect |
| 23. #1423 | C22 | semidirect |
| 24. C4⋊C32 | C22 | semidirect |
| 25. #1415 | C22 | semidirect |
| 26. #1415 | C22 | semidirect |
| 27. C22⋊C32 | C22 | semidirect |
| 28. C4×C32 | C22 | semidirect |
| 47. D32 | D4 | direct (N × H) |
| 78. D16 | C2×D4 | semidirect |
| 92. C32 | C2×D4 | semidirect |
| 141. D4 | D32 | direct (N × H) |
| 163. C22 | C2×D32 | semidirect |
| 168. C4 | C2×D32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D32 | 2 |
| D4×D16 | 2 |
| #13725 | 2 |
| #13694 | 2 |
| #13685 | 2 |
| #13661 | 2 |
| C4×D32 | 2 |
| D4×C32 | 2 |