Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
443 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 118 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×C42 ◁ C2×C4×C8 ◁ C2×C4×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. #33735 | C2 | semidirect |
| 2. C2×#2246 | C2 | semidirect |
| 3. #13729 | C2 | semidirect |
| 4. #13723 | C2 | semidirect |
| 5. #13686 | C2 | semidirect |
| 6. #13685 | C2 | semidirect |
| 7. #13664 | C2 | semidirect |
| 8. #13661 | C2 | semidirect |
| 9. #13640 | C2 | semidirect |
| 14. C2×SD64 | C22 | semidirect |
| 15. C2×D32 | C22 | semidirect |
| 16. #2253 | C22 | semidirect |
| 17. #2249 | C22 | semidirect |
| 26. C4×Q32 | C22 | semidirect |
| 28. C4×D16 | C22 | semidirect |
| 30. C4⋊C32 | C22 | semidirect |
| 31. #1415 | C22 | semidirect |
| 32. #1415 | C22 | semidirect |
| 34. #1397 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33735 | 2 |
| C2×#2246 | 2 |
| #13729 | 2 |
| #13723 | 2 |
| #13686 | 2 |
| #13685 | 2 |
| #13664 | 2 |
| #13661 | 2 |
| #13640 | 2 |