Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
381 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 112 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 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×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 |
|---|---|---|
| 2. C2×#2216 | C2 | semidirect |
| 3. #13716 | C2 | semidirect |
| 4. #13715 | C2 | semidirect |
| 5. #13712 | C2 | semidirect |
| 6. #13699 | C2 | semidirect |
| 7. #13698 | C2 | semidirect |
| 8. #13666 | C2 | semidirect |
| 9. #13665 | C2 | semidirect |
| 10. #13662 | C2 | semidirect |
| 11. #13553 | C2 | semidirect |
| 17. C2×Q64 | C22 | semidirect |
| 18. C2×SD64 | C22 | semidirect |
| 34. #1424 | C22 | semidirect |
| 35. #1423 | C22 | semidirect |
| 36. #1416 | C22 | semidirect |
| 37. #1416 | C22 | semidirect |
| 38. #1416 | C22 | semidirect |
| 39. #1415 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33600 | 2 |
| C2×#2216 | 2 |
| #13716 | 2 |
| #13715 | 2 |
| #13712 | 2 |
| #13699 | 2 |
| #13698 | 2 |
| #13666 | 2 |
| #13665 | 2 |
| #13662 | 2 |
| #13553 | 2 |