Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
458 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 458 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 | C2×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #19668 | C2 | semidirect |
| 5. #12449 | C2 | semidirect |
| 6. #12426 | C2 | semidirect |
| 7. #12422 | C2 | semidirect |
| 8. #12420 | C2 | semidirect |
| 9. #12014 | C2 | semidirect |
| 10. #11953 | C2 | semidirect |
| 11. #11950 | C2 | semidirect |
| 32. #1713 | C22 | semidirect |
| 33. #1710 | C22 | semidirect |
| 34. #1707 | C22 | semidirect |
| 35. #1707 | C22 | semidirect |
| 43. #1370 | C22 | semidirect |
| 44. #1367 | C22 | semidirect |
| 45. #1365 | C22 | semidirect |
| 46. #1364 | C22 | semidirect |
| 47. #1362 | C22 | semidirect |
| 48. #1362 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33670 | 2 |
| #33661 | 2 |
| #19668 | 2 |
| #17854 | 2 |
| #12449 | 2 |
| #12426 | 2 |
| #12422 | 2 |
| #12420 | 2 |
| #12014 | 2 |
| #11953 | 2 |
| #11950 | 2 |