Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
458 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C2×#493 | C22 |
| 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 |
|---|---|---|
| 1. #33803 | C2 | semidirect |
| 4. #17972 | C2 | semidirect |
| 5. #12486 | C2 | semidirect |
| 6. #12484 | C2 | semidirect |
| 7. #12419 | C2 | semidirect |
| 8. #12417 | C2 | semidirect |
| 9. #12014 | C2 | semidirect |
| 10. #11973 | C2 | semidirect |
| 23. #2227 | C22 | semidirect |
| 24. #2225 | C22 | semidirect |
| 37. #1591 | C22 | semidirect |
| 39. C8×Q16 | C22 | semidirect |
| 43. #1370 | C22 | semidirect |
| 44. #1367 | C22 | semidirect |
| 45. #1365 | C22 | semidirect |
| 46. #1364 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33803 | 2 |
| #33774 | 2 |
| #19652 | 2 |
| #17972 | 2 |
| #12486 | 2 |
| #12484 | 2 |
| #12419 | 2 |
| #12417 | 2 |
| #12014 | 2 |
| #11973 | 2 |
| #11964 | 2 |