Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
507 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 507 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 | #19629 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ 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. #33820 | C2 | semidirect |
| 2. #33774 | C2 | semidirect |
| 4. #18086 | C2 | semidirect |
| 5. #12492 | C2 | semidirect |
| 6. #12486 | C2 | semidirect |
| 7. #12425 | C2 | semidirect |
| 8. #12419 | C2 | semidirect |
| 9. #12018 | C2 | semidirect |
| 10. #11948 | C2 | semidirect |
| 11. #11931 | C2 | semidirect |
| 20. #2227 | C22 | semidirect |
| 26. #2207 | C22 | semidirect |
| 33. #1591 | C22 | semidirect |
| 35. C8×Q16 | C22 | semidirect |
| 39. #1370 | C22 | semidirect |
| 40. #1367 | C22 | semidirect |
| 41. #1365 | C22 | semidirect |
| 42. #1365 | C22 | semidirect |
| 45. #1361 | C22 | semidirect |
| 46. #1361 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33820 | 2 |
| #33774 | 2 |
| #19629 | 2 |
| #18086 | 2 |
| #12492 | 2 |
| #12486 | 2 |
| #12425 | 2 |
| #12419 | 2 |
| #12018 | 2 |
| #11948 | 2 |
| #11931 | 2 |