Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
524 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 144 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×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ 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. #33733 | C2 | semidirect |
| 2. C2×D4×C16 | C2 | semidirect |
| 3. #19255 | C2 | semidirect |
| 4. #12467 | C2 | semidirect |
| 5. #12451 | C2 | semidirect |
| 6. #12430 | C2 | semidirect |
| 7. #12022 | C2 | semidirect |
| 8. #11981 | C2 | semidirect |
| 13. #2252 | C22 | semidirect |
| 14. #2246 | C22 | semidirect |
| 15. #2240 | C22 | semidirect |
| 32. #1371 | C22 | semidirect |
| 33. #1371 | C22 | semidirect |
| 34. #1364 | C22 | semidirect |
| 35. #1361 | C22 | semidirect |
| 151. C2×D4 | D16 | semidirect |
| 197. C16 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33733 | 2 |
| C2×D4×C16 | 2 |
| #19255 | 2 |
| #12467 | 2 |
| #12451 | 2 |
| #12430 | 2 |
| #12022 | 2 |
| #11981 | 2 |