Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
454 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 126 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 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33853 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ 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. Q8×Q32 | C2 | semidirect |
| 3. #19986 | C2 | semidirect |
| 4. #12503 | C2 | semidirect |
| 5. #12452 | C2 | semidirect |
| 6. #12439 | C2 | semidirect |
| 7. #12419 | C2 | semidirect |
| 8. #12373 | C2 | semidirect |
| 9. #12354 | C2 | semidirect |
| 10. C16×SD16 | C2 | semidirect |
| 11. C8×Q32 | C2 | semidirect |
| 124. Q32 | SD16 | semidirect |
| 220. Q8 | #508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Q32 | 2 |
| #33853 | 2 |
| #19986 | 2 |
| #12503 | 2 |
| #12452 | 2 |
| #12439 | 2 |
| #12419 | 2 |
| #12373 | 2 |
| #12354 | 2 |
| C16×SD16 | 2 |
| C8×Q32 | 2 |