Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
443 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 443 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 | #19975 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ 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. #33929 | C2 | semidirect |
| 2. Q162 | C2 | semidirect |
| 4. #12509 | C2 | semidirect |
| 5. #12476 | C2 | semidirect |
| 6. #12447 | C2 | semidirect |
| 7. #12425 | C2 | semidirect |
| 8. #12384 | C2 | semidirect |
| 9. #12354 | C2 | semidirect |
| 10. C16×Q16 | C2 | semidirect |
| 11. C8×SD32 | C2 | semidirect |
| 177. Q16 | SD32 | semidirect |
| 178. Q16 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33929 | 2 |
| Q162 | 2 |
| #19975 | 2 |
| #12509 | 2 |
| #12476 | 2 |
| #12447 | 2 |
| #12425 | 2 |
| #12384 | 2 |
| #12354 | 2 |
| C16×Q16 | 2 |
| C8×SD32 | 2 |