Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
441 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 118 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 | #12424 | 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. #33940 | C2 | semidirect |
| 2. #20021 | C2 | semidirect |
| 3. D8×Q16 | C2 | semidirect |
| 4. #12509 | C2 | semidirect |
| 5. #12479 | C2 | semidirect |
| 6. #12446 | C2 | semidirect |
| 8. #12381 | C2 | semidirect |
| 9. #12352 | C2 | semidirect |
| 10. #12054 | C2 | semidirect |
| 11. C8×SD32 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33940 | 2 |
| #20021 | 2 |
| D8×Q16 | 2 |
| #12509 | 2 |
| #12479 | 2 |
| #12446 | 2 |
| #12424 | 2 |
| #12381 | 2 |
| #12352 | 2 |
| #12054 | 2 |
| C8×SD32 | 2 |