Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
440 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 440 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 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ 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. #33870 | C2 | semidirect |
| 2. #20019 | C2 | semidirect |
| 3. #19971 | C2 | semidirect |
| 4. #12510 | C2 | semidirect |
| 5. #12469 | C2 | semidirect |
| 6. #12447 | C2 | semidirect |
| 7. #12426 | C2 | semidirect |
| 8. #12375 | C2 | semidirect |
| 9. #12347 | C2 | semidirect |
| 10. C8×D16 | C2 | semidirect |
| 11. C16×D8 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33870 | 2 |
| #20019 | 2 |
| #19971 | 2 |
| #12510 | 2 |
| #12469 | 2 |
| #12447 | 2 |
| #12426 | 2 |
| #12375 | 2 |
| #12347 | 2 |
| C8×D16 | 2 |
| C16×D8 | 2 |