Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
371 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #11672 | C2 | semidirect |
| 5. #9718 | C2 | semidirect |
| 6. #7296 | C2 | semidirect |
| 7. #7163 | C2 | semidirect |
| 12. #2226 | C4 | semidirect |
| 13. #2226 | C4 | semidirect |
| 14. #2208 | C4 | semidirect |
| 15. #2208 | C4 | semidirect |
| 194. M16 | SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33661 | 2 |
| M16×D8 | 2 |
| C4×#357 | 2 |
| #11672 | 2 |
| #9718 | 2 |
| #7296 | 2 |
| #7163 | 2 |