Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
399 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1519 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8220 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×#349 | C2 | semidirect |
| 5. #10813 | C2 | semidirect |
| 6. #7257 | C2 | semidirect |
| 7. #7256 | C2 | semidirect |
| 9. C2×#443 | C4 | semidirect |
| 15. #1975 | C4 | semidirect |
| 16. #1946 | C4 | semidirect |
| 20. #1772 | C4 | semidirect |
| 27. #1342 | C4 | semidirect |
| 28. #1341 | C4 | semidirect |
| 29. #1340 | C4 | semidirect |
| 30. #1339 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21549 | 2 |
| C4×#349 | 2 |
| #11688 | 2 |
| #11685 | 2 |
| #10813 | 2 |
| #7257 | 2 |
| #7256 | 2 |