Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
449 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 449 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33771 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33500 | C2 | semidirect |
| 3. #19344 | C2 | semidirect |
| 4. #19203 | C2 | semidirect |
| 5. #12480 | C2 | semidirect |
| 6. #12476 | C2 | semidirect |
| 7. #12442 | C2 | semidirect |
| 8. #12439 | C2 | semidirect |
| 9. #12036 | C2 | semidirect |
| 10. #11961 | C2 | semidirect |
| 11. #11948 | C2 | semidirect |
| 23. #2183 | C22 | semidirect |
| 25. Q8×C16 | C22 | semidirect |
| 31. #1676 | C22 | semidirect |
| 34. #1668 | C22 | semidirect |
| 39. #1372 | C22 | semidirect |
| 40. #1371 | C22 | semidirect |
| 41. #1365 | C22 | semidirect |
| 42. #1365 | C22 | semidirect |
| 45. #1361 | C22 | semidirect |
| 46. #1361 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33771 | 2 |
| #33500 | 2 |
| #19344 | 2 |
| #19203 | 2 |
| #12480 | 2 |
| #12476 | 2 |
| #12442 | 2 |
| #12439 | 2 |
| #12036 | 2 |
| #11961 | 2 |
| #11948 | 2 |