Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
493 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #19267 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ 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. C2×#2240 | C2 | semidirect |
| 2. #33479 | C2 | semidirect |
| 3. #19271 | C2 | semidirect |
| 5. #12479 | C2 | semidirect |
| 6. #12471 | C2 | semidirect |
| 7. #12023 | C2 | semidirect |
| 8. #11971 | C2 | semidirect |
| 9. #11961 | C2 | semidirect |
| 18. #2183 | C22 | semidirect |
| 19. #2169 | C22 | semidirect |
| 27. #1674 | C22 | semidirect |
| 29. #1673 | C22 | semidirect |
| 38. #1372 | C22 | semidirect |
| 39. #1364 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2240 | 2 |
| #33479 | 2 |
| #19271 | 2 |
| #19267 | 2 |
| #12479 | 2 |
| #12471 | 2 |
| #12023 | 2 |
| #11971 | 2 |
| #11961 | 2 |