Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
539 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 142 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×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33582 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ 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. #33629 | C2 | semidirect |
| 3. C2×#1706 | C2 | semidirect |
| 4. #17886 | C2 | semidirect |
| 5. #12486 | C2 | semidirect |
| 6. #12485 | C2 | semidirect |
| 7. #12004 | C2 | semidirect |
| 8. #11941 | C2 | semidirect |
| 9. #11931 | C2 | semidirect |
| 17. #2227 | C22 | semidirect |
| 18. #2226 | C22 | semidirect |
| 32. #1591 | C22 | semidirect |
| 34. #1589 | C22 | semidirect |
| 38. #1370 | C22 | semidirect |
| 39. #1365 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33629 | 2 |
| #33582 | 2 |
| C2×#1706 | 2 |
| #17886 | 2 |
| #12486 | 2 |
| #12485 | 2 |
| #12004 | 2 |
| #11941 | 2 |
| #11931 | 2 |