Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
530 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 530 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×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33771 | 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. #33772 | C2 | semidirect |
| 3. C2×#1709 | C2 | semidirect |
| 4. #17956 | C2 | semidirect |
| 5. #12495 | C2 | semidirect |
| 6. #12428 | C2 | semidirect |
| 7. #12012 | C2 | semidirect |
| 8. #11962 | C2 | semidirect |
| 9. #11961 | C2 | semidirect |
| 20. #2208 | C22 | semidirect |
| 31. #1589 | C22 | semidirect |
| 33. C8×D8 | C22 | semidirect |
| 37. #1370 | C22 | semidirect |
| 38. #1367 | C22 | semidirect |
| 39. #1362 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33772 | 2 |
| #33771 | 2 |
| C2×#1709 | 2 |
| #17956 | 2 |
| #12495 | 2 |
| #12428 | 2 |
| #12012 | 2 |
| #11962 | 2 |
| #11961 | 2 |