Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
587 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 134 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 |
| 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. #33783 | C2 | semidirect |
| 2. C2×#1712 | C2 | semidirect |
| 3. #17959 | C2 | semidirect |
| 4. #12492 | C2 | semidirect |
| 5. #12491 | C2 | semidirect |
| 6. #12425 | C2 | semidirect |
| 7. #12424 | C2 | semidirect |
| 8. #12013 | C2 | semidirect |
| 9. #11964 | C2 | semidirect |
| 17. #2207 | C22 | semidirect |
| 18. #2206 | C22 | semidirect |
| 26. #1591 | C22 | semidirect |
| 28. #1589 | C22 | semidirect |
| 29. C8×Q16 | C22 | semidirect |
| 31. C8×D8 | C22 | semidirect |
| 35. #1370 | C22 | semidirect |
| 36. #1367 | C22 | semidirect |
| 37. #1361 | C22 | semidirect |
| 38. #1361 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33783 | 2 |
| C2×#1712 | 2 |
| #17959 | 2 |
| #12492 | 2 |
| #12491 | 2 |
| #12425 | 2 |
| #12424 | 2 |
| #12013 | 2 |
| #11964 | 2 |