Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
578 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 150 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 |
| 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×#2246 | C2 | semidirect |
| 2. C16×#41 | C2 | semidirect |
| 3. #19267 | C2 | semidirect |
| 4. #12439 | C2 | semidirect |
| 5. #12430 | C2 | semidirect |
| 6. #12022 | C2 | semidirect |
| 7. #11961 | C2 | semidirect |
| 15. Q8×C16 | C22 | semidirect |
| 17. D4×C16 | C22 | semidirect |
| 23. #1668 | C22 | semidirect |
| 25. #1667 | C22 | semidirect |
| 28. #1371 | C22 | semidirect |
| 29. #1361 | C22 | semidirect |
| 132. #41 | D16 | semidirect |
| 187. C16 | AGL(1,ℤ/8) | semidirect |
| 195. Q8 | C2×D16 | semidirect |
| 206. D4 | C2×D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2246 | 2 |
| C16×#41 | 2 |
| #19267 | 2 |
| #12439 | 2 |
| #12430 | 2 |
| #12022 | 2 |
| #11961 | 2 |