Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
646 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 154 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. #33582 | C2 | semidirect |
| 2. C2×#1712 | C2 | semidirect |
| 3. C8×#135 | C2 | semidirect |
| 4. #12425 | C2 | semidirect |
| 5. #12424 | C2 | semidirect |
| 6. #12004 | C2 | semidirect |
| 7. #11931 | C2 | semidirect |
| 15. #2207 | C22 | semidirect |
| 17. #2206 | C22 | semidirect |
| 22. C8×Q16 | C22 | semidirect |
| 24. C8×D8 | C22 | semidirect |
| 28. #1370 | C22 | semidirect |
| 29. #1361 | C22 | semidirect |
| 82. #135 | D8 | semidirect |
| 166. Q16 | C2×D8 | semidirect |
| 177. D8 | C2×D8 | semidirect |
| 247. C8 | #509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33582 | 2 |
| C2×#1712 | 2 |
| C8×#135 | 2 |
| #12425 | 2 |
| #12424 | 2 |
| #12004 | 2 |
| #11931 | 2 |