Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
398 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 116 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 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33734 | C2 | semidirect |
| 3. #13730 | C2 | semidirect |
| 4. #13722 | C2 | semidirect |
| 5. #13690 | C2 | semidirect |
| 6. #13665 | C2 | semidirect |
| 8. #13640 | C2 | semidirect |
| 14. C2×Q64 | C22 | semidirect |
| 16. #2252 | C22 | semidirect |
| 21. #2247 | C22 | semidirect |
| 26. C4×Q32 | C22 | semidirect |
| 28. C4⋊C32 | C22 | semidirect |
| 29. #1416 | C22 | semidirect |
| 32. #1397 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33734 | 2 |
| C2×#2249 | 2 |
| #13730 | 2 |
| #13722 | 2 |
| #13690 | 2 |
| #13665 | 2 |
| #13664 | 2 |
| #13640 | 2 |