Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
347 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 84 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 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19269 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33505 | C2 | semidirect |
| 3. #13255 | C2 | semidirect |
| 4. #11958 | C2 | semidirect |
| 5. #8666 | C2 | semidirect |
| 6. #7299 | C2 | semidirect |
| 7. #7244 | C2 | semidirect |
| 21. #1670 | C4 | semidirect |
| 22. #1669 | C4 | semidirect |
| 23. #1668 | C4 | semidirect |
| 24. #1667 | C4 | semidirect |
| 237. Q8 | #360 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33505 | 2 |
| #19269 | 2 |
| #13255 | 2 |
| #11958 | 2 |
| #8666 | 2 |
| #7299 | 2 |
| #7244 | 2 |