Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
317 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ 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 |
|---|---|---|
| 4. #11936 | C2 | semidirect |
| 5. #9755 | C2 | semidirect |
| 6. #7275 | C2 | semidirect |
| 7. #7153 | C2 | semidirect |
| 13. #2227 | C4 | semidirect |
| 14. #2226 | C4 | semidirect |
| 15. #2225 | C4 | semidirect |
| 16. #2224 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33626 | 2 |
| #17899 | 2 |
| #12817 | 2 |
| #11936 | 2 |
| #9755 | 2 |
| #7275 | 2 |
| #7153 | 2 |