Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
608 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22009 | C2 | semidirect |
| 4. #9599 | C2 | semidirect |
| 5. #9554 | C2 | semidirect |
| 6. C4×#334 | C2 | semidirect |
| 7. #8181 | C2 | semidirect |
| 8. C8×#97 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22009 | 2 |
| #21275 | 2 |
| #9665 | 2 |
| #9599 | 2 |
| #9554 | 2 |
| C4×#334 | 2 |
| #8181 | 2 |
| C8×#97 | 2 |