Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
489 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 146 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×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2158 | C2 | semidirect |
| 4. #13518 | C2 | semidirect |
| 5. #13496 | C2 | semidirect |
| 6. #13171 | C2 | semidirect |
| 7. #12872 | C2 | semidirect |
| 8. #12842 | C2 | semidirect |
| 9. #12837 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33432 | 2 |
| C2×#2158 | 2 |
| #27674 | 2 |
| #13518 | 2 |
| #13496 | 2 |
| #13171 | 2 |
| #12872 | 2 |
| #12842 | 2 |
| #12837 | 2 |