Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1180 conjugacy classes of subgroups; only the 536 normal subgroups are shown.
This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 536 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 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C23 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33962 | C2 | semidirect |
| 3. Q8×Q32 | C2 | semidirect |
| 4. #33930 | C2 | semidirect |
| 96. C2×Q8 | Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62360 | 2 |
| #33962 | 2 |
| Q8×Q32 | 2 |
| #33930 | 2 |
| #33815 | 2 |
| #33734 | 2 |
| C2×Q8×C16 | 2 |