Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
625 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 625 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #379 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12679 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13558 | C2 | semidirect |
| 3. C2×#1400 | C2 | semidirect |
| 12. C2×M64 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3407 | 2 |
| #13558 | 2 |
| C2×#1400 | 2 |