Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
215 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1934 |
| prime radical | C23 | #488 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11950 | 2 |
| #11672 | 2 |
| C16×M16 | 2 |
| #9725 | 2 |
| #9634 | 2 |
| #7298 | 2 |
| #7172 | 2 |