Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
338 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 108 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 | #1443 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #8128 | 2 |
| #8121 | 2 |
| #8080 | 2 |
| #8062 | 2 |
| #8007 | 2 |
| #7968 | 2 |
| #7934 | 2 |