Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
289 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #381 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12844 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C23 ▷ 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 |
|---|---|---|
| 3. #7469 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33324 | 2 |
| C2×#1400 | 2 |
| #7469 | 2 |