Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
251 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×#99 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | C2×#1322 |
| prime radical | C24 | M32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7364 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33397 | 2 |
| C2×#1400 | 2 |
| #7364 | 2 |