Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
203 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1345 |
| commutator | C2×M16 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C2 | #7122 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ #1729 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 26. #338 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11295 | 2 |
| #7190 | 2 |
| #7185 | 2 |