Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
186 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #95 |
| commutator | C23 | C4×C16 |
| Frattini, cosocle | C23×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C2×M64 | C4 | semidirect |
| 13. C22⋊C32 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13553 | 2 |
| #12611 | 2 |