Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
167 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #99 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C2×#504 | C22 |
| socle | C2 | #7199 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×#504 ▷ 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 |
| 7. C22×C32 | C4 | semidirect |
| 12. #1400 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13556 | 2 |
| #13554 | 2 |
| #12613 | 2 |