Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
208 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #367 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×#367 |
| prime radical | C23 | C2×SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×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 |
|---|---|---|
| 13. #1426 | C4 | direct (N × H) |
| 14. #1397 | C4 | semidirect |
| 15. C4×C32 | C4 | semidirect |
| 64. C32 | C42 | semidirect |
| 119. C4 | #1426 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1426 | 2 |
| C4×M64 | 2 |
| C4×#368 | 2 |
| C4×#367 | 2 |
| #7474 | 2 |