Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
137 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1368 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | #346 | C23 |
| socle | C2 | C2×#1368 |
| prime radical | C23 | #365 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7457 | C2 | semidirect |
| 22. M64 | C8 | semidirect |
| 26. C2×C32 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1398 | 2 |
| C2×#1369 | 2 |
| #11999 | 2 |
| #7457 | 2 |