Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
358 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #354 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #384 | C23 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7367 | C2 | direct (N × H) |
| 22. C2×#97 | C8 | semidirect |
| 68. #97 | C2×C8 | semidirect |
| 180. C2 | #7367 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1836 | 2 |
| C2×#1325 | 2 |
| #7367 | 2 |