Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
353 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7124 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21350 | 2 |
| C4×#349 | 2 |
| #7124 | 2 |