Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
163 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1333 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #7466 | C2 | semidirect |
| 3. #7156 | C2 | semidirect |
| 14. C2×Q32 | C8 | semidirect |
| 16. C2×D16 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13242 | 2 |
| #7466 | 2 |
| #7156 | 2 |