Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
172 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C16 | D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×C4⋊C16 |
| prime radical | C23 | C4⋊C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7458 | C2 | semidirect |
| 4. #7455 | C2 | semidirect |
| 70. C2×C8 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13639 | 2 |
| C2×C8×C16 | 2 |
| #7458 | 2 |
| #7455 | 2 |