Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
566 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12783 | C2 | semidirect |
| 3. C4×C22⋊C16 | C2 | semidirect |
| 4. C2×#1315 | C2 | semidirect |
| 55. C22⋊C16 | D4 | semidirect |
| 287. C4 | #1315 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21187 | 2 |
| #12783 | 2 |
| C4×C22⋊C16 | 2 |
| C2×#1315 | 2 |