Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
414 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 132 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 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13516 | C2 | semidirect |
| 3. #13505 | C2 | semidirect |
| 6. #13176 | C2 | semidirect |
| 7. #13110 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27840 | 2 |
| #13516 | 2 |
| #13505 | 2 |
| #13492 | 2 |
| #13176 | 2 |
| #13110 | 2 |
| #12913 | 2 |
| #12836 | 2 |