Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
778 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #10676 | C2 | semidirect |
| 5. #9825 | C2 | semidirect |
| 6. #8816 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33172 | 2 |
| #27960 | 2 |
| C2×#1845 | 2 |
| #10676 | 2 |
| #9825 | 2 |
| #8816 | 2 |
| C8×C22⋊C8 | 2 |