Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
777 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 777 conjugacy classes of subgroups — too many to draw. Only its 186 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×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#1846 | C2 | semidirect |
| 4. #11355 | C2 | semidirect |
| 5. #11271 | C2 | semidirect |
| 6. #11239 | C2 | semidirect |
| 9. #10822 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23631 | 2 |
| #21046 | 2 |
| C2×#1846 | 2 |
| #11355 | 2 |
| #11271 | 2 |
| #11239 | 2 |
| #11107 | 2 |
| #11106 | 2 |
| #10822 | 2 |