Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
518 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 518 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 | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20827 | C2 | semidirect |
| 3. #13536 | C2 | semidirect |
| 4. #13535 | C2 | semidirect |
| 5. #13488 | C2 | semidirect |
| 6. #13487 | C2 | semidirect |
| 7. #13399 | C2 | semidirect |
| 9. #13258 | C2 | semidirect |
| 10. #13184 | C2 | semidirect |
| 11. #12785 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33177 | 2 |
| #20827 | 2 |
| #13536 | 2 |
| #13535 | 2 |
| #13488 | 2 |
| #13487 | 2 |
| #13399 | 2 |
| #13398 | 2 |
| #13258 | 2 |
| #13184 | 2 |
| #12785 | 2 |