Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
490 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 124 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 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #21737 | C2 | semidirect |
| 3. #13535 | C2 | semidirect |
| 4. #13458 | C2 | semidirect |
| 6. #13367 | C2 | semidirect |
| 7. #13142 | C2 | semidirect |
| 8. #12897 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33248 | 2 |
| #21737 | 2 |
| #13535 | 2 |
| #13458 | 2 |
| #13413 | 2 |
| #13367 | 2 |
| #13142 | 2 |
| #12897 | 2 |