Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
906 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 190 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 | #33171 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#499 | C2 | semidirect |
| 2. C2×#2244 | C2 | semidirect |
| 4. #13365 | C2 | semidirect |
| 5. #13284 | C2 | semidirect |
| 6. #12991 | C2 | semidirect |
| 7. #12599 | C2 | semidirect |
| 51. #499 | D4 | semidirect |
| 259. C22 | #2244 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#499 | 2 |
| C2×#2244 | 2 |
| #33171 | 2 |
| #13365 | 2 |
| #13284 | 2 |
| #12991 | 2 |
| #12599 | 2 |