Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
497 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3205 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #29254 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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 |
|---|---|---|
| 1. #21735 | C2 | semidirect |
| 3. #13532 | C2 | semidirect |
| 4. #13373 | C2 | semidirect |
| 6. #13182 | C2 | semidirect |
| 7. #12944 | C2 | semidirect |
| 8. #12881 | C2 | semidirect |
| 41. #495 | Q8 | semidirect |
| 42. #494 | Q8 | semidirect |
| 43. #493 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21735 | 2 |
| #21707 | 2 |
| #13532 | 2 |
| #13373 | 2 |
| #13259 | 2 |
| #13182 | 2 |
| #12944 | 2 |
| #12881 | 2 |