Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
242 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1936 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9556 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13164 | C2 | semidirect |
| 5. #11404 | C2 | semidirect |
| 6. #7182 | C2 | semidirect |
| 7. #7162 | C2 | semidirect |
| 49. #362 | C8 | semidirect |
| 50. #361 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13197 | 2 |
| #13164 | 2 |
| #12646 | 2 |
| C2×#1336 | 2 |
| #11404 | 2 |
| #7182 | 2 |
| #7162 | 2 |