Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
343 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 343 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, socle | C2 | #8921 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ 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. #13187 | C2 | semidirect |
| 3. #13180 | C2 | semidirect |
| 6. #7181 | C2 | semidirect |
| 7. #7180 | C2 | semidirect |
| 25. #1359 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23196 | 2 |
| #13187 | 2 |
| #13180 | 2 |
| #11922 | 2 |
| #9756 | 2 |
| #7181 | 2 |
| #7180 | 2 |