Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
646 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #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. #21707 | C2 | semidirect |
| 3. #13209 | C2 | semidirect |
| 4. #13011 | C2 | semidirect |
| 5. #12891 | C2 | semidirect |
| 6. #12592 | C2 | semidirect |
| 12. #2249 | C4 | semidirect |
| 13. #2248 | C4 | semidirect |
| 14. #2247 | C4 | semidirect |
| 15. #2246 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33713 | 2 |
| #21707 | 2 |
| #13209 | 2 |
| #13011 | 2 |
| #12891 | 2 |
| #12592 | 2 |