Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
354 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #525 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12848 | C2 | semidirect |
| 4. #12845 | C2 | semidirect |
| 6. #7202 | C2 | semidirect |
| 7. #7199 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33490 | 2 |
| #21592 | 2 |
| #12848 | 2 |
| #12845 | 2 |
| #11468 | 2 |
| #7202 | 2 |
| #7199 | 2 |