Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4802 conjugacy classes of subgroups; only the 1410 normal subgroups are shown.
This group has 4802 conjugacy classes of subgroups — too many to draw. Only its 1410 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #14170 | C2 | direct (N × H) |
| 7. C22×#125 | C4 | semidirect |
| 26. C2×#125 | C2×C4 | semidirect |
| 174. C2 | #14170 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#125 | 2 |
| C22×#376 | 2 |
| #14170 | 2 |
| C2×#1445 | 2 |
| C2×#1436 | 2 |