Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
466 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12785 | C2 | semidirect |
| 4. C4⋊#338 | C2 | semidirect |
| 5. #7255 | C2 | semidirect |
| 349. C4 | #1341 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#469 | 2 |
| #12785 | 2 |
| C2×#1341 | 2 |
| C4⋊#338 | 2 |
| #7255 | 2 |