Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
457 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 194 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 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13107 | C2 | semidirect |
| 4. C4×#364 | C2 | semidirect |
| 6. #12593 | C2 | semidirect |
| 7. #12541 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20474 | 2 |
| #13128 | 2 |
| #13107 | 2 |
| C4×#364 | 2 |
| #12602 | 2 |
| #12593 | 2 |
| #12541 | 2 |