Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
306 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#1312 | C2 | semidirect |
| 5. #9594 | C2 | semidirect |
| 6. #7171 | C2 | semidirect |
| 7. #7154 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#357 | 2 |
| C2×#1331 | 2 |
| C2×#1312 | 2 |
| #9657 | 2 |
| #9594 | 2 |
| #7171 | 2 |
| #7154 | 2 |