Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
271 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1852 |
| prime radical | C24 | #107 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ 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 |
|---|---|---|
| 2. #12033 | C2 | semidirect |
| 3. #12032 | C2 | semidirect |
| 6. #7288 | C2 | semidirect |
| 7. #7288 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #17815 | 2 |
| #12033 | 2 |
| #12032 | 2 |
| #11459 | 2 |
| #8599 | 2 |
| #7288 | 2 |