Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
229 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #102 |
| commutator | C2×C8 | C2×C16 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C23 | D4⋊C8 |
| prime radical | C22×D8 | C8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #7151 | C2 | semidirect |
| 3. #7136 | C2 | semidirect |
| 38. C2×D8 | C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #9657 | 2 |
| #7151 | 2 |
| #7136 | 2 |