Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
347 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1533 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #8030 |
| prime radical | C24 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7267 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2165 | 2 |
| #20769 | 2 |
| #7267 | 2 |