Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
678 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #10654 | C2 | semidirect |
| 5. #10653 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21987 | 2 |
| #21351 | 2 |
| C2×#1844 | 2 |
| #10654 | 2 |
| #10653 | 2 |
| #8040 | 2 |
| #7875 | 2 |