Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1460 conjugacy classes of subgroups; only the 594 normal subgroups are shown.
This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 594 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×#95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #8130 | C2 | direct (N × H) |
| 3. C2×#1309 | C2 | semidirect |
| 13. #1309 | C22 | semidirect |
| 163. C2 | #8130 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2920 | 2 |
| #8130 | 2 |
| C2×#1309 | 2 |