Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
453 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 144 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 | #535 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2580 |
| prime radical | C23 | #525 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 35. #500 | Q8 | semidirect |
| 38. #498 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21733 | 2 |
| #21715 | 2 |
| #10668 | 2 |
| #10371 | 2 |
| #9749 | 2 |
| #9627 | 2 |
| #8879 | 2 |
| #8780 | 2 |