Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
539 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 539 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 | #3147 | C22 |
| 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 |
|---|---|---|
| 1. #21715 | C2 | semidirect |
| 3. #10666 | C2 | semidirect |
| 4. #10370 | C2 | semidirect |
| 6. #9628 | C2 | semidirect |
| 7. #8880 | C2 | semidirect |
| 8. #8780 | C2 | semidirect |
| 40. #494 | Q8 | semidirect |
| 41. #493 | Q8 | semidirect |
| 42. #492 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21715 | 2 |
| #21707 | 2 |
| #10666 | 2 |
| #10370 | 2 |
| #9750 | 2 |
| #9628 | 2 |
| #8880 | 2 |
| #8780 | 2 |