Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
850 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 850 conjugacy classes of subgroups — too many to draw. Only its 202 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 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 27. #533 | Q8 | semidirect |
| 44. #423 | Q8 | semidirect |
| 47. #384 | Q8 | semidirect |
| 48. #382 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1994 | 2 |
| #11387 | 2 |
| #11341 | 2 |
| #11244 | 2 |
| #11212 | 2 |
| #10839 | 2 |