Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
358 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33731 | C2 | semidirect |
| 4. #13274 | C2 | semidirect |
| 5. C2×#1365 | C2 | semidirect |
| 7. #7176 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33731 | 2 |
| #28731 | 2 |
| #19195 | 2 |
| #13274 | 2 |
| C2×#1365 | 2 |
| #11586 | 2 |
| #7176 | 2 |