Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
166 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1281 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | #1726 | C22 |
| socle | C2 | #7018 |
| prime radical | C23 | H(3,ℤ/4) |
| LCS | G ▷ C2×C8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ #1726 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 31. #322 | C8 | semidirect |
| 33. C82 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12881 | 2 |
| #8780 | 2 |
| #8137 | 2 |