Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
125 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1365 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7121 |
| prime radical | C23 | #360 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 22. #367 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13169 | 2 |
| #7467 | 2 |
| #7155 | 2 |