Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
258 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C22 | #1322 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7365 | C2 | direct (N × H) |
| 177. C2 | #7365 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2151 | 2 |
| C2×#1400 | 2 |
| #7365 | 2 |