Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
182 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 182 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×C32 | D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C22 | C22⋊C32 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M128 | C2 | semidirect |
| 3. C22×C64 | C2 | semidirect |
| 4. #7514 | C2 | direct (N × H) |
| 15. M128 | C22 | semidirect |
| 19. C2×C64 | C22 | semidirect |
| 134. C2 | #7514 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2258 | 2 |
| C2×M128 | 2 |
| C22×C64 | 2 |
| #7514 | 2 |