Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
296 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 296 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, socle | C23 | #357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4⋊C32 | C2 | semidirect |
| 3. C2×#1372 | C2 | semidirect |
| 4. #7446 | C2 | direct (N × H) |
| 17. C4⋊C32 | C22 | semidirect |
| 19. #1372 | C22 | semidirect |
| 183. C2 | #7446 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2249 | 2 |
| C2×C4⋊C32 | 2 |
| C2×#1372 | 2 |
| #7446 | 2 |