Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
422 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 214 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C82 | D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C22⋊C8 |
| prime radical | C2×#41 | C42 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C82 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C8×M32 | C2 | semidirect |
| 4. C2×C8×C16 | C2 | semidirect |
| 5. #7141 | C2 | semidirect |
| 38. #350 | C8 | direct (N × H) |
| 67. M32 | C2×C8 | semidirect |
| 71. C2×C16 | C2×C8 | semidirect |
| 143. C8 | #350 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#131 | 2 |
| C4×#350 | 2 |
| C8×M32 | 2 |
| C2×C8×C16 | 2 |
| #7141 | 2 |