Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
149 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C2×D32 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ C8×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. #12081 | C2 | semidirect |
| 3. C8×D16 | C2 | semidirect |
| 4. #7461 | C2 | semidirect |
| 5. #7383 | C2 | semidirect |
| 6. #7383 | C2 | semidirect |
| 7. #7331 | C2 | semidirect |
| 53. D16 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D32 | 2 |
| #12081 | 2 |
| C8×D16 | 2 |
| #7461 | 2 |
| #7383 | 2 |
| #7331 | 2 |