Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
17105 conjugacy classes of subgroups.
The subgroup lattice has 351 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×C42 |
| commutator | C4 | C24×C42 |
| Frattini, cosocle | C2×C8 | C26 |
| socle | C23 | C22×#97 |
| prime radical | C23×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#33329 | C2 | direct (N × H) |
| 4. C22×#2110 | C2 | semidirect |
| 5. C23×#347 | C2 | semidirect |
| 9. C2×#3407 | C4 | semidirect |
| 11. C22×#504 | C4 | semidirect |
| 19. #33329 | C22 | direct (N × H) |
| 20. C2×#2110 | C22 | semidirect |
| 23. C22×#347 | C22 | semidirect |
| 42. #3407 | C2×C4 | semidirect |
| 44. C2×#504 | C2×C4 | semidirect |
| 45. C22×M32 | C2×C4 | semidirect |
| 46. C22×M32 | C2×C4 | semidirect |
| 49. C23×C16 | C2×C4 | semidirect |
| 63. #2110 | C23 | semidirect |
| 66. C2×#347 | C23 | semidirect |
| 116. #504 | C22×C4 | semidirect |
| 118. C2×M32 | C22×C4 | semidirect |
| 119. C2×M32 | C22×C4 | semidirect |
| 123. C22×C16 | C22×C4 | semidirect |
| 212. M32 | C23×C4 | semidirect |
| 215. C2×C16 | C23×C4 | semidirect |
| 333. C22 | #33329 | direct (N × H) |
| 348. C2 | C2×#33329 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#3407 | 2 |
| C22×#2875 | 2 |
| C2×#33329 | 2 |
| C22×#2110 | 2 |
| C23×#347 | 2 |