Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
77177 conjugacy classes of subgroups.
The subgroup lattice has 562 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×C42 |
| commutator | C4 | C24×C42 |
| Frattini, cosocle | C22×C4 | C26 |
| socle | C23 | C22×#95 |
| prime radical | C24×#41 | C22 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#20338 | C2 | direct (N × H) |
| 4. C23×H(3,ℤ/4) | C2 | semidirect |
| 8. C22×#568 | C4 | semidirect |
| 10. C23×#131 | C4 | semidirect |
| 14. C2×#3428 | C4 | semidirect |
| 19. C22×C4×#41 | C4 | semidirect |
| 21. #20338 | C22 | direct (N × H) |
| 23. C22×H(3,ℤ/4) | C22 | semidirect |
| 41. C2×#568 | C2×C4 | semidirect |
| 45. C22×#131 | C2×C4 | semidirect |
| 48. C23×M16 | C2×C4 | semidirect |
| 49. C23×M16 | C2×C4 | semidirect |
| 52. C24×C8 | C2×C4 | semidirect |
| 56. #3428 | C2×C4 | semidirect |
| 65. C2×C4×#41 | C2×C4 | semidirect |
| 72. C22×#117 | C2×C4 | semidirect |
| 73. C22×#117 | C2×C4 | semidirect |
| 78. C23×C42 | C2×C4 | semidirect |
| 80. C2×H(3,ℤ/4) | C23 | semidirect |
| 138. C2×#131 | C22×C4 | semidirect |
| 148. C22×M16 | C22×C4 | semidirect |
| 149. C22×M16 | C22×C4 | semidirect |
| 155. C23×C8 | C22×C4 | semidirect |
| 163. C4×#41 | C22×C4 | semidirect |
| 177. C2×#117 | C22×C4 | semidirect |
| 178. C2×#117 | C22×C4 | semidirect |
| 192. C22×C42 | C22×C4 | semidirect |
| 301. C2×M16 | C23×C4 | semidirect |
| 308. C22×C8 | C23×C4 | semidirect |
| 320. #117 | C23×C4 | semidirect |
| 336. C2×C42 | C23×C4 | semidirect |
| 539. C22 | #20338 | direct (N × H) |
| 559. C2 | C2×#20338 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#568 | 2 |
| C22×#3428 | 2 |
| C2×#20338 | 2 |
| C23×H(3,ℤ/4) | 2 |