Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1438 conjugacy classes of subgroups.
The subgroup lattice has 296 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C7×C8 | C22 |
| commutator | C2 | C22×C4×C7×C8 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle, prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C7×C8×C4⋊C4 | C2 | direct (N × H) |
| 13. C2×C7×C4⋊C8 | C4 | semidirect |
| 16. C2×C4×C7×C8 | C4 | semidirect |
| 23. C2×C8×C4⋊C4 | C7 | direct (N × H) |
| 40. C2×C7×C4⋊C4 | C8 | direct (N × H) |
| 50. C7×C4⋊C8 | C2×C4 | semidirect |
| 52. C4×C7×C8 | C2×C4 | semidirect |
| 58. C8×C4⋊C4 | C2×C7 | direct (N × H) |
| 85. C2×C7×C8 | C4⋊C4 | direct (N × H) |
| 91. C7×C4⋊C4 | C2×C8 | direct (N × H) |
| 104. C2×C4⋊C8 | C4×C7 | semidirect |
| 107. C2×C4×C8 | C4×C7 | semidirect |
| 125. C2×C4×C7 | C4×C8 | semidirect |
| 145. C7×C8 | C2×C4⋊C4 | direct (N × H) |
| 162. C2×C4⋊C4 | C7×C8 | direct (N × H) |
| 172. C4⋊C8 | C2×C4×C7 | semidirect |
| 174. C4×C8 | C2×C4×C7 | semidirect |
| 189. C4×C7 | C2×C4×C8 | semidirect |
| 219. C2×C8 | C7×C4⋊C4 | direct (N × H) |
| 225. C4⋊C4 | C2×C7×C8 | direct (N × H) |
| 232. C2×C7 | C8×C4⋊C4 | direct (N × H) |
| 249. C2×C4 | C4×C7×C8 | semidirect |
| 269. C8 | C2×C7×C4⋊C4 | direct (N × H) |
| 270. C7 | C2×C8×C4⋊C4 | direct (N × H) |
| 284. C4 | C2×C4×C7×C8 | semidirect |
| 289. C2 | C7×C8×C4⋊C4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×C4⋊C8 | 2 |
| C22×C4×C7×C8 | 2 |
| C2×C4×C7×C4⋊C4 | 2 |
| C7×C8×C4⋊C4 | 2 |
| C2×C7×#336 | 2 |
| C2×C8×C4⋊C4 | 7 |