Why we cannot use Gauss's Law to find the Electric Field of a finite charged sheet?

In order to calculate the electric field using only the Gauss law (rather than by solving the Poisson equation) one usually exploits the symmetry of the problem. In textbook cases this usually means that one can guess such a Gaussian surface, that the electric field has the same magnitude and direction in respect to the surface everywhere on the surface. This is not the case for a finite charge sheet, where the electric field lines at the edges of the sheet are likely to have different shape then those leaving closer to the center of the sheet - this is known as fringe effects (I couldn't find an image for a charged sheet, but for parallel plate capacitors the effect has been well studied.)