The Lemlich law provides a simple estimate of the relative conductivity of a three-dimensional foam, as one third of its liquid fraction. This is based on an expression for the conductivity of a network of uniform wires (conducting lines). We show this to be an exact upper bound for the conductivity, orientationally averaged in the case of anisotropic systems. We discuss the dependance of conductivity on the geometry of the network structure and establish two necessary and sufficient conditions to maximize the conductivity. We note the connection between this problem and that of line-length minimization and also that between anisotropic conductivity and stress for a two-dimensional foam. These results are illustrated by various numerical simulations of network conductivities. The theorems presented in this paper may also be applied to the thermal conductivity and the permeability of a network.