We construct analytical solutions describing black holes and black strings in the Einstein SU(N)- nonlinear sigma model in (3 + 1) dimensions. This construction is carried out using the maximal embedding ansatz of SU(2) together with the Euler parametrization of the SU(N) group, in such a way that the nonlinear sigma model equations are automatically satisfied for arbitrary values of the flavor number N while the Einstein equations can be solved analytically. In particular, we construct black holes with spherical and flat horizons as well as black strings that present the geometry of a three-dimensional charged Banados-Teitelboim-Zanelli black hole on the transverse section of the string. These configurations are not trivial embeddings of SU(2) into SU(N), which allow us to explicitly show the role that the flavor number plays on the geometry and thermodynamics of the black holes and black strings. Finally, we perform a thermal comparison between these configurations.