  Based on variational methods, we study the spectral problem for the
subelliptic $p$-Laplacian arising from smooth H\"ormander vector fields. We
derive the smallest eigenvalue, prove its simplicity and isolatedness,
establish the positivity of the first eigenfunction and show H\"older
regularity of eigenfunctions. Moreover, we determine the best constant for the
$L^{p}$-Poincar\'e-Friedrichs inequality for H\"ormander vector fields as a
byproduct.
