  Let $G$ be a subgroup of the three dimensional projective group
$\mathrm{PGL}(3,q)$ defined over a finite field $\mathbb{F}_q$ of order $q$,
viewed as a subgroup of $\mathrm{PGL}(3,K)$ where $K$ is an algebraic closure
of $\mathbb{F}_q$. For the seven nonsporadic, maximal subgroups $G$ of
$\mathrm{PGL}(3,q)$, we investigate the (projective, irreducible) plane curves
defined over $K$ that are left invariant by $G$. For each, we compute the
minimum degree $d(G)$ of $G$-invariant curves, provide a classification of all
$G$-invariant curves of degree $d(G)$, and determine the first gap
$\varepsilon(G)$ in the spectrum of the degrees of all $G$-invariant curves. We
show that the curves of degree $d(G)$ belong to a pencil depending on $G$,
unless they are uniquely determined by $G$. We also point out that
$G$-invariant curves of degree $d(G)$ have particular geometric features such
as Frobenius nonclassicality and an unusual variation of the number of
$\mathbb{F}_{q^i}$-rational points. For most examples of plane curves left
invariant by a large subgroup of $\mathrm{PGL}(3,q)$, the whole automorphism
group of the curve is linear, i.e., a subgroup of $\mathrm{PGL}(3,K)$. Although
this appears to be a general behavior, we show that the opposite case can also
occur for some irreducible plane curves, that is, the curve has a large group
of linear automorphisms, but its full automorphism group is nonlinear.
