There is an extension due to Schutzenberger. One needs to interpret Lagrange correctly. A partial function on a set $X$ is a function defined on a subset of $X$ to $X$. The composition of partial functions is defined where it makes sense.
The set of all partial functions on X is a semigroup and so we can define the notion of an action of a semigroup S$S$ by partial functions on the right of X. Say the action is transitive if for all $x\neq y$ in X there is s$s$ in S with xs=y$xs=y$.
Define an endomorphism of this action to be a totally defined map $f:X\to X$ such that $f(xs)=f(x)s$ (where equality means both sides are either undefined or agree).
Theorem.Theorem. If X$X$ is finite and $T$ is a semigroup of endomorphisms of the action of S$S$ on X$X$, then the size of X$X$ is divisible by the size of T$T$.
The proof is trivial.: Show T$T$ is a group acting freely on X$X$.
This generalizes Lagrange by taking G=X$S=G=X$ with the regular right action and T$T=H$ a subgroup acting on the left.
Schutzenberger used this to generalize the monomial representation of groups to semigroups.