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 $S$ by partial functions on the right of X. Say the action is transitive if for all $x\neq y$ in X there is $s$ in S with $xs=y$.
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. If $X$ is finite and $T$ is a semigroup of endomorphisms of the action of $S$ on $X$, then the size of $X$ is divisible by the size of $T$.
The proof is trivial: Show $T$ is a group acting freely on $X$.
This generalizes Lagrange by taking $S=G=X$ with the regular right action and $T=H$ a subgroup acting on the left.
Schutzenberger used this to generalize the monomial representation of groups to semigroups.