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 $X$. 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 $x\neq y$ 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 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 G=X with the regular right action and T a subgroup acting on the left. 

Schutzenberger used this to generalize the monomial representation of groups to semigroups.