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The Abelian Sandpile Model is a mathematical model for diffusion that has captivated physicists and mathematicians since its introduction in 1987. The model is built on a directed graph with a distinguished vertex (called the sink) which is accessible from every other vertex. At each iteration of the dynamical system, a grain of sand is dropped on a random vertex, and once the number of grains on a non-sink vertex reaches its out-degree, the vertex topples, sending one grain along each of its outward edges. This can cause some or all of its neighbors to topple as well, forming an avalanche. The sink swallows all grains of sand sent to it and never topples.
Any two states of the system can be added by adding the number of grains on each vertex pointwise and allowing the system to avalanche if necessary, giving the system a monoid structure. Remarkably, the recurrent configurations of sand (those that appear in the dynamical system infinitely often with probability 1) have the algebraic structure of a group, and many questions about the evolution of the system have algebraic answers. Much of the work on the sandpile model to date has assumed the underlying graph is undirected. We study directed graphs and give a combinatorial description of every maximal subgroup of the monoid.