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We systematically investigate the complexity of counting subgraph patternsmodulo fixed integers. For example, it is known that the parity of the numberof $k$-matchings can be determined in polynomial time by a simple reduction tothe determinant. We generalize this to an $n^{f(t,s)}$-time algorithm tocompute modulo $2^t$ the number of subgraph occurrences of patterns that are$s$ vertices away from being matchings. This shows that the knownpolynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carryover into the setting of counting modulo $2^t$.  Complementing our algorithm, we also give a simple and self-contained proofthat counting $k$-matchings modulo odd integers $q$ is Mod_q-W[1]-complete andprove that counting $k$-paths modulo $2$ is Parity-W[1]-complete, answering anopen question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).

Modular counting of subgraphs: Matchings, matching-splittable graphs, and paths