Thank you both!

I just want to add, I personally know a "Subcontracting Manager" - I think it depends on the company and/or perhaps the role. I would see a Subcontracting Manager interacting with the subcontractors and subcontracting companies, whereas a Subcontracts Manager, I could see focused more on the company's policies and paperwork for subcontractors but perhaps not interacting directly with them. I would think "Subcontracts Manager/Subcontract Manager" would be the same thing, just preference as whether to use a plural or singular title.

If I was talking about a title for someone's job, I might say "Joe is the subcontracts manager". If I was talking about someone's role on one specific project that had only one subcontract, I might say that "On my project, Joe is the subcontract manager". For a title it is probably better to say "subcontracts manager." The difference is whether someone is in charge of more than one subcontract, or just one. In general, I think that the department would be called "subcontracts", so a manager would be the "subcontracts manager", because they would have more than one. Usually people would not say the "subcontracting manager" as a formal title, but during normal conversation, that might say "Joe works in subcontracting". I do not think that these are necessarily grammar rules, maybe they are just common business expressions.