Made a template with tasks and subtasks. Made a project from that template. The project is included in the tasks but not subtasks.

Welcome to the Asana Community Forum @maxdoes :wave:

By default subtasks do not belong to any project.
You would have to add them to the project so they show up as task in the project as well.

Here is the relevant feedback request for what you would like to set up: Allow to automatically add subtasks to project

A workaround is using Flowsana for example.