Attribute Routing Inheritance

I can't find a way to combine action + controller + base-controller automatically, but it is possible to achieve what you're looking for like this:

[Route("admin")]
public class AdminController : Controller { }

public class ProductsAdminController : AdminController
{ 
    [Route("products/list")]
    public IActionResult Index()
    {
        ...
    }
}

This approach ends up generating a URL of /admin/products/list, with the obvious downside that products is something that needs to be repeated for each action. This might be an acceptable compromise; that's up to you to decide. You could make it a bit better with a constant, like this:

private const string RoutePrefix = "products";

[Route(RoutePrefix + "/list")]
public IActionResult Index()
{
    ...
}

It's not pretty as it's just a workaround, but worth considering if you don't want to go with Chris Pratt's Areas suggestion.

As far as I'm aware, that never would have worked. You could use areas, and by applying the [Area] attribute to the base controller, you'd get the result you describe:

[Area("admin")]
public class AdminController : Controller

[Route("products")]
public class ProductsAdminController : AdminController

The same would work in Core, as well.