User accessible category filter and search for sub categories

I’m sorry if I posted this in the wrong section…
I want my site to be set up so users can click on a particular category and then be shown a page of all the posts in the sub categories within that category. I know how I can do that, but the next thing I want is to have a search field with a drop-down menu that lets the user choose from sub categories within that particular category and filter out the posts according to their search. Is this possible? Thanks.