This code displays a list of all child or sub categories of the parent category like this in the sidebar:
You can change the hook to any other if you want the child category posts listed within the content area however by default, posts from child categories are already include in the parent category archive.
Only the child categories of the current parent category are displayed.
The code hooks each child category in before the sidebar widget area.
It displays the child category name, a description for the child category ( if added ) and the post count for each child category.
The code is easy to modify which is supported for logged in members: