If you install the useful URL-Helper add-on this will give you the ability to translate segments in your URL to category IDs.
For example, once URL-Helper is installed you can access the Category ID of the last segment using the variable {last_segment_category_id}
Once you have the category ID you can use the EE category tools to get your list of child categories.
HTH
You should be able to do this using the Channel Categories tag. See the documentation on link provided for information and examples.
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