Listing Subcats on a Category Page

I am a php newbee. I am tring to get a list of Child Categories listed on a Category page. If I have a Category called ‘Books’ I want it to write the Subcats of ‘Books’ on the page. I have the below code that gets me the category number and writes it .. which i dont need.

I want the wp_list_categories to only list the children of the category they are in

<?php
$cat = get_the_category(); $cat = $cat[0]; echo $cat->category_parent;
wp_list_categories('child_of=21');
?>