Show subcategory in category page

I’m wondering how to customize category page in my child theme, but I can’t find how.

I’m trying to show subcategories in category page.

this should be the code, but I can’t imagine where to put it.
There is not a category.php

if (is_category()) {$this_category = get_category($cat); }

if($this_category->category_parent)
    $this_category = wp_list_categories('orderby=id&show_count=0&title_li=&use_desc_for_title=1&child_of='.$this_category->category_parent."&echo=0");
else
    $this_category = wp_list_categories('orderby=id&depth=1&show_count=0&title_li=&use_desc_for_title=1&child_of='.$this_category->cat_ID."&echo=0");

if ($this_category) { ?>
    <ul>
        <?php echo $this_category; ?>
    </ul>