Get Wordpress child category from specific parent - wordpress

Question

I'm building a small thumbnail gallery with posts from a category that has the ID 406.
Some of the posts are in multiple categories and I'm not sure how to grab the category name that is a child of 406. $post_cat[0]->name; returns a category, but I only need it to return children of 406.
$thumbnails = get_posts('posts_per_page=10&cat=406');
foreach ($thumbnails as $thumbnail) {
$args = array(
'type' => 'post',
'child_of' => 0,
'parent' => 406,
);
$categories = get_categories ( $args );
foreach ( $categories as $category) {
$cat_name = $category->name;
$cat_slug = $category->slug;
}
echo $cat_name;
echo $cat_slug;
}
*EDIT**
$thumbnails = get_posts('posts_per_page=10&cat=406');
foreach ($thumbnails as $thumbnail) {
$post_cat = get_the_category( $thumbnail->ID );
echo '<li><a class="side-thumb" href="' . get_permalink( $thumbnail->ID ) . '" title="' . esc_attr( $thumbnail->post_title ) . '">';
if ( has_post_thumbnail($thumbnail->ID)) {
echo get_the_post_thumbnail($thumbnail->ID, 'thumbnail');
$upload_dir = wp_upload_dir();
$art_image = ''.$upload_dir['basedir'].'/icons/'.$post_cat[0]->slug.'.png';
if (file_exists($art_image)) {
echo '<p class="artist-latest"><img src="http://site.com/wp-content/uploads/icons/'.$post_cat[0]->slug.'.png" alt="'.$post_cat[0]->slug.'"/>'.$post_cat[0]->name.'</p>';
} else{
echo '<p class="artist-latest">'.$post_cat[0]->name.'</p>';
}
} else {
}
echo '</a></li>';
}

Answer 1

So now I'm fetching a list of category and its child, then I fetch posts having parent category assigned to it, after I get posts array, I loop into it and get all the categories assigned to the post, If there are additional categories other than the required category and its child I skip the post, otherwise do whatever I want to do.
$category = get_category_by_slug( 'category-name' );
$args = array(
'type' => 'post',
'child_of' => $category->term_id,
'orderby' => 'name',
'order' => 'ASC',
'hide_empty' => FALSE,
'hierarchical' => 1,
'taxonomy' => 'category',
);
$child_categories = get_categories($args );
$category_list = array();
$category_list[] = $category->term_id;
if ( !empty ( $child_categories ) ){
foreach ( $child_categories as $child_category ){
$category_list[] = $child_category->term_id;
}
}
$posts_args = array(
'posts_per_page' => 10,
'cat' => $category->term_id,
'post_type' => 'post',
'post_status' => 'publish',
'suppress_filters' => true
);
$posts = new WP_Query ( $posts_args );
if ( $posts->have_posts() ){
while ( $posts->have_posts() ){
$posts->the_post();
$category_array = array();
$post_categories = get_the_category ( get_the_ID() );
if ( !empty ( $post_categories ) ){
foreach ( $post_categories as $post_category ) {
$category_array[] = $post_category->term_id;
}
}
//Checks if post has an additional category
$result = array_diff( $category_array, $category_list );
if ( empty ( $result ) ) { ?>
<li>
<a href="<?php the_permalink(); ?>" class="side-thumb" title="<?php the_title(); ?>"> dfdf<?php
if ( has_post_thumbnail() ){
echo get_the_post_thumbnail();
}
//Put your default icon code here
?>
</a>
</li> <?php
}
}
}
wp_reset_postdata();
Above answer is as per your explanation, but I think what you want is, like you have a category Artists and there are subcategories each having a specific name, So for Artist gallery you want to display list of all the artists with their pictures.

Answer 2

I also needed to get list of child categories of selected parent category with url,
I found following code so useful
<?php
$categories = get_categories( array(
'orderby' => 'name',
'parent' => 5
) );
foreach ( $categories as $category ) {
printf( '<li><a class="btn btn-light btn-light-custom mt-2" href="%1$s">%2$s</a></li>',
esc_url( get_category_link( $category->term_id ) ),
esc_html( $category->name )
);
}
?>
Replace "5" with your selected parent category ID