Here is the code to show all the subcategories inside an category in magento
<?php
$children = Mage::getModel('catalog/category')->getCategories(10);
foreach ($children as $category) {
echo $category->getName();
}
?>
Here 10 is the id of the parent category
and to get the url for that category use
<?php
$url=Mage::getModel('catalog/category')->load($category->getId())->getURL();
?>
You can also get the image of an category by adding this inside the for loop
<?php
$imageurl=Mage::getModel('catalog/category')->load($category->getId())->getImageUrl();
?>
Inorder to get the child categories of a category when include in navigation set to no
we have to use the below code to get the children categories
$_categories = Mage::getModel('catalog/category')->load(10)->getChildrenCategories();
foreach ($_categories as $category) {
echo $category->getName();
}