How to get submenu of admin menu?

You can use the global variable $submenu.

Example to list the child menu of parent post menu:-

function admin_init_callback() {
    global $submenu;
    $sumenu_list = $submenu['edit.php'];
    var_dump($sumenu_list); //array of submenu
}
add_action('admin_init', 'admin_init_callback', 999);

Profi660 EDIT:
Not needed to be used with admin_init hook.
I used this in my plugin page and worked very well.