Remove index.php from Permalink

The following is my friend’s site: http://q-pidot.com/index/index.php

We wish to remove the ‘index.php’ at the end (not the intitial index, that’s the sub directory), however whenever editing the custom permalinks, removing index.php crashes the site, showing only Not Found.

Searching a bit on it, most I can find is checking the htaccess file if it is as such:

<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>

And it is.

Any help we could receive to fix this problem would be appreciated.