Contacts on my iPhone 6S not appearing on iCloud on my Windows 10 PC
If your contacts are stored on Google then you will not be able to see your contacts in iCloud because that is not where they are stored.  Unfortunately syncing groups between Google and your iPhone is not possible without an app to do so.  Here is one such app that you can use that will allow you to sync your Google groups with iPhone.

Google Contact Sync

Otherwise you will have to organize the groups on your phone directly.