Is there a possibility to list all the groups, a contact is in, on the card. This would make the organisation of my contacts much easier
My solution doesn't list the groups on the contact card itself, but it may still solve your situation:
By holding down the "Option" key when you have a contact selected and the "Groups" pane showing, the groups which contain the selected contact are highlighted.
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Unfortunately, the answer to your question is no. The default Contacts app does not allow you to do this. I believe there's some third-party apps that will do this but you'll have to search the App Store to find one.