Menno,
Just did some research and found out 2 things. First, you need to allow subscribers to your profile. You can do this by clicking the "Subscriptions" link from your own profile and at the top of the next page should be a question asking:
Want Subscribers of Your Own?
Now you can include more people in your public updates, without adding them as friends.
Make sure you've enabled that and the button should work.
Unfortunately, the subscribe button only works for Profiles. I don't believe it will work for Groups. Groups, in general, don't have a lot of features. Facebook widely recommends you move groups to "Fan Pages", which can user's can Like.. which is similar to subscribing. I don't believe there are any similar widgets for user's to Like or Subscribe to a Group, just so you know.
Hope that helps, and good luck!
Alex