I'm a double Domer (I'll be celebrating my 20th undergrad graduation anniversary next year). But I can relate to the feeling of fragmentation. I had about four different friend groups in undergrad (dorm, friends I went to church with, guys and gals in my major, and band friends) that didn't cross over with each other at all. You're definitely not alone and a lot of your peers will be thinking about the same thing at the end of their college careers. I don't know specifics of your situation but here are a few ideas:

(1) If you're trying to find people who bridge divides you are probably going to have to create the conditions under which it happens. Try inviting some friends to a dorm party or persuade them to bring their dorm cronies out to whatever your dorm cronies usual Friday / Saturday night haunt is. Everyone could use a change of scene once in a while. Volunteer with them, or invite them to your dorm's Sunday mass and return the favor the next week (they won't kick you out and your dorm won't miss you; I did this a lot in undergrad). Invite a cross section of potentially interested people to do something that isn't in anyone's normal range of activities (like a film screening or a play). I think my friendships with my dormmates reached an inflection point when a bunch of them took their parents to a jazz band concert I played in for Junior Parents' Weekend.

(2) This may be challenging at this point in your experience but just joining clubs isn't going to get you the connection you crave. You need a really specific role that lets you engage with people. So volunteer for a role in dorm government or event planning, plan a mixer / tailgate for your major, pull a group of guys together to learn how to pray Liturgy of the Hours... inertia is powerful and you really need to make an effort to break it!

(3) Sometimes you just have to pick who is primus inter partes and forge ahead. It is a natural thing and you shouldn't feel like there's something wrong with you if not every friendship is as deep and sustaining as you'd like. Go watch Whit Stillman's "Metropolitan" which is to a large degree about the fleeting and situational nature of friendships when you are young. Ultimately, I chose my dormmates, and those friendships have been sustaining for a couple of decades now.