There are some smaller Lutheran groups that do as well. That's why there were some Lutheran fraternities and sororities.

My wife is a Latter Day Saint and is a member of Alpha Phi Omega which is a co-ed community service fraternity.

I know people who are both Latter Day Saints and Masons and (from the wikipedia article Mormonism and Freemasonry)

From 1925 to 1984 the Grand Lodge of Utah prohibited Latter-day Saints from joining, but no other Grand Lodge followed this ban and Mormons were free to join Lodges outside Utah. In 1984 the Grand Lodge of Utah officially dropped its anti-Mormon position and allowed LDS church members to join. Today there is no formal obstacle in Utah or in any other Grand Lodge preventing Mormons from becoming Freemasons.
The presidency of the LDS Church has not made an official statement as to whether or not Freemasonry is compatible with Mormonism. However Don LeFevre, a past spokesman for the church has said the church "...strongly advises its members not to affiliate with organizations that are secret, oath-bound, or would cause them to lose interest in church activities."[8] There are many LDS Masons in Utah and other Grand Lodges who serve and have served in various leadership positions, including Grand Masters, other Grand Officers, and Worshipful Masters.

More conservative Baptists would also not look kindly on membership. Some Baptist churches do not allow you to be a member of the church if you are in a lodge, Mason, etc.

The Catholic church doesn't allow Masonic membership, but has no issue with GLOs.

I haven't seen one that prohibits Greek or lodge membership.
I'm sure they're out there, but the more mainstream, the less likely.
Of course, there are 4011 different kinds of Baptist churches!

Over the years, I've met various fraternity members who happen to be Mormon and none expressed any conflict with being Mormon and being Greek. Also the University of Utah has Greek Life and roughly 50% of the students are Mormon. So statically, half of the UofU GLOs might be made up of Mormons.

Former Utah Governor Jon Huntsman, Jr. is a Sigma Chi (Penn) and a Mormon. And the Mormon Tabernacle Choir has recorded fraternity songs.

The Sweetheart of Sigma Chi by The Mormon Tabernacle Choir :D

God does not think like us. But He did say you shall have no other god before Me. So don't make the Sorority or Fraternity an idol god....God hates idolatry. Just enjoy the brotherhood and sisterhood while honoring God.

The University of Utah also had church sponsored GLOs on campus until they disbanded them a few years ago in favor of the Institute. The sorority was Lambda Delta Sigma and the fraternity was Sigma Gamma Chi. I remember learning about Lambda Delta Sigma in an edition of Baird's from the 1960s or 70s.

My mom was raised in the Wisconsin Synod. Girl/Boy Scouting was not allowed so I can't fathom them being down with GLOs. Course she signed me up for Girl Scouts and never expressed religious objections to my membership.

No and no.

No and no.