Rabbi Yaakov Robinson from Chicago gave a shiur on the topic of breakaway minyanim. You can listen to the shiur here. He specifically mentions that he is not discussing the topic of home minyanim (which he says has its own issues), but rather, he is discussing creating a new shul. Here is my summary of the shiur:
The RIVaSh, one of the rishonim, says that it is asur to prevent someone from building a shul. The ReMA paskens this as halacha. The ReEM, another rishon brings as halacha that this only applies when the first shul does not have enough room for everybody. He argues that it is better for as many people as possible to daven together, so a shul should only be split up if there is not enough room to hold everyone. Most achronim see this as a machlokes rishonim.
Everybody agrees that it is not permitted to created a breakaway minyan when it is not done le-shem shamayim. Rabbi Robinson seems to say that a dispute within the shul does not constitute a sufficient reason to create a breakaway. He does suggest, however, that personal preferences such as nusach could be considered le-shem shamayim. This is not necessarily sufficient to create a new shul, however.
Also, if the new shul's existence would cause financial or any other type of damage to the original shul, everybody agrees that it is forbidden to break away. Rav Moshe Feinstein takes this to an extreme:
A rav of a shul came to Rav Moshe Feinstein after a group of people broke off from his shul. The group had broken away from the original shul because when this new rav had come, he had changed the nusach and the niggunim of the davening, in such a way which was so confusing to the members of the shul that they weren't able to daven properly. Also, the rav of the shul had cursed, scared, and embarrassed members of the shul on multiple occasions. The rav argued that there had actually only been two times when he had spoken strongly: once when he had to give some musser to the congregation, and another time when he said that his words had been misunderstood and he had not intended to be insulting. Furthermore, he said that without the other people remaining with his shul it would suffer financially, and not be able to continue operating. Rav Moshe Feinstein said that it was completely asur for the group to split off. He explains that just the fact that the break-away would cause financial difficulties for the original shul is sufficient to prevent the new minyan from breaking off. It was asked of Rav Dovid Feinstein, the son of Rav Moshe Feinstein, if anything was missing from that story since it seems like the breakaway group had a lot of good reasons for separating. Rav Dovid Feinstein said that nothing was missing from the story and that Rav Moshe held that strongly about causing financial loss.
The Ksav Sofer says that it is only permitted to break away in very very unusual circumstances. Rabbi Robinson gives the example of a shul taking down the mechitza or hiring a female rabbi or doing something else that is totally against halacha. It seems that many of the acharonim and modern poskim agree with this opinion.