Yes, many. The primary historical division has been between Sunnis and Shi'ites. This started out as essentially about who should be the leader of Muslims, someone appointed by the community (Sunnis), or Muhammed's descendents though his daughter Fatima and her husband/Muhammed's cousin Ali (the Shi'ites). Today, most Muslims are Sunnis, but Shi'ites make up the majority of Muslims in Lebanon, Iraq, Bahrain and by far, in Iran. One of the more famous historical Shi'ite-run regimes was the Fatimid Caliphate, based in Egypt, in the middle-Medieval era (so you can see that areas of Sunnis and Shi'ites have moved around a bit: Iran only started to become Shi'ite largely in the 17th century). However, Shi'ites can also be subdivided in a way that Sunnis can't, into sects based on who they believe to be the appropriate list of Imams who are divinely appointed to rule Muslims. The largest group are Twelvers (because they believe there were 12 Imams, the last one vanishing, but he'll return one day), but there are also Ismailis (who were significant historically) and Zaidis (the kind currently fighting the Yemenese government). Although this started out as a division based on who should lead Islam, it has developed over time in a multitude of theological ways. Similar to how Catholicism and Protestantism started out as being about the rule of the Pope and Church hierarchy, but developed many theological differences.* There are also smaller sects, like the Alawites (whose theology isn't well understood by outsiders) and the Ahmaddis (a sect from the Indian subcontinent who proselytised and emigrated heavily in the west, so are fairly common in places like Britain, the home of their leader, Kalif Masih V). These tend to be groups that started from the interpretation of particular Islamic leaders. There are also sects that used to be somewhat significant, but are fairly small now, like Zikris and Khajirites. Sunnis have always been the largest group however.

Finally, to complicate this, are the Sufi orders. These are organisations (mostly Sunni, but some other) that Muslims can join that stress a mystical understanding of Allah. They're not really a sect, so much as an organised form of religious practice.

There are also subsets, the ones you've most likely heard of are probably Wahhabis or Salafists. They are movements within Sunni Islam that stress a particular interpretation of the Quran. The comparison I'd use perhaps is Evangelical* (it's not that these are similar in theology or whatever, I just mean in terms of how to define it as a sect). It's not a sect per-se, but Evangelical Protestantism is a kind of movement within Protestantism.

I should say, of course, that I'm talking about self-defined Muslims. Many Sunnis might say Shi'ites are not real Muslims, and vice-versa, but that's not my place to say (as a point of clarification, I am not a Muslim, simply someone with an interest in religious history).

Now, as well as this division, there are also schools of jurisprudence, which are essentially interpretations of the Qu'ran. I don't have a great understanding of them, frankly, to understand them you'd probably be better off asking in https://www.reddit.com/r/islam. Some are shared between sects, some are different, but they tend to be most common in different areas.

• I don't really like comparing Islam's history and Christianity's history, because it can be misleading. For example, every now and then you'll see western commentators say Islam needs a reformation like the Protestant reformation, against ISIS and so on, which shows a pretty flawed understanding of Islam (and for that matter, the Reformation). But I assume from your question you're much more familiar with Christianity than Islam, so this seems the easiest way to explain it.

Very much so, though "subsect" doesn't do them (or it) justice. The primary divide is obviously the 7th-century schism between the majority Sunni and minority Shia traditions, originating in the disputed succession to the Prophet. But Sunni Islam is is in turn subdivided into four (sometimes five) subsequent schools of interpretation, while besides internal differences over its own line of succession to the rank of Imam (7th-9th centuries), the Shia branch is conventionally considered to embrace a handful of autonomous but loosely affiliated sects, often of limited geographical extent.

I'm sure others can answer in more detail and add some of the theology. The important point is that far from it being a monolithic block ossified by its founding text, as with Christianity power struggles and differences of religious interpretation have been bubbling away in Islam since its emergence, with hard-liners and modernisers in the last centuries pursuing radically different visions.

[edit: Pretty much what /u/The_Manchurian said more fully, in other words :)]