Yes, there were, but it's not always clear how they carried out commands or who exactly they were. Check out this previous comment I made about the military officers of Edward I. Unfortunately, we can't say for sure whether this system was also followed in later periods. It's tempting to say that there were lower-level officers of a sort, but the structure and composition of English armies changed in quite significant ways from Edward I to Edward III; armies were raised from contracted retinues and hiring companies rather than from shire levies. The captains of the retinues and companies were not necessarily aristocrats, although some at least were knighted. Others might have been raised to knighthood through military service and were not from especially notable backgrounds. Sir John Hawkwood, to use a very famous example, was the son of a well-off landowner and may have been working as a tailor in London for a time. It's not clear how and when Hawkwood was knighted, but he either managed to wrangle a knighthood or at least was skilled enough as a mercenary that no one was willing to call him on it. It's cases like Hawkwood's that show that the class system of the medieval period was not strictly "peasant/aristocrat;" there were significant demographics who were not aristocrats but owned property or had some degree of wealth.

The tactical behavior of the retinue or company as a fighting unit is too unclear to say what sort of command structure might have been in place. Of course, these units all have captains listed in their muster rolls, but it doesn't make sense for them to be the only leaders present in the unit. Since retinues usually contained both archers and men-at-arms, surely there would need to be at least two different officers maintaining order for these two categories of soldier, who would fight and be deployed in different ways? It seems likely that retinues were split up, because the archers were generally split off from the men-at-arms. How exactly the retinues were subdivided is unclear and we are without evidence to say who might have been in command of the archers. There is certainly enough evidence to say that aristocratic titles were not an absolute necessity for military command, especially in smaller units. But the specifics of these arrangements are unclear.