Two sorts of men were required to do military service : the leudes, vassals or sub-vassals, who were obliged to as a function of their fiefs ; and the free men, Franks, Romans, and Gauls, who served under the count, and were led by him and his officers.
They called free men those who on the one hand had no benefices or fiefs, and who on the other hand were not subjected to villeinage ; the lands they possessed were what were called allodial lands.
The counts assembled the free men and led them into war  ; they had under them officers whom they called vicars  ; and as all free men were divided into centenæ, who formed what was called a borough, the counts also had under them officers called centenarii, who led the free men of the borough  or their centenæ to war.
This division by hundreds is subsequent to the Franks’ settlement in the Gauls. It was done by Clotaire and Childebert with an eye to obliging every district to answer for the robberies committed there : we see this in the decrees of those princes.  A similar policy is still observed today in England.
The bishops were rather conflicted : they were not themselves quite in agreement  on their situation ; they asked Charlemagne not to oblige them any longer to go to war, and when they obtained that, they complained that they were being made to lose the public’s consideration, and the prince was obliged to justify his intentions. However that may be, at times when they no longer went to war, I do not see their vassals being led to war by the counts ; we see on the contrary the kings or bishops choosing one of the fidèles to lead them. 
In a capitulary of Louis the Debonaire,  the king distinguishes three sorts of vassals : the king’s, the bishops’, and the count’s. The vassals of a leude  or lord were led into war by the count only when some responsibility in the king’s household prevented those leudes from leading them in person.
But who was leading the leudes to war ? We cannot doubt that it was the king, who was always at the head of the fidèles. That is why, in the capitularies, we always see an opposition between the king’s vassals and those of the bishops.  Our courageous, proud, and magnanimous kings were not in the army to ride at the head of this ecclesiastical militia ; those were not the men they would choose to conquer or die with them.
But these leudes took with them their vassals and sub-vassals, and that appears clearly in the capitulary where Charlemagne orders that every free man who has four manors, either in his holdings, or in someone’s benefice, should go against the enemy, or follow his lord.  Clearly what Charlemagne means is that the man who had but one estate of his own was to participate in the count’s militia, and he who held a lord’s benefice accompanied him.
Yet the abbé Dubos pretends that when it is spoken in the capitularies of the men who depended on a particular lord, that means only serfs, and bases himself on the law of the Visigoths and the practice of that people.  It would be better to base oneself on the capitularies themselves ; the one which I have just cited formally states the opposite. The treaty between Charles the Bald and his brothers likewise speaks of the free men who can choose a lord or the king ; and this provision is consistent with many others.
We can therefore say that there were three sorts of militias : that of the leudes or fidèles of the king, who themselves had other fidèles under them ; that of the bishops or other ecclesiastics and their vassals ; and finally that of the count, at the head of the free men.
I am not saying that the vassals could not be subordinate to the count, as those who have a particular command are subordinate to him who has a more general command.
We even see that the count and the king’s envoys could make them pay the ban, which is to say a fine, when they had not fulfilled the engagements of their fief.
Similarly, if the king’s vassals  committed plunders, they were subjected to correction by the count, unless they preferred to submit to correction by the king.