Boeotian League organized its merged armies by specified numbers of the Boeotian city-states’ unities — see below. The Boeotian army gathered men from eleven subdivisions of the Boeotian League, which had seven Boeotarchs (Paus. 9.13.6-7) leading the army. We can see them in action in Coroneia and Acraephium where they hold the leadership of the army.
The full army size could have been in between twenty to thirty thousand men strong — see Mantineia. However, this should have been an exception. We know that the Boeotian army gathered specific numbers of elite units from all the members of the League.
The Hellenica Oxyrhynchia (xi.4) stated clearly that each of the eleven μέρη (members) into which the Boeotia has been subdivided, for League purposes, bound to contribute “περί χιλίους μέν ὁπλίτας ἱππέας δὲ ἐκατόν” to the federal army. Therefore the total strength should be of 11,000 hoplites and 1,000 cavalry with whatever numbers of light troops.
We can further support the numbers by pointing the numbers of Thespian hoplites that have been at service on a specified time. Approximately one thousand Thespian hoplites were always ready for war. In Plataia there were eight hundredth but a number of seven hundredth had already fought and died at Thermopylae, in total one thousand and thirty hundredth thespain hoplites.
In Delion, as Thucydides pointed out, seven thousand hoplites, five hundredth peltasts, one thousand cavalry and a number of more than ten thousand light-armed troops — the light infantry of the Boeotian army, the hamippoi, worked with cooperation with the prodromoi and the hippeis (see Thuc. 5.57; Xenoph. 7.5.24) — have been used by the League. (4.93) The five hundredth peltasts were possible a contribution by the μέρη in-place of an equivalent number of hoplites. In this case, the total numbers of hoplites riches the seven thousand and five hundredth men approximately. In Delion all the Boeotian army as well of Athens were πανδημεί — in full strength (Thuc. 4.90). Hence the question, why seven hundredth and not ten thousand?
Thus where are the three thousand hoplites? The difference of such numbers when placed in a hoplites’ phalanx is of importance. The League’s army structure took very seriously units’ numbers. In case of the Delion, we must consider that three thousand and five hundredth hoplites were at home — so that brings the Hoplites’ number in use at ten thousand. Take for instance the Spartan League (Thuc. 2.10;2.47;3.15;5.57) in which each ally was to contribute two thirds of its forces, except in the case where the war was to be waged in the territories of some member of the alliance in which emergency this particular ally must turn out πανστρατιά (full force). In Tanagra, in addition, we witnessed a full force of one thousand hoplites ready to be provided for the League plus all the rest 2/3 = 6,666 = 7,666 hoplites (peltasts substitutes hoplites) ( see foo[1])
1see Seymour P.A. “Note on the Boeotian League” CR 36 (3/4) p. 1922