Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 24?
Each side must be less than $12$. The positive solutions of $a+b+c+d=24$ number $\binom{23}3=1771$; those with a side at least $12$ number $4\binom{12}3=880$, so $891$ tuples are allowed.
The rotation by $180^\circ$ fixes $(a,b,a,b)$ with $a+b=12$, $1\le a\le11$: $11$ tuples. Each rotation by $90^\circ$ or $270^\circ$ fixes only $(6,6,6,6)$.
By Burnside's lemma, the number of classes is $\dfrac{891+11+1+1}4=\boxed{226}$.
