On generalized hexagons of order (3,t) and (4,t) containing a subhexagon

We prove that there are no semi-finite generalized hexagons with q+1 points on each line containing the known generalized hexagons of order q as full subgeometries when q is equal to 3 or 4, thus contributing to the existence problem of semi-finite generalized polygons posed by Tits. The case when q is equal to 2 was treated by us in an earlier work, for which we give an alternate proof. For the split Cayley hexagon of order 4 we obtain the stronger result that it cannot be contained as a proper full subgeometry in any generalized hexagon.