The lutetium‐rich intermetallic compounds Lu21T14–xInx (T=Rh, Pd, Pt) with x values ranging from 1.57 to 4.70 were synthesized from the elements in sealed tantalum ampules in an induction furnace with a special annealing sequence. Two samples from each solid solution were studied by powder X‐ray diffraction and the structures of Lu21Rh10.36In3.55, Lu21Rh12.43In1.57, Lu21Pd9.74In4.26, Lu21Pd10.19In3.81, Lu21Pt9.26In4.70 and Lu21Pt9.93In4.07 were refined from single crystal X‐ray diffractometer data. All phases crystallize with coloring variants of the Y3Rh2 type structure, space group I4/mcm (No. 140). The Lu21T14–xInx phases are built up from a tetragonal rod packing of columns that consist of a condensation of square prisms and square antiprisms of the lutetium atoms that are centered by the indium and transition metal atoms. These columns condense with the substructure of coordination number 9 and 11 polyhedra around the Pt2 and In1/Pt5 atoms. The striking features of the ternary samples concern small defects on the transition metal 16 l sites and T/In mixing on the sites 16 l and 4c.