We have identified three multiply imaged galaxies in Hubble Space Telescope images of the redshift z = 0.68 cluster responsible for the large-separation quadruply lensed quasar, SDSS J1004+4112. Spectroscopic redshifts have been secured for two of these systems using the Keck I 10 m telescope. The most distant lensed galaxy, at z = 3.332, forms at least four images, and an Einstein ring encompassing 3.1 times more area than the Einstein ring of the lensed QSO images at z = 1.74, due to the greater source distance. For a second multiply imaged galaxy, we identify Lyα emission at a redshift of z = 2.74. The cluster mass profile can be constrained from near the center of the brightest cluster galaxy, where we observe both a radial arc and the fifth image of the lensed quasar, to the Einstein radius of the highest redshift galaxy, ~110 kpc. Our preliminary modeling indicates that the mass approximates an elliptical body, with an average projected logarithmic gradient of sime-0.5. The system is potentially useful for a direct measurement of world models in a previously untested redshift range.