The structure of the three-dimensional (3D) random field Ising magnet is studied by ground-state calculations. We investigate the percolation of the minority-spin orientation in the paramagnetic phase above the bulk phase transition, located at [Δ/J]c ≃ 2.27, where Δ is the standard deviation of the Gaussian random fields (J = 1). With an external field H there is a disorder-strength-dependent critical field ± Hc(Δ) for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. Hc ∼ (Δ - Δp)δ, where Δp = 2.43 ± 0.01 and δ = 1.31 ± 0.03, implying a critical line for Δc < Δ ≤ Δp. When, with zero external field, Δ is decreased from a large value there is a transition from the simultaneous up- and down-spin spanning, with probability π↑↓ = 1.00 to π↑↓ = 0. This is located at Δ = 2.32 ± 0.01, i.e., above Δc. The spanning cluster has the fractal dimension of standard percolation, Df = 2.53 at H = Hc(Δ). We provide evidence that this is asymptotically true even at H = 0 for Δc < Δ ≤ Δp beyond a crossover scale that diverges as Δc is approached from above. Percolation implies extra finite-size effects in the ground states of the 3D random field Ising model.