  We investigate the band structure of the three-dimensional Hofstadter model
on cubic lattices, with an isotropic magnetic field oriented along the diagonal
of the cube with flux $\Phi=2 \pi \cdot m /n$, where $m,n$ are co-prime
integers. Using reduced exact diagonalization in momentum space, we show that,
at fixed $m$, there exists an integer $n(m)$ associated with a specific value
of the magnetic flux, that we denote by $\Phi_c(m) \equiv 2 \pi \cdot m/n(m)$,
separating two different regimes. The first one, for fluxes $\Phi<\Phi_c(m)$,
is characterized by complete band overlaps, while the second one, for
$\Phi>\Phi_c(m)$, features isolated band touching points in the density of
states and Weyl points between the $m$- and the $(m+1)$-th bands. In the
Hasegawa gauge, the minimum of the $(m+1)$-th band abruptly moves at the
critical flux $\Phi_c(m)$ from $k_z=0$ to $k_z=\pi$. We then argue that the
limit for large $m$ of $\Phi_c(m)$ exists and it is finite: $\lim_{m\to \infty}
\Phi_c(m) \equiv \Phi_c$. Our estimate is $\Phi_c/2\pi=0.1296(1)$. Based on the
values of $n(m)$ determined for integers $m\leq60$, we propose a mathematical
conjecture for the form of $\Phi_c(m)$ to be used in the large-$m$ limit. The
asymptotic critical flux obtained using this conjecture is $\Phi_c^{{\rm
(conj)}}/2\pi=7/54$.
