  The free-streaming length of dark matter depends on fundamental dark matter
physics, and determines the abundance and concentration of dark matter halos on
sub-galactic scales. Using the image positions and flux ratios from eight
quadruply-imaged quasars, we constrain the free-streaming length of dark matter
and the amplitude of the subhalo mass function (SHMF). We model both main
deflector subhalos and halos along the line of sight, and account for warm dark
matter (WDM) free-streaming effects on the mass function and mass-concentration
relation. By calibrating the scaling of the SHMF with host halo mass and
redshift using a suite of simulated halos, we infer a global normalization for
the SHMF. We account for finite-size background sources, and marginalize over
the mass profile of the main deflector. Parameterizing dark matter
free-streaming through the half-mode mass $m_{\rm{hm}}$, we constrain the
thermal relic particle mass $m_{\rm{DM}}$ corresponding to $m_{\rm{hm}}$. At
$95 \%$ CI: $m_{\rm{hm}} < 10^{7.8} M_{\odot}$ ($m_{\rm{DM}} > 5.2 \
\rm{keV}$). We disfavor $m_{\rm{DM}} = 4.0 \rm{keV}$ and $ m_{\rm{DM}} = 3.0
\rm{keV}$ with likelihood ratios of 7:1 and 30:1, respectively, relative to the
peak of the posterior distribution. Assuming cold dark matter, we constrain the
projected mass in substructure between $10^6 - 10^{9} M_{\odot}$ near lensed
images. At $68 \%$ CI, we infer $2.0 - 6.1 \times 10^{7} M_{\odot}
\rm{kpc^{-2}}$, corresponding to mean projected mass fraction
$\bar{f}_{\rm{sub}} = 0.035_{-0.017}^{+0.021}$. At $95 \%$ CI, we obtain a
lower bound on the projected mass of $0.6 \times 10^{7} M_{\odot}
\rm{kpc^{-2}}$, corresponding to $\bar{f}_{\rm{sub}} > 0.005$. These results
agree with the predictions of cold dark matter.
