  We measure the hyperfine structure of the $5p^65d$ ${}^2D_{3/2}$, $5p^65d$
${}^2D_{5/2}$, $5p^64f$ ${}^2F^o_{5/2}$, and $5p^64f$ ${}^2F^o_{7/2}$ levels in
doubly-ionized lanthanum (La III; La$^{2+}$) in a hollow cathode lamp using
optogalvanic spectroscopy. Analysis of the observed spectra allows us to
determine the hyperfine $A$ coefficients for these levels to be
$A_{D3/2}=412(4)$ MHz, $A_{D5/2}=20(5)$ MHz, $A_{F5/2}=319(2)$ MHz, and
$A_{F7/2}=155(4)$ MHz; and provide estimates for the hyperfine $B$ coefficients
as $B_{D3/2}=105(29)$ MHz, $B_{D5/2}=157(40)$ MHz, $B_{F5/2}=-2(53)$ MHz, and
$B_{F7/2}=171(51)$ MHz.
