Let
G
>
Mod
2
G>\operatorname {Mod}_2
be the Goeritz subgroup of the genus-2 mapping class group. We show that finitely-generated, purely pseudo-Anosov subgroups of
G
G
are convex cocompact in
Mod
2
\operatorname {Mod}_2
, addressing a case of a general question of Farb–Mosher. We also give a simple criterion to determine if a Goeritz mapping class is pseudo-Anosov, which we use to give very explicit convex-cocompact subgroups. In our analysis, a central role is played by the primitive disk complex
P
\mathcal {P}
. In particular, we (1) establish a version of the Masur–Minksy distance-formula for
P
\mathcal {P}
, (2) classify subsurfaces
X
⊂
S
X\subset S
that are infinite-diameter holes of
P
\mathcal {P}
, and (3) show that
P
\mathcal {P}
is quasi-isometric to a coned-off Cayley graph for
G
G
.