  Compactifying the 6-dimensional (2,0) superconformal field theory, of type
ADE, on a Riemann surface, $C$, with codimension-2 defect operators at points
on $C$, yields a 4-dimensional $\mathcal{N}=2$ superconformal field theory. An
outstanding problem is to classify the 4D theories one obtains, in this way,
and to understand their properties. In this paper, we turn our attention to the
$E_6$ (2,0) theory, which (unlike the A- and D-series) has no realization in
terms of M5-branes. Classifying the 4D theories amounts to classifying all of
the 3-punctured spheres ("fixtures"), and the cylinders that connect them, that
can occur in a pants-decomposition of $C$. We find 904 fixtures: 19
corresponding to free hypermultiplets, 825 corresponding to isolated
interacting SCFTs (with no known Lagrangian description) and 60 "mixed
fixtures", corresponding to a combination of free hypermultiplets and an
interacting SCFT. Of the 825 interacting fixtures, we list only the 139
"interesting" ones. As an application, we study the strong coupling limits of
the Lagrangian field theories: $E_6$ with 4 hypermultiplets in the $27$ and
$F_4$ with 3 hypermultiplets in the $26$.
