Curly braces in Hasse diagram

I don't know how to add a brace emcompassing only part of the height of the figure with tikz, but it's easy to obtain it with pstricks:

\documentclass[border=3mm]{standalone}

\usepackage{amsmath, pstricks-add}
\usepackage{auto-pst-pdf}

\begin{document}
\psset{linejoin=1}
\everypsbox{\footnotesize}
$ \begin{psmatrix}[rowsep=1.3cm, colsep=2.5cm]
    \pnode[-4pt, 1.5ex]{K1}\phantom{K₁} & K₁ \\
    & k₁ \\
    \pnode[-4pt, -0.5ex]{K0}K₀ \\
    k₀
    \psset{nodesep=4pt, labelsep=2pt}
    \ncline{3,1}{1,2}\ncline{4,1}{2,2}\nbput[nrot=:U]{\text{degree $ p $}}
    \ncline{1,2}{2,2}\ncline{4,1}{3,1}\naput[nrot=:U]{\text{degree $ 2 $}}
    \psset{braceWidthInner=3pt,braceWidthOuter=3pt, braceWidth =0.8pt, nodesepB=-4pt, nodesepA=-1pt}
    \psbrace(K1)(K0){H}
  \end{psmatrix} $

\end{document}

A simple cases environment will do:

\documentclass[border=3mm]{standalone}

\usepackage{amsmath, tikz, tikz-cd}

\begin{document}

$ H \begin{cases}
    \begin{tikzpicture}[baseline= (a).base]
      \node[scale=.85] (a) at (0,0){
        \begin{tikzcd}[every arrow/.append style={dash}]
          & & K_1
          \arrow[ddll]
          \arrow[d]\\
          & & k_1
          \arrow[ddll,sloped,pos=0.7,"\text{degree $p$}"]\\
          K_0 \arrow[d,swap,"\text{degree 2} \, \,"] & & \\
          k_0 & &
        \end{tikzcd}
      };
    \end{tikzpicture}
  \end{cases} $

\end{document}

With \left\lbrace and \right.

\documentclass[border=3mm]{standalone}

\usepackage{amsmath,tikz, tikz-cd}

\begin{document}
$H \left\lbrace 
    \begin{tikzpicture}[baseline= (a).base]
    \node[scale=.85] (a) at (0,0){
        \begin{tikzcd}[every arrow/.append style={dash}]
        & & K_1 
        \arrow[ddll]
        \arrow[d]\\
        & & k_1
        \arrow[ddll,sloped,pos=0.7,"\text{degree $p$}"]\\
        K_0 \arrow[d,swap,"\text{degree 2} \, \,"] & & \\
        k_0 & &
        \end{tikzcd}
    };
    \end{tikzpicture}
\right.$    
\end{document}