\documentclass[12pt]{article}
\thispagestyle{empty}
\usepackage{tikz}
\usepackage{amsfonts}
\begin{document}
\begin{tikzpicture}[node distance=2cm]
 \node (Q)                  {$\mathbb{Q}$};
 \node (Q6)  [above of=Q]   {$\mathbb{Q}(\sqrt{6})$};
 \node (Q2)  [right of=Q6]  {$\mathbb{Q}(\sqrt{2})$};
 \node (Q3)  [left of=Q6]   {$\mathbb{Q}(\sqrt{3})$};
 \node (Q23) [above of=Q6]  {$\mathbb{Q}(\sqrt{2}, \sqrt{3})$};
 \node (1f)  [right of=Q2]  {$\{1, f\}$};
 \node (1fg) [right of=1f]  {$\{1, fg\}$};
 \node (1g)  [right of=1fg] {$\{1, g\}$};
 \node (G)   [below of=1fg] {$\{1, f, g, fg\}$};
 \node (1)   [above of=1fg] {$\{1\}$};
 \draw (Q)   -- (Q2);
 \draw (Q)   -- (Q3);
 \draw (Q)   -- (Q6);
 \draw (Q2)  -- (Q23);
 \draw (Q3)  -- (Q23);
 \draw (Q6)  -- (Q23);
 \draw (G)   -- (1f);
 \draw (G)   -- (1fg);
 \draw (G)   -- (1g);
 \draw (1f)  -- (1);
 \draw (1fg) -- (1);
 \draw (1g)  -- (1);
\end{tikzpicture}
\end{document}