\documentclass[12pt]{article}
\thispagestyle{empty}
\usepackage{tikz}
\usepackage{amsfonts}
\begin{document}
\begin{tikzpicture}[node distance=2cm]
 \node (Qw-o)                       {$\mathbb{Q}(\omega, \theta)$};
 \node (Qo)   [below of=Qw-o]       {$\mathbb{Q}(\theta)$};
 \node (Qwo)  [right of=Qo]         {$\mathbb{Q}(\omega \theta)$};
 \node (Qw2o) [right of=Qwo]        {$\mathbb{Q}(\omega^2 \theta)$};
 \node (Qw)   [below left of=Qo]    {$\mathbb{Q}(\omega)$};
 \node (Q)    [below right of=Qw]   {$\mathbb{Q}$};
 \node (1ff2) [below right of=Qw2o] {$\{1, f, f^2\}$};
 \node (1g)   [above right of=1ff2] {$\{1, g\}$};
 \node (1gf)  [right of=1g]         {$\{1, gf\}$};
 \node (1gf2) [right of=1gf]        {$\{1, gf^2\}$};
 \node (G)    [below right of=1ff2] {$\{1, f, f^2, g, gf, gf^2\}$};
 \node (1)    [above of=1g]         {$\{1\}$};
 \draw (Q)    -- (Qo);
 \draw (Q)    -- (Qw);
 \draw (Q)    -- (Qwo);
 \draw (Q)    -- (Qw2o);
 \draw (Qo)   -- (Qw-o);
 \draw (Qw)   -- (Qw-o);
 \draw (Qwo)  -- (Qw-o);
 \draw (Qw2o) -- (Qw-o);
 \draw (G)    -- (1ff2);
 \draw (G)    -- (1g);
 \draw (G)    -- (1gf);
 \draw (G)    -- (1gf2);
 \draw (1ff2) -- (1);
 \draw (1g)   -- (1);
 \draw (1gf)  -- (1);
 \draw (1gf2) -- (1);
\end{tikzpicture}
\end{document}