Suppose I take a number field $ K $, not necessarily Galois, with class number $ h_k $ (over $ \mathbb{Q} $). Write $ \overline{K} $ for the normal closure of $ K $. What, if anything, can be said about the relation between $ h_K $ and $ h_{\overline{K}} $? A question dealing with one particular case of interest is presented at the end.
In general I know there is no real relationship between the class number of a field $ E $, and of an extension $ F \supset E $. Probably the best example of this is $ \mathbb{Q} \subset \mathbb{Q}(\sqrt{-d}) $, with $ d $ squarefree. We have that $ h_{\mathbb{Q}} = 1 $, but $ h_{\mathbb{Q}(\sqrt{-d})} \to \infty $ as $ d \to \infty $. In particular there can be no upper bound on $ h_F $ as a function of only $ h_E $ and $ [F\colon E] $.
However, from class field theory we do get the following observation. Let $ H(E) $ denote the Hilbert class field of $ E $. Then the fields $ E \subset F $ fit into the following diagram. \begin{array}{cc} & H(F) & \\ \quad \huge\diagup & {\huge|} h_{F} \\ H(E) & F \\ {\huge|} h_E & {\huge\diagup} [F\colon E] \\ E \end{array} We conclude that $ h_E \mid h_F [F \colon E] $. But of course, in the previous case this tell us the trivial result that $ 1 \mid h_F [F \colon E] $. Can anything more precise be said in the case $ K \subset \overline{K} $?
Initially, I wondered/hoped if there was some bound like $ h_{\overline{K}} \mid [\overline{K}\colon K] h_k^{[\overline{K} \colon K]} $, but after some numerical experimentations with Magma, I find this doesn't hold. For example with $ K = \mathbb{Q}(\theta) $, where $ \theta $ is a root of $ 2 + 7x + 2x^2 + 8x^3 $, we get that $ h_K = 2 $ and $ h_{\overline{K}} = 88 $. Somehow a factor of 11 suddenly appears. But maybe this has to do with the fact that the discriminant $ \Delta_{\overline{K}} $ contains a factor of 11?
This open-ended question is motivated by a specific observation which occurred when I looked at the the pure cubic fields $ K = \mathbb{Q}(\sqrt[3]{n}) $. Here the situation seems more 'rigid'. In this case $ \overline{K} = \mathbb{Q}(\sqrt[3]{n}, \sqrt{-3}) $, and in all cube-free cases from $ n = 1, \ldots, 1000 $, I seem to find that $ h_{\overline{K}} \mid h_K^2 $. But moreover $$ h_K^2 / h_{\overline{K}} \in \{ 1, 3 \} \, . $$
Question: Is this true generally for $ K = \mathbb{Q}(\sqrt[3]{n}) $ that $ h_{\overline{K}} \mid h_K^2 $? Even more, do we have $ h_K^2 / h_{\overline{K}} \in \{ 1, 3 \} $. If so, how can this be proven? Is this a shadow of some more general result?
Extra: If it is true that for $ K = \mathbb{Q}(\sqrt[3]{n}) $, we have $ h_K^2 / h_{\overline{K}} \in \{ 1, 3 \} $. Is there anything about $ n $, or $ K $, which lets us predict whether the case $ h_K^2 / h_{\overline{K}} = 1 $ occurs, or the case $ h_K^2 / h_{\overline{K}} = 3 $?
Your guess is not far from the truth: in the case of an $S_3$-extension, the class number of the normal closure is, up to a bounded factor involving a unit index, the product of the class numbers of the subfields over which the normal closure is cyclic. Since the quadratic subfield has class number $1$ in the pure cubic case, your guess is true here. In your other example, the additional factor $11$ must come from the class number of the quadratic subfield ${\mathbb Q}(\sqrt{{\rm disc}(K)})$ of the normal closure. Similar remarks apply in the general case. For details see the literature; looking for "class number formulas" should give you enough papers to start with.