Class numbers of orders Consider an order $R$ in a number field $L$. Let $C_R$ be the set of $R$-fractional ideals modulo $L^\times$. Let $O$ be the maximal order in $L$, and $C_O$ be the class group of $O$. 
My question: Is there a formula that relates $\# C_R$ with $\# C_O$, that involves perhaps the conductor of $R$?
A note: Beware that in the definition of $C_R$ I really consider fractional ideals: I do not ask that they are non-singular, so $C_R$ is not the Picard group of the order $R$ (In fact, it is not even a group!). 
 A: The ideal class semigroups mentioned in the question got studied in this setting (orders of number fields) and in other and more general ones by various authors in recent years.
A starting references is: 

Zanardo, P.; Zannier, U. The class semigroup of orders in number fields.
  Math. Proc. Cambridge Philos. Soc. 115 (1994), 379–391. 

They show that this semigroup is a Clifford semigroup for orders of quadratic fields, but for degree greater 2, there always is some order such that the respective class semigroup is not a Clifford semigroup. 
There is also an older paper on this theme (that it appears was overlooked in recent literature for some time but reapperas in still more recent one) 

Dade, E. C.; Taussky, O.; Zassenhaus, H. On the theory of orders, in paricular on the semigroup of ideal classes and genera of an order in an algebraic number field.  Math. Ann. 148 (1962) 31–64.

Starting from these two papers and looking for papers that quote them in MathSciNet for example one will find several more recent contributions. Investigating these semiclass groups; but it seems (but I donot have a good overview) the emphasis is more to generalize to more general structures (say, other domains than just orders) than more details in the number-theoretic setting. 
Yet, in particular the early papers (I do not know for the recent ones) contain also some explicit information on these semigroups, but as commented earlier I would not know of in some sense simple descriptions (but again my knowledge is superficial here).
A: $\DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Spec}{Spec}$
EDIT: This answer is wrong; please un-accept it.  I erroneously believed that a fact about quadratic fields held in general.  See David Speyer's counterexample in the comment section.
Your "class monoid" $C_R$ may [EDIT: in the quadratic case] be described as the disjoint union of the Picard groups of all intermediate orders between $R$ and $O$, plus the additional data of how to multiply classes in different Picard groups.  Specifically, for a fractional ideal $I$ of $R$, let $R_I$ be the maximum order for which $I$ (as a set) remains a fractional ideal.  In other words, $R_I$ is the multiplier ring
$$ R_I = \{x \in L : xI \subseteq I\}. $$
With a little work, you can show that $I$ is non-singular (= invertible = locally principal) in $R_I$ [EDIT: $I$ may not be invertible in $R_I$ when $[L:\mathbb{Q}]>2$], and in no proper suborder.  Therefore,
$$\#C_R = \sum_{R \subseteq R' \subseteq O} \#{\Pic{R'}}.$$
As you may already know, there are standard theorems relating $\Pic{R'}$ to $C_O$ and some other data.  The idea is to look at the long exact sequence in sheaf cohomology associated to the short exact sequence of sheaves on $\Spec{R'}$
$$1 \to \mathcal{O}_{R'}^\times \to i_*\mathcal{O}_O^\times \to i_*\mathcal{O}_O^\times/\mathcal{O}_{R'}^\times \to 1,$$
or see Neukirch's Algebraic Number Theory, Ch. 1 Sec. 12 for non-sheafy details.  For the purposes of counting, the upshot is that
$$ \#\Pic{R'} = \frac{h_L}{(O^\times : R'^\times)} \frac{\#(O/\mathfrak{f}_{R'})^\times}{\#(R'/\mathfrak{f}_{R'})^\times}, $$
where $\mathfrak{f}_{R'}$ is the conductor of $R'$, and $h_L = \#C_O$.
