You cannot embed the Poincare group or the Lorentz group into a compact Lie group G. Indeed, denote the Lie algebra of G as g and the Lorentz algebra as l=o(1,3)≅sl2(C).
The Killing form on g is non-positive-definite, but then so is its restriction to l. Restriction of the Killing form on g to l is l-invariant and is therefore proportional to the Killing form on l, since the latter is a simple real Lie algebra. Finally, the Killing form on l has signature (3,3), contradiction.
By the same reasoning, you cannot mod out by a discrete subgroup of the Lorentz group and get a compact group: the Lie algebra does not change, so the Killing form cannot become non-positive-definite.
On the other hand, there are well-known 'compactifications' of the translation group. You can either mod out R/Z=S1 or immerse R→T2 as an irrational winding depending on what kind of compactifications you are interested in.
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