Congruence condition summary for groups of order 2^n

This article gives relevant information on


 * Every possible collection of groups satisfying a universal congruence condition relative to the prime 2, for small orders.
 * Conditional versions of congruence conditions, e.g., those obtained by restricting the size or putting some other constraint on the ambient group.

Order 1
The only nontrivial collection of groups of order 1 is the singleton collection comprising the trivial group, and this satisfies a universal congruence condition and an existence condition.

Order 2
The only nontrivial collection of groups of order 1 is the singleton collection comprising cyclic group:Z2, and this satisfies a universal congruence condition and an existence condition.

Order 4
There are two groups of order 4: cyclic group:Z4 and Klein four-group. There are thus $$2^2 - 1 = 3$$ possible collections of groups. We note which of these satisfy the congruence condition:

Order 8
There are five groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8. There are thus $$2^5 - 1 = 31$$ possible collections of groups. Instead of listing all 31, we simply note the ones that do satisfy a universal congruence condition or a congruence condition to an interesting restricted class: