Congruence condition summary for groups of order 2^n

This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of order 2^n.
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Universal and conditional congruence conditions by order

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:

Collection Does it satisfy a universal congruence condition? Restricted class of groups in which it satisfies a congruence condition Explanation
cyclic group:Z4 only No Nothing interesting (?) Doesn't even satisfy a congruence condition in abelian groups, such as direct product of Z4 and Z2.
Klein four-group only No abelian groups congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group explains why it is true in abelian groups. The smallest failure among non-abelian groups occurs in dihedral group:D8.
cyclic group:Z4 and Klein four-group Yes all groups congruence condition on number of subgroups of given prime power order

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:

Collection Does it satisfy a universal congruence condition? Restricted class of groups in which it satisfies a congruence condition Explanation
elementary abelian group:E8 only No abelian groups congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group explains why it is true in abelian groups. The smallest failure among non-abelian groups occurs in direct product of D8 and Z2
direct product of Z4 and Z2, elementary abelian group:E8 Yes all groups congruence condition on number of abelian subgroups of order eight and exponent dividing four
cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8 (i.e., all the abelian groups of order 8) Yes all groups congruence condition on number of abelian subgroups of prime-cube order
cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8 Yes all groups congruence condition on number of subgroups of given prime power order