Cp × Cp × Cp. There are also two non-abelian groups. For p ≠ 2, one is a semi-direct product of Cp × Cp with Cp, and the other is a semi-direct product of Cp2 with Cp. The first one can be described in other terms as group UT(3,p) of unitriangular matrices over finite field with p elements, also called the Heisenberg group mod p. For p = 2, both the semi-direct products mentioned above are isomorphic to the dihedral group Dih4 of order 8. The other non-abelian group of order 8 is the quaternion group Q8. Prevalence Among groups The number of isomorphism classes of groups of order pn grows as , and these are dominated by the classes that are two-step nilpotent. Because of this rapid growth, there is a folklore conjecture asserting that almost all finite groups are 2-groups: the fraction of isomorphism classes of 2-groups among isomorphism classes of groups of order at most n is thought to tend to 1 as n tends to infinity. For instance, of the 49 910 529 484 different groups of order at most 2000, 49 487 365 422, or just over 99%, are 2-groups of order 1024. Within a group Every finite group whose order is divisible by p contains a subgroup which is a non-trivial p-group, namely a cyclic group of order p generated by an element of order p obtained from Cauchy's theorem. In fact, it contains a p-group of maximal possible order: if where p does not divide m, then G has a subgroup P of order called a Sylow p-subgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and any p-subgroup of G is contained in a Sylow p-subgroup. This and other properties are proved in the Sylow theorems. Application to structure of a group p-groups are fundamental tools in understanding the structure of groups and in the classification of finite simple groups. p-groups arise both as subgroups and as quotient groups. As subgroups, for a given prime p one has the Sylow p-subgroups P (largest p-subgroup not unique but all conjugate) and the p-core (the unique largest normal p-subgroup), and various others. As quotients, the largest p-group quotient is the quotient of G by the p-residual subgroup These groups are related (for different primes), possess important properties such as the focal subgroup theorem, and allow one to determine