Classification of groups of order eight times a prime

We split the classification of groups of order $8p$, $p$ prime into five cases. Some of these cases themselves have their own sub-cases, which you can see on their pages.

• Classification of groups of order 16 ($p=2$, which is also classified as a group of prime-fourth order.)
• Classification of groups of order eight times a prime congruent to 1 modulo 8
• Classification of groups of order eight times a prime congruent to 3 modulo 8
• Classification of groups of order eight times a prime congruent to 5 modulo 8
• Classification of groups of order eight times a prime congruent to 7 modulo 8

Summary of results

Number of groups up to isomorphism

• In the case $p=2$, that is, the groups of order 16, there are 14 groups.
• In the case $p=3$, that is, the groups of order 24, there are 15 groups.
• In the case $p=7$, that is, the groups of order 56, there are 13 groups.

All other cases depend only on the value of the prime modulo $8$, and the value is constant on the residue classes:

• When $p\equiv 1mod8$, there are 15 groups.
• When $p\equiv 3,7mod8$ (aside from the cases $p=3,7$ dealt with above), there are 12 groups.
• When $p\equiv 5mod8$, there are 14 groups.

List of groups

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See also

• Classification of groups of order a power of two times a prime