Groups of order 336

Factorization and useful forms
336 has prime factors 2, 3, and 7, and prime factorization:

$$\! 336 = 2^4 \cdot 3^1 \cdot 7^1 = 16 \cdot 3 \cdot 7$$

Other useful expressions for the number are:

$$\! 336 = 7^3 - 7 = 2(2^3 - 1)(2^3 - 2)(2^3 - 2^2)$$

GAP implementation
gap> SmallGroupsInformation(336);

There are 228 groups of order 336. They are sorted by their Frattini factors. 1 has Frattini factor [ 42, 1 ]. 2 has Frattini factor [ 42, 2 ]. 3 has Frattini factor [ 42, 3 ]. 4 has Frattini factor [ 42, 4 ]. 5 has Frattini factor [ 42, 5 ]. 6 has Frattini factor [ 42, 6 ]. 7 - 22 have Frattini factor [ 84, 7 ]. 23 - 47 have Frattini factor [ 84, 8 ]. 48 - 55 have Frattini factor [ 84, 9 ]. 56 has Frattini factor [ 84, 10 ]. 57 has Frattini factor [ 84, 11 ]. 58 - 73 have Frattini factor [ 84, 12 ]. 74 - 89 have Frattini factor [ 84, 13 ]. 90 - 105 have Frattini factor [ 84, 14 ]. 106 - 113 have Frattini factor [ 84, 15 ]. 114 has Frattini factor [ 168, 42 ]. 115 - 117 have Frattini factor [ 168, 45 ]. 118 - 120 have Frattini factor [ 168, 46 ]. 121 - 130 have Frattini factor [ 168, 47 ]. 131 - 133 have Frattini factor [ 168, 48 ]. 134 - 136 have Frattini factor [ 168, 49 ]. 137 - 163 have Frattini factor [ 168, 50 ]. 164 - 167 have Frattini factor [ 168, 51 ]. 168 - 170 have Frattini factor [ 168, 52 ]. 171 - 173 have Frattini factor [ 168, 53 ]. 174 - 183 have Frattini factor [ 168, 54 ]. 184 - 193 have Frattini factor [ 168, 55 ]. 194 - 203 have Frattini factor [ 168, 56 ]. 204 - 207 have Frattini factor [ 168, 57 ]. 208 - 228 have trivial Frattini subgroup.

For the selection functions the values of the following attributes are precomputed and stored: IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, LGLength, FrattinifactorSize and FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library. IdSmallGroup is available for this size.