  We use the SmallGroups Library to find the finite subgroups of U(3) of order
smaller than 512 which possess a faithful three-dimensional irreducible
representation. From the resulting list of groups we extract those groups that
can not be written as direct products with cyclic groups. These groups are the
basic building blocks for models based on finite subgroups of U(3). All
resulting finite subgroups of SU(3) can be identified using the well known list
of finite subgroups of SU(3) derived by Miller, Blichfeldt and Dickson at the
beginning of the 20th century. Furthermore we prove a theorem which allows to
construct infinite series of finite subgroups of U(3) from a special type of
finite subgroups of U(3). This theorem is used to construct some new series of
finite subgroups of U(3). The first members of these series can be found in the
derived list of finite subgroups of U(3) of order smaller than 512. In the last
part of this work we analyse some interesting finite subgroups of U(3),
especially the group S_4(2)\cong A_4\rtimes Z_4, which is closely related to
the important SU(3)-subgroup S_4.
