  Symmetries of three-dimensional periodic scalar fields are described by 230
space groups (SGs). Symmetries of three-dimensional periodic (pseudo-) vector
fields, however, are described by the spin-space groups (SSGs), which were
initially used to describe the symmetries of magnetic orders. In SSGs, the
real-space and spin degrees of freedom are unlocked in the sense that an
operation could have different spacial and spin rotations. SSGs gives a
complete symmetry description of magnetic structures, and have natural
applications in the band theory of itinerary electrons in magnetically ordered
systems with weak spin-orbit coupling.\textit{Altermagnetism}, a concept raised
recently that belongs to the symmetry-compensated collinear magnetic orders but
has non-relativistic spin splitting, is well described by SSGs. Due to the vast
number and complicated group structures, SSGs have not yet been systematically
enumerated. In this work, we exhaust SSGs based on the invariant subgroups of
SGs, with spin operations constructed from three-dimensional (3D) real
representations of the quotient groups for the invariant subgroups. For
collinear and coplanar magnetic orders, the spin operations can be reduced into
lower dimensional real representations. As the number of SSGs is infinite, we
only consider SSGs that describe magnetic unit cells up to 12 times crystal
unit cells. We obtain 157,289 non-coplanar, 24,788 coplanar-non-collinear, and
1,421 collinear SSGs. The enumerated SSGs are stored in an online database at
\url{https://cmpdc.iphy.ac.cn/ssg} with a user-friendly interface. We also
develop an algorithm to identify SSG for realistic materials and find SSGs for
1,626 magnetic materials. Our results serve as a solid starting point for
further studies of symmetry and topology in magnetically ordered materials.
