A direct solver for symmetric sparse matrices from finite element problems is presented. The solver is supposed to work as a local solver of domain decomposition methods for hybrid parallelization on cluster systems of multi-core CPUs, and then it is required to run on shared memory computers and to have an ability of kernel detection. Symmetric pivoting with a given threshold factorizes a matrix with a decomposition introduced by a nested bisection and selects suspicious null pivots from the threshold. The Schur complement constructed from the suspicious null pivots is examined by a factorization with 1x1 and 2x2 pivoting and by a robust kernel detection algorithm based on measurement of residuals with orthogonal projections onto supposed image spaces. A static data structure from the nested bisection and a block sub-structure for Schur complements at all bisection-levels can use level 3 BLAS routines efficiently. Asynchronous task execution for each block can reduce idle time of processors drastically and as a result, the solver has high parallel efficiency. Competitive performance of the developed solver to Intel Pardiso on shared memory computers is shown by numerical experiments.