The rate of subcores in cosmic-ray air showers has been measured near sea level with a close-packed 35 m2 array of spark chambers. A large transition effect was observed from wood beams in the original roof. After the first year, the spark chambers were affixed directly to a new styrofoam sandwich roof of thickness 20 cm (2.2 g cm-2) in order to minimise transition effects. The measured rate is 0.038+or-0.01 per shower of size N>or=105 particles for subcores of size n25>or=50, where n25 is the net number of subcore particles within a circle of radius 25 cm. The dependence on subscore size goes approximately as n25-3.4. The higher rates of subcores reported in other experiments are probably due to transition effects. If pt is calculated by assuming subcore production by pi 0's, the authors find no cases of p1>5 GeV/c. The problem of abstracting realistic information from the subcore data in order to test theories of high-energy interactions at 1015 eV is treated. Simulation results that predict significant improvement in data acquisition at higher altitudes are presented.