PSL(3,4): |P| = 20160, k = 21, |Q| = 960, |phibar(Q)| = 60, |Q_t| by o(t): {2: [64], 3: [48], 5: [80]}, M = 80, |P|/M = 252
  [lattice] processed 1/8 classes, closures 7, 0s
  [lattice] processed 11/83 classes, closures 4086, 10s
  [lattice] processed 21/87 classes, closures 5898, 16s
  [lattice] processed 31/88 classes, closures 7992, 23s
  [lattice] processed 41/90 classes, closures 10041, 29s
  [lattice] processed 51/90 classes, closures 12667, 36s
  [lattice] processed 61/92 classes, closures 14186, 40s
  [lattice] processed 71/92 classes, closures 15415, 44s
  [lattice] processed 81/93 classes, closures 16761, 49s
  [lattice] processed 91/95 classes, closures 18303, 53s
  [lattice] processed 95/95 classes, closures 18575, 54s
classes of subgroups: 95; subgroups: 44877; time 54s
normal subgroups (orders): [1, 20160]
order histogram of classes: {1: 1, 2: 1, 3: 1, 4: 10, 5: 1, 6: 1, 7: 1, 8: 9, 9: 1, 10: 1, 12: 7, 16: 15, 18: 1, 21: 1, 24: 6, 32: 5, 36: 3, 48: 5, 60: 7, 64: 1, 72: 1, 80: 2, 96: 2, 160: 2, 168: 3, 192: 1, 360: 3, 960: 2, 20160: 1}

classes with |S| >= M = 80:  (order, conjugates, [(orbit length, |S_d|, induced order on the lines through d, good at d)], good)
   20160      1  [(21, 960, 60, False)]  not good
     960     21  [(20, 48, 12, False), (1, 960, 60, False)]  not good
     960     21  [(5, 192, 12, False), (16, 60, 60, False)]  not good
     360     56  [(15, 24, 6, False), (6, 60, 60, False)]  not good
     360     56  [(15, 24, 6, False), (6, 60, 60, False)]  not good
     360     56  [(6, 60, 60, False), (15, 24, 6, False)]  not good
     192    105  [(16, 12, 12, False), (1, 192, 12, False), (4, 48, 12, False)]  not good
     168    120  [(7, 24, 6, False), (14, 12, 12, False)]  not good
     168    120  [(7, 24, 6, False), (14, 12, 12, False)]  not good
     168    120  [(7, 24, 6, False), (14, 12, 12, False)]  not good
     160    126  [(16, 10, 10, False), (5, 32, 4, True)]  GOOD
     160    126  [(20, 8, 4, True), (1, 160, 10, False)]  GOOD
      96    210  [(3, 32, 4, True), (16, 6, 6, False), (2, 48, 12, False)]  GOOD
      96    210  [(8, 12, 12, False), (12, 8, 4, True), (1, 96, 6, False)]  GOOD
      80    126  [(16, 5, 5, True), (5, 16, 4, True)]  GOOD
      80    126  [(20, 4, 4, True), (1, 80, 5, True)]  GOOD

RESULT PSL(3,4): M = 80; M_G = largest order of a good subgroup over all 95 classes = 160; classes with |S| > M: 14, of which good: 4; good classes of order M: 2; |P|/M = 252, |P|/M_G = 126; strict inequality: True
number of subgroups of order >= M that are good at the base point, by order: {80: 252, 96: 150, 160: 150}
total time 54s
