There are X spiders, Y dragonflies and (18-x-y) cicadas.

From the total logarithm of wings, we can get:

2y+( 18-x-y)=20

y=x+2

Judging from the total number of legs:

8x+6y+6( 18-x-y)= 1 18

therefore

8x+6(x+2)+6( 18-x-x-2)= 1 18

14x+ 12+96- 12x = 1 18

2x= 10

x=5

So y=x+2=7,

18-x-y=6

So there are five spiders, seven dragonflies and six cicadas.