Fano threefolds with $\rho=2$
ID  $\mathrm{K}_X^3$  $\mathrm{h}^{1,2}$  index  description  blowups  blowdowns  rational  unirational  moduli  $\mathrm{Aut}^0$  

21  4  22  1 
blowup of 111 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system 
111  no  ?  36  $0$  
22  6  20  1 
double cover of 234 with branch locus a $(2,4)$divisor 
no  yes  33  $0$  
23  8  11  1 
blowup of 112 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system 
112  no  yes  23  $0$  
24  10  10  1 
blowup of 117 in the intersection of two cubics

117  yes  yes  21  $0$  
25  12  6  1 
blowup of 113 in a plane cubic 
113  no  yes  16  $0$  
26  12  9  1 
Verra 3fold

no  yes 

$0$  
27  14  5  1 
blowup of 116 in the intersection of two divisors from $\mathcal{O}_Q(2)$ 
116  yes  yes  14  $0$  
28  14  9  1  no  yes 

$0$  
29  16  5  1 
complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

117  yes  yes  13  $0$  
210  16  3  1 
blowup of 114 in an elliptic curve which is an intersection of 2 hyperplanes 
114  yes  yes  11  $0$  
211  18  5  1 
blowup of 113 in a line 
113  no  yes  12  $0$  
212  20  3  1 
intersection of 3 $(1,1)$divisors in $\mathbb{P}^3\times\mathbb{P}^3$

117  yes  yes  9  $0$  
213  20  2  1 
blowup of 116 in a curve of degree 6 and genus 2 
116  yes  yes  8  $0$  
214  20  1  1 
blowup of 115 in an elliptic curve which is an intersection of 2 hyperplanes 
115  yes  yes  7  $0$  
215  22  4  1  117  yes  yes 

$0$  
216  22  2  1 
blowup of 114 in a conic 
114  yes  yes  7  $0$  
217  24  1  1 
blowup of 116 in an elliptic curve of degree 5 
116, 117  yes  yes  5  $0$  
218  24  2  1 
double cover of 234 with branch locus a divisor of degree $(2,2)$ 
*  yes  yes  6  $0$  
219  26  2  1 
blowup of 114 in a line 
114, 117  yes  yes  5  $0$  
220  26  0  1 
blowup of 115 in a twisted cubic 
115  yes  yes  3 


221  28  0  1 
blowup of 116 in a twisted quartic 
116  yes  yes  2 


222  30  0  1 
blowup of 115 in a conic 
115, 117  yes  yes  1 


223  30  1  1  116  yes  yes 

$0$  
224  30  0  1 
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$ 
*  yes  yes  1 


225  32  1  1 
blowup of 117 in an elliptic curve which is an intersection of 2 quadrics

*  117  yes  yes  1  $0$  
226  34  0  1 
blowup of 115 in a line 
115, 116  yes  yes  0 


227  38  0  1 
blowup of 117 in a twisted cubic 
*  117  yes  yes  0  $\mathrm{PGL}_2$  
228  40  1  1 
blowup of 117 in a plane cubic 
117  yes  yes  1  $\mathbb{G}_{\mathrm{a}}^3\rtimes\mathbb{G}_{\mathrm{m}}$  
229  40  0  1 
blowup of 116 in a conic 
*  116  yes  yes  0  $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$  
230  46  0  1 
blowup of 117 in a conic 
*  117  yes  yes  0  $\mathrm{PSO}_{5;1}$  
231  46  0  1 
blowup of 116 in a line 
*  116  yes  yes  0  $\mathrm{PSO}_{5;2}$  
232  48  0  2 
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

*  yes  yes  0  $\mathrm{PGL}_3$  
233  54  0  1 
blowup of 117 in a line 
*  117  yes  yes  0  $\mathrm{PGL}_{4;2}$  
234  54  0  1 
$\mathbb{P}^1\times\mathbb{P}^2$ 
*  yes  yes  0  $\mathrm{PGL}_2\times\mathrm{PGL}_3$  
235  56  0  2 
$\mathrm{Bl}_p\mathbb{P}^3$

*  yes  yes  0  $\mathrm{PGL}_{4;1}$  
236  62  0  1 
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ 
*  yes  yes  0  $\mathrm{Aut}(\mathbb{P}(1,1,1,2))$ 