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Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative |
|---|---|---|---|---|---|---|
| 2P | ||||||
| $C_{16}\times D_4:C_2$ | 1A | $1$ | $1$ | $C_{16}\times D_4:C_2$ | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2A | $2$ | $1$ | $C_{16}\times D_4:C_2$ | 1A | $ \left(\begin{array}{rr} 17 & 16 \\ 0 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2B | $2$ | $1$ | $C_{16}\times D_4:C_2$ | 1A | $ \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2C | $2$ | $1$ | $C_{16}\times D_4:C_2$ | 1A | $ \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2D | $2$ | $2$ | $C_2\times C_4\times C_{16}$ | 1A | $ \left(\begin{array}{rr} 1 & 17 \\ 0 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2E | $2$ | $2$ | $C_2\times C_4\times C_{16}$ | 1A | $ \left(\begin{array}{rr} 17 & 0 \\ 16 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2F | $2$ | $2$ | $C_2\times C_4\times C_{16}$ | 1A | $ \left(\begin{array}{rr} 1 & 1 \\ 0 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2G | $2$ | $2$ | $C_2\times C_4\times C_{16}$ | 1A | $ \left(\begin{array}{rr} 17 & 16 \\ 16 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2H | $2$ | $2$ | $C_2\times C_4\times C_{16}$ | 1A | $ \left(\begin{array}{rr} 7 & 31 \\ 16 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 2I | $2$ | $2$ | $C_2\times C_4\times C_{16}$ | 1A | $ \left(\begin{array}{rr} 7 & 15 \\ 16 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4A1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2A | $ \left(\begin{array}{rr} 23 & 24 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4A-1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2A | $ \left(\begin{array}{rr} 7 & 8 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4B1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2A | $ \left(\begin{array}{rr} 23 & 8 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4B-1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2A | $ \left(\begin{array}{rr} 7 & 24 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4C1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2B | $ \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4C-1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2B | $ \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4D1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2B | $ \left(\begin{array}{rr} 23 & 16 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4D-1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2B | $ \left(\begin{array}{rr} 7 & 16 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4E1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2C | $ \left(\begin{array}{rr} 1 & 8 \\ 0 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4E-1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2C | $ \left(\begin{array}{rr} 1 & 24 \\ 0 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4F1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2C | $ \left(\begin{array}{rr} 17 & 24 \\ 0 & 1 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4F-1 | $4$ | $1$ | $C_{16}\times D_4:C_2$ | 2C | $ \left(\begin{array}{rr} 17 & 8 \\ 0 & 1 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4G | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2A | $ \left(\begin{array}{rr} 23 & 15 \\ 0 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4H | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2A | $ \left(\begin{array}{rr} 7 & 24 \\ 16 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4I | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2A | $ \left(\begin{array}{rr} 17 & 1 \\ 16 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4J | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2A | $ \left(\begin{array}{rr} 23 & 31 \\ 0 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4K | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2A | $ \left(\begin{array}{rr} 7 & 8 \\ 16 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4L | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2A | $ \left(\begin{array}{rr} 17 & 17 \\ 16 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4M1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2C | $ \left(\begin{array}{rr} 17 & 9 \\ 0 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4M-1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2C | $ \left(\begin{array}{rr} 17 & 25 \\ 0 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4N1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2C | $ \left(\begin{array}{rr} 1 & 24 \\ 16 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4N-1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2C | $ \left(\begin{array}{rr} 1 & 8 \\ 16 & 17 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4O1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2B | $ \left(\begin{array}{rr} 7 & 23 \\ 0 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4O-1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2B | $ \left(\begin{array}{rr} 7 & 7 \\ 0 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4P1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2B | $ \left(\begin{array}{rr} 23 & 0 \\ 16 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4P-1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2B | $ \left(\begin{array}{rr} 23 & 16 \\ 16 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4Q1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2B | $ \left(\begin{array}{rr} 1 & 25 \\ 16 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4Q-1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2B | $ \left(\begin{array}{rr} 1 & 9 \\ 16 & 31 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4R1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2C | $ \left(\begin{array}{rr} 23 & 7 \\ 16 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 4R-1 | $4$ | $2$ | $C_2\times C_4\times C_{16}$ | 2C | $ \left(\begin{array}{rr} 23 & 23 \\ 16 & 25 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8A1 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E1 | $ \left(\begin{array}{rr} 31 & 12 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8A-1 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E-1 | $ \left(\begin{array}{rr} 31 & 20 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8A3 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E-1 | $ \left(\begin{array}{rr} 31 & 4 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8A-3 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E1 | $ \left(\begin{array}{rr} 31 & 28 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8B1 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E1 | $ \left(\begin{array}{rr} 15 & 28 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8B-1 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E-1 | $ \left(\begin{array}{rr} 15 & 4 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8B3 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E-1 | $ \left(\begin{array}{rr} 15 & 20 \\ 0 & 7 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8B-3 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4E1 | $ \left(\begin{array}{rr} 15 & 12 \\ 0 & 23 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8C1 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4F1 | $ \left(\begin{array}{rr} 25 & 12 \\ 0 & 1 \end{array}\right) $ |
| $C_{16}\times D_4:C_2$ | 8C-1 | $8$ | $1$ | $C_{16}\times D_4:C_2$ | 4F-1 | $ \left(\begin{array}{rr} 9 & 20 \\ 0 & 1 \end{array}\right) $ |