← #31762376
Subgroups — #31762376
Order 17152.
2026 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16864213 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22573360 | C2 | 1 | 8 | yes | maximal, complemented |
| 3 | #22573356 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #22573343 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#16863836 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #22573312 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C67×#3584 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#14524090 | — | 2 | 1 | no | — |
| 9 | #16864213 | — | 2 | 4 | no | — |
| 10 | #16864213 | C22 | 1 | 8 | yes | complemented |
| 11 | C2×#14524076 | — | 2 | 3 | no | — |
| 12 | C2×#14524065 | C22 | 1 | 3 | yes | complemented |
| 13 | C22×Dic268 | C22 | 1 | 3 | yes | complemented |
| 14 | #16863837 | C22 | 1 | 12 | yes | complemented |
| 15 | #16863836 | C22 | 1 | 12 | yes | complemented |
| 16 | #16863836 | C22 | 1 | 24 | yes | complemented |
| 17 | #16863832 | C22 | 1 | 12 | yes | complemented |
| 18 | #16863831 | C22 | 1 | 4 | yes | complemented |
| 19 | #16863826 | C22 | 1 | 24 | yes | complemented |
| 20 | C2×#14523989 | C22 | 1 | 12 | yes | complemented |
| 21 | C2×#14523987 | C22 | 1 | 12 | yes | complemented |
| 22 | #16863813 | C22 | 1 | 8 | yes | complemented |
| 23 | C2×#14523983 | C22 | 1 | 6 | yes | complemented |
| 24 | #585×C67 | — | 2 | 1 | no | — |
| 25 | #578×C67 | C22 | 1 | 4 | yes | complemented |
| 26 | #577×C67 | — | 2 | 4 | no | — |
| 27 | #576×C67 | C22 | 1 | 4 | yes | complemented |
| 28 | #575×C67 | — | 2 | 3 | no | — |
| 29 | C2×#135×C67 | C22 | 1 | 6 | yes | — |
| 30 | C2×#131×C67 | C22 | 1 | 1 | yes | characteristic |
| 31 | #14524090 | — | 2 | 8 | no | — |
| 32 | #14524090 | — | 4 | 4 | no | — |
| 33 | C2×#11761628 | — | 2 | 3 | no | — |
| 34 | C2×#11761627 | — | 4 | 3 | no | — |
| 35 | C22×Dic134 | — | 2 | 3 | no | — |
| 36 | C2×Q8×D67 | — | 4 | 1 | no | — |
| 37 | #14524076 | — | 2 | 24 | no | — |
| 38 | #14524065 | — | 2 | 6 | no | — |
| 39 | #14524065 | C23 | 1 | 12 | yes | — |
| 40 | C2×Dic268 | — | 2 | 3 | no | — |
| 41 | C2×Dic268 | — | 2 | 6 | no | — |
| 42 | C2×Dic268 | C23 | 1 | 12 | yes | — |
| 43 | C2×Dic268 | C23 | 1 | 3 | yes | — |
| 44 | C2×#11761617 | — | 2 | 3 | no | — |
| 45 | C2×D536 | C23 | 1 | 1 | yes | characteristic |
| 46 | #14523989 | C23 | 1 | 24 | yes | — |
| 47 | #14523987 | C23 | 1 | 24 | yes | — |
| 48 | #14523986 | C23 | 1 | 8 | yes | — |
| 49 | #14523984 | C23 | 1 | 8 | yes | — |
| 50 | #14523983 | C23 | 1 | 4 | yes | — |
| 51 | #14523983 | C23 | 1 | 12 | yes | — |
| 52 | C2×C67⋊C16 | C23 | 1 | 12 | yes | — |
| 53 | 2−1+4×C67 | — | 2 | 4 | no | — |
| 54 | 2−1+4×C67 | — | 4 | 1 | no | — |
| 55 | H(5,𝔽2)×C67 | — | 2 | 4 | no | — |
| 56 | H(5,𝔽2)×C67 | — | 4 | 3 | no | — |
| 57 | C2×#41×C67 | — | 2 | 6 | no | — |
| 58 | C2×#41×C67 | — | 4 | 4 | no | — |
| 59 | C2×#41×C67 | D4 | 1 | 1 | yes | characteristic |
| 60 | #137×C67 | — | 2 | 12 | no | — |
| 61 | AGL(1,ℤ/8)×C67 | — | 2 | 12 | no | — |
| 62 | #135×C67 | — | 2 | 6 | no | — |
| 63 | #135×C67 | — | 2 | 6 | no | — |
| 64 | #135×C67 | C23 | 1 | 12 | yes | — |
| 65 | #135×C67 | C23 | 1 | 4 | yes | — |
| 66 | C2×Q16×C67 | C23 | 1 | 6 | yes | — |
| 67 | C2×SD16×C67 | — | 2 | 6 | no | — |
| 68 | C2×D8×C67 | C23 | 1 | 6 | yes | — |
| 69 | #131×C67 | D4 | 1 | 4 | yes | — |
| 70 | #131×C67 | C23 | 1 | 4 | yes | — |
| 71 | C2×M16×C67 | D4 | 1 | 3 | yes | — |
| 72 | C22×C8×C67 | C23 | 1 | 3 | yes | — |
| 73 | #11761628 | — | 2 | 12 | no | — |
| 74 | #11761628 | — | 4 | 6 | no | — |
| 75 | #11761627 | — | 4 | 24 | no | — |
| 76 | C2×Dic134 | — | 2 | 3 | no | — |
| 77 | C2×Dic134 | — | 2 | 12 | no | — |
| 78 | C2×Dic134 | — | 4 | 3 | no | — |
| 79 | C2×Dic134 | — | 4 | 6 | no | — |
| 80 | Q8×D67 | — | 4 | 8 | no | — |
| 81 | C2×D268 | — | 2 | 1 | no | — |
| 82 | C2×C22⋊D67 | — | 4 | 3 | no | — |
| 83 | C22×Dic67 | — | 4 | 3 | no | — |
| 84 | C2×C4×D67 | — | 4 | 3 | no | — |
| 85 | Dic268 | — | 2 | 3 | no | — |
| 86 | Dic268 | — | 2 | 12 | no | — |
| 87 | Dic268 | C24 | 1 | 6 | yes | — |
| 88 | #11761617 | — | 2 | 12 | no | — |
| 89 | D536 | — | 2 | 1 | no | — |
| 90 | D536 | C24 | 1 | 2 | yes | — |
| 91 | C67⋊C16 | C24 | 1 | 8 | yes | — |
| 92 | #41×C67 | — | 2 | 4 | no | — |
| 93 | #41×C67 | — | 2 | 12 | no | — |
| 94 | #41×C67 | — | 4 | 4 | no | — |
| 95 | #41×C67 | — | 4 | 12 | no | — |
| 96 | #41×C67 | — | 4 | 6 | no | — |
| 97 | #41×C67 | — | 4 | 6 | no | — |
| 98 | #41×C67 | D8 | 1 | 4 | yes | — |
| 99 | #41×C67 | C2×D4 | 1 | 4 | yes | — |
| 100 | C2×Q8×C67 | — | 2 | 6 | no | — |
| 101 | C2×Q8×C67 | — | 4 | 4 | no | — |
| 102 | C2×Q8×C67 | D8 | 1 | 1 | yes | characteristic |
| 103 | C2×D4×C67 | — | 2 | 6 | no | — |
| 104 | C2×D4×C67 | — | 4 | 6 | no | — |
| 105 | C2×D4×C67 | — | 4 | 12 | no | — |
| 106 | C2×D4×C67 | D8 | 1 | 3 | yes | — |
| 107 | C22×C4×C67 | — | 4 | 6 | no | — |
| 108 | C22×C4×C67 | C2×D4 | 1 | 3 | yes | — |
| 109 | Q16×C67 | — | 2 | 6 | no | — |
| 110 | Q16×C67 | C24 | 1 | 4 | yes | — |
| 111 | SD16×C67 | — | 2 | 4 | no | — |
| 112 | SD16×C67 | — | 2 | 12 | no | — |
| 113 | D8×C67 | — | 2 | 6 | no | — |
| 114 | D8×C67 | C24 | 1 | 4 | yes | — |
| 115 | M16×C67 | C2×D4 | 1 | 12 | yes | — |
| 116 | C2×C8×C67 | C2×D4 | 1 | 3 | yes | — |
| 117 | C2×C8×C67 | C2×D4 | 1 | 6 | yes | — |
| 118 | C2×C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 119 | C2×C8×C67 | C24 | 1 | 6 | yes | — |
| 120 | C23×C67 | — | 4 | 6 | no | — |
| 121 | C23×C67 | C2×D8 | 1 | 3 | yes | — |
| 122 | C22×D67 | — | 4 | 1 | no | — |
| 123 | Q8×C67 | — | 2 | 4 | no | — |
| 124 | Q8×C67 | — | 4 | 6 | no | — |
| 125 | Q8×C67 | C2×D8 | 1 | 4 | yes | — |
| 126 | D4×C67 | — | 2 | 4 | no | — |
| 127 | D4×C67 | — | 4 | 4 | no | — |
| 128 | D4×C67 | — | 4 | 12 | no | — |
| 129 | D4×C67 | — | 4 | 6 | no | — |
| 130 | D4×C67 | C2×D8 | 1 | 12 | yes | — |
| 131 | C2×C4×C67 | — | 4 | 6 | no | — |
| 132 | C2×C4×C67 | — | 4 | 4 | no | — |
| 133 | C2×C4×C67 | — | 4 | 12 | no | — |
| 134 | C2×C4×C67 | C2×D8 | 1 | 6 | yes | — |
| 135 | C2×C4×C67 | C2×D8 | 1 | 3 | yes | — |
| 136 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C67 | C22×D4 | 1 | 6 | yes | — |
| 138 | C22⋊D67 | — | 4 | 12 | no | — |
| 139 | C2×Dic67 | — | 4 | 3 | no | — |
| 140 | C2×Dic67 | — | 4 | 12 | no | — |
| 141 | D268 | — | 2 | 2 | no | — |
| 142 | D268 | — | 4 | 1 | no | — |
| 143 | C4×D67 | — | 4 | 12 | no | — |
| 144 | Dic134 | — | 2 | 6 | no | — |
| 145 | Dic134 | — | 4 | 12 | no | — |
| 146 | Dic134 | — | 4 | 3 | no | — |
| 147 | C8×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 148 | C8×C67 | C22×D4 | 1 | 6 | yes | — |
| 149 | C8×C67 | C25 | 1 | 1 | yes | characteristic |
| 150 | C22×C67 | — | 4 | 3 | no | — |
| 151 | C22×C67 | — | 4 | 4 | no | — |
| 152 | C22×C67 | — | 16 | 6 | no | — |
| 153 | C22×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 154 | C22×C67 | C22×D8 | 1 | 6 | yes | — |
| 155 | C2×D67 | — | 4 | 2 | no | — |
| 156 | C2×D67 | — | 8 | 2 | no | — |
| 157 | C4×C67 | — | 4 | 4 | no | — |
| 158 | C4×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 159 | C4×C67 | C22×D8 | 1 | 6 | yes | — |
| 160 | C4×C67 | C23×D4 | 1 | 1 | yes | characteristic |
| 161 | Dic67 | — | 4 | 6 | no | — |
| 162 | #62637 | — | 67 | 1 | no | maximal, Sylow |
| 163 | C2×C67 | — | 2 | 1 | no | — |
| 164 | C2×C67 | — | 2 | 6 | no | — |
| 165 | C2×C67 | — | 8 | 4 | no | — |
| 166 | C2×C67 | C23×D8 | 1 | 1 | yes | characteristic |
| 167 | D67 | — | 8 | 2 | no | — |
| 168 | #3584 | D67 | 1 | 1 | yes | characteristic, complemented |
| 169 | C2×#578 | — | 67 | 1 | no | — |
| 170 | #3417 | — | 67 | 8 | no | — |
| 171 | #3416 | — | 67 | 8 | no | — |
| 172 | #3414 | — | 67 | 6 | no | — |
| 173 | C2×#510 | — | 67 | 6 | no | — |
| 174 | #3407 | — | 67 | 1 | no | — |
| 175 | C67 | #62637 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 176 | #585 | — | 2 | 1 | no | — |
| 177 | C2×2−1+4 | — | 134 | 1 | no | — |
| 178 | #578 | — | 67 | 8 | no | — |
| 179 | #578 | — | 134 | 4 | no | — |
| 180 | #578 | C2×D67 | 1 | 4 | yes | complemented |
| 181 | #577 | — | 2 | 4 | no | — |
| 182 | #576 | C2×D67 | 1 | 4 | yes | complemented |
| 183 | #575 | — | 2 | 3 | no | — |
| 184 | C2×#137 | — | 134 | 3 | no | — |
| 185 | C2×#135 | — | 67 | 3 | no | — |
| 186 | C2×#135 | C2×D67 | 1 | 6 | yes | — |
| 187 | C22×Q16 | — | 67 | 3 | no | — |
| 188 | C2×#131 | C2×D67 | 1 | 1 | yes | characteristic |
| 189 | #510 | — | 67 | 12 | no | — |
| 190 | #510 | — | 67 | 24 | no | — |
| 191 | #510 | — | 67 | 12 | no | — |
| 192 | #509 | — | 67 | 4 | no | — |
| 193 | #509 | — | 67 | 12 | no | — |
| 194 | #508 | — | 67 | 24 | no | — |
| 195 | C2×Q32 | — | 67 | 12 | no | — |
| 196 | C2×SD32 | — | 67 | 12 | no | — |
| 197 | #504 | — | 67 | 8 | no | — |
| 198 | C2×M32 | — | 67 | 6 | no | — |
| 199 | 2−1+4 | — | 2 | 4 | no | — |
| 200 | 2−1+4 | — | 4 | 1 | no | — |
| 201 | 2−1+4 | — | 268 | 4 | no | — |
| 202 | 2−1+4 | — | 134 | 8 | no | — |
| 203 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 204 | H(5,𝔽2) | — | 4 | 3 | no | — |
| 205 | C2×#41 | — | 2 | 6 | no | — |
| 206 | C2×#41 | — | 4 | 4 | no | — |
| 207 | C2×#41 | — | 268 | 3 | no | — |
| 208 | C2×#41 | — | 134 | 3 | no | — |
| 209 | C2×#41 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 210 | C22×Q8 | — | 268 | 1 | no | — |
| 211 | C22×Q8 | — | 134 | 3 | no | — |
| 212 | #137 | — | 2 | 12 | no | — |
| 213 | #137 | — | 134 | 24 | no | — |
| 214 | AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 215 | #135 | — | 2 | 6 | no | — |
| 216 | #135 | — | 2 | 6 | no | — |
| 217 | #135 | — | 67 | 12 | no | — |
| 218 | #135 | — | 134 | 6 | no | — |
| 219 | #135 | C22×D67 | 1 | 4 | yes | — |
| 220 | #135 | C22×D67 | 1 | 12 | yes | — |
| 221 | C2×Q16 | — | 67 | 3 | no | — |
| 222 | C2×Q16 | — | 67 | 12 | no | — |
| 223 | C2×Q16 | — | 134 | 3 | no | — |
| 224 | C2×Q16 | — | 134 | 6 | no | — |
| 225 | C2×Q16 | C22×D67 | 1 | 6 | yes | — |
| 226 | C2×SD16 | — | 2 | 6 | no | — |
| 227 | C2×SD16 | — | 134 | 3 | no | — |
| 228 | C2×D8 | — | 67 | 1 | no | — |
| 229 | C2×D8 | C22×D67 | 1 | 6 | yes | — |
| 230 | #131 | C22⋊D67 | 1 | 4 | yes | — |
| 231 | #131 | C22×D67 | 1 | 4 | yes | — |
| 232 | C2×M16 | C22⋊D67 | 1 | 3 | yes | — |
| 233 | C22×C8 | C22×D67 | 1 | 3 | yes | — |
| 234 | Q32 | — | 67 | 24 | no | — |
| 235 | SD32 | — | 67 | 8 | no | — |
| 236 | SD32 | — | 67 | 24 | no | — |
| 237 | D16 | — | 67 | 8 | no | — |
| 238 | M32 | — | 67 | 4 | no | — |
| 239 | M32 | — | 67 | 12 | no | — |
| 240 | C2×C16 | — | 67 | 12 | no | — |
| 241 | #41 | — | 2 | 4 | no | — |
| 242 | #41 | — | 2 | 12 | no | — |
| 243 | #41 | — | 4 | 4 | no | — |
| 244 | #41 | — | 4 | 12 | no | — |
| 245 | #41 | — | 4 | 6 | no | — |
| 246 | #41 | — | 4 | 6 | no | — |
| 247 | #41 | — | 268 | 24 | no | — |
| 248 | #41 | — | 268 | 6 | no | — |
| 249 | #41 | — | 134 | 12 | no | — |
| 250 | #41 | #11761613 | 1 | 4 | yes | — |
| 251 | #41 | C2×C22⋊D67 | 1 | 4 | yes | — |
| 252 | C2×Q8 | — | 2 | 6 | no | — |
| 253 | C2×Q8 | — | 4 | 4 | no | — |
| 254 | C2×Q8 | — | 268 | 8 | no | — |
| 255 | C2×Q8 | — | 268 | 3 | no | — |
| 256 | C2×Q8 | — | 268 | 6 | no | — |
| 257 | C2×Q8 | — | 134 | 3 | no | — |
| 258 | C2×Q8 | — | 134 | 12 | no | — |
| 259 | C2×Q8 | #11761613 | 1 | 1 | yes | characteristic |
| 260 | C2×D4 | — | 2 | 6 | no | — |
| 261 | C2×D4 | — | 4 | 6 | no | — |
| 262 | C2×D4 | — | 4 | 12 | no | — |
| 263 | C2×D4 | — | 268 | 3 | no | — |
| 264 | C2×D4 | — | 134 | 1 | no | — |
| 265 | C2×D4 | #11761613 | 1 | 3 | yes | — |
| 266 | C22×C4 | — | 4 | 6 | no | — |
| 267 | C22×C4 | — | 268 | 3 | no | — |
| 268 | C22×C4 | — | 268 | 3 | no | — |
| 269 | C22×C4 | C2×C22⋊D67 | 1 | 3 | yes | — |
| 270 | Q16 | — | 2 | 6 | no | — |
| 271 | Q16 | — | 67 | 6 | no | — |
| 272 | Q16 | — | 134 | 3 | no | — |
| 273 | Q16 | — | 134 | 12 | no | — |
| 274 | Q16 | C23×D67 | 1 | 4 | yes | — |
| 275 | SD16 | — | 2 | 4 | no | — |
| 276 | SD16 | — | 2 | 12 | no | — |
| 277 | SD16 | — | 134 | 12 | no | — |
| 278 | D8 | — | 2 | 6 | no | — |
| 279 | D8 | — | 67 | 2 | no | — |
| 280 | D8 | — | 134 | 1 | no | — |
| 281 | D8 | C23×D67 | 1 | 4 | yes | — |
| 282 | M16 | C2×C22⋊D67 | 1 | 12 | yes | — |
| 283 | C2×C8 | C2×C22⋊D67 | 1 | 3 | yes | — |
| 284 | C2×C8 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 285 | C2×C8 | C23×D67 | 1 | 6 | yes | — |
| 286 | C2×C8 | C23×D67 | 1 | 1 | yes | characteristic |
| 287 | C16 | — | 67 | 8 | no | — |
| 288 | C23 | — | 4 | 6 | no | — |
| 289 | C23 | — | 268 | 1 | no | — |
| 290 | C23 | C2×#11761613 | 1 | 3 | yes | — |
| 291 | Q8 | — | 2 | 4 | no | — |
| 292 | Q8 | — | 4 | 6 | no | — |
| 293 | Q8 | — | 268 | 3 | no | — |
| 294 | Q8 | — | 268 | 12 | no | — |
| 295 | Q8 | — | 134 | 6 | no | — |
| 296 | Q8 | C2×#11761613 | 1 | 4 | yes | — |
| 297 | D4 | — | 2 | 4 | no | — |
| 298 | D4 | — | 4 | 12 | no | — |
| 299 | D4 | — | 4 | 4 | no | — |
| 300 | D4 | — | 4 | 6 | no | — |
| 301 | D4 | — | 268 | 12 | no | — |
| 302 | D4 | — | 268 | 1 | no | — |
| 303 | D4 | — | 134 | 2 | no | — |
| 304 | D4 | C2×#11761613 | 1 | 12 | yes | — |
| 305 | C2×C4 | — | 4 | 6 | no | — |
| 306 | C2×C4 | — | 4 | 4 | no | — |
| 307 | C2×C4 | — | 4 | 12 | no | — |
| 308 | C2×C4 | — | 268 | 3 | no | — |
| 309 | C2×C4 | — | 268 | 12 | no | — |
| 310 | C2×C4 | — | 268 | 12 | no | — |
| 311 | C2×C4 | C2×#11761613 | 1 | 6 | yes | — |
| 312 | C2×C4 | C2×#11761613 | 1 | 3 | yes | — |
| 313 | C2×C4 | C22×C22⋊D67 | 1 | 1 | yes | characteristic |
| 314 | C2×C4 | C22×C22⋊D67 | 1 | 6 | yes | — |
| 315 | C8 | C22×C22⋊D67 | 1 | 1 | yes | characteristic |
| 316 | C8 | C22×C22⋊D67 | 1 | 6 | yes | — |
| 317 | C8 | C24×D67 | 1 | 1 | yes | characteristic |
| 318 | C22 | — | 4 | 3 | no | — |
| 319 | C22 | — | 4 | 4 | no | — |
| 320 | C22 | — | 268 | 2 | no | — |
| 321 | C22 | — | 16 | 6 | no | — |
| 322 | C22 | — | 536 | 2 | no | — |
| 323 | C22 | C22×#11761613 | 1 | 6 | yes | — |
| 324 | C22 | C22×#11761613 | 1 | 1 | yes | characteristic |
| 325 | C4 | — | 4 | 4 | no | — |
| 326 | C4 | — | 268 | 6 | no | — |
| 327 | C4 | C22×#11761613 | 1 | 1 | yes | characteristic |
| 328 | C4 | C22×#11761613 | 1 | 6 | yes | — |
| 329 | C4 | C23×C22⋊D67 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | — | 2 | 6 | no | — |
| 332 | C2 | — | 8 | 4 | no | — |
| 333 | C2 | — | 536 | 2 | no | — |
| 334 | C2 | C23×#11761613 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta