← #31010130
Subgroups — #31010130
Order 17152.
702 conjugacy classes of subgroups.
The subgroup lattice has 298 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16863836 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×#16863776 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22562259 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #22562254 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #22561965 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22561928 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C4×#135×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C22×Dic268 | — | 2 | 1 | no | — |
| 9 | C22×Dic268 | C22 | 1 | 2 | yes | — |
| 10 | C2×#14524043 | — | 2 | 1 | no | — |
| 11 | #16863836 | — | 2 | 8 | no | — |
| 12 | C2×#14523989 | C22 | 1 | 4 | yes | — |
| 13 | C2×#14523987 | C22 | 1 | 4 | yes | — |
| 14 | C2×#14523983 | C22 | 1 | 2 | yes | — |
| 15 | #16863776 | — | 2 | 2 | no | — |
| 16 | #16863776 | C22 | 1 | 4 | yes | — |
| 17 | C4⋊(C67⋊C16) | C22 | 1 | 4 | yes | — |
| 18 | #16863199 | C22 | 1 | 8 | yes | — |
| 19 | C2×#135×C67 | C22 | 1 | 1 | yes | characteristic |
| 20 | C4×#41×C67 | — | 2 | 1 | no | — |
| 21 | C4×Q16×C67 | C22 | 1 | 2 | yes | — |
| 22 | C4×SD16×C67 | — | 2 | 2 | no | — |
| 23 | C4×D8×C67 | C22 | 1 | 2 | yes | — |
| 24 | #427×C67 | C22 | 1 | 1 | yes | characteristic |
| 25 | #416×C67 | — | 2 | 1 | no | — |
| 26 | C2×C4×C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 27 | C22×Dic134 | — | 2 | 2 | no | — |
| 28 | C2×Dic268 | — | 2 | 4 | no | — |
| 29 | C2×Dic268 | — | 2 | 2 | no | — |
| 30 | C2×Dic268 | — | 2 | 4 | no | — |
| 31 | C2×Dic268 | — | 2 | 4 | no | — |
| 32 | C2×Dic268 | — | 4 | 4 | no | — |
| 33 | C2×Dic268 | C23 | 1 | 4 | yes | — |
| 34 | #14524043 | — | 2 | 4 | no | — |
| 35 | #14524043 | — | 4 | 1 | no | — |
| 36 | #14524043 | — | 4 | 1 | no | — |
| 37 | C2×C4⋊Dic67 | — | 4 | 2 | no | — |
| 38 | #14523989 | — | 2 | 8 | no | — |
| 39 | #14523987 | — | 2 | 8 | no | — |
| 40 | #14523983 | — | 2 | 4 | no | — |
| 41 | C2×C67⋊C16 | C23 | 1 | 4 | yes | — |
| 42 | C2×#41×C67 | — | 2 | 1 | no | — |
| 43 | #135×C67 | — | 2 | 2 | no | — |
| 44 | #135×C67 | D4 | 1 | 4 | yes | — |
| 45 | C2×Q16×C67 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×SD16×C67 | — | 2 | 1 | no | — |
| 47 | C2×D8×C67 | C23 | 1 | 1 | yes | characteristic |
| 48 | C22×C8×C67 | D4 | 1 | 1 | yes | characteristic |
| 49 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 50 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 51 | C4×D4×C67 | — | 2 | 2 | no | — |
| 52 | C4×D4×C67 | — | 4 | 2 | no | — |
| 53 | #117×C67 | — | 2 | 1 | no | — |
| 54 | #117×C67 | — | 4 | 1 | no | — |
| 55 | C2×C42×C67 | — | 4 | 1 | no | — |
| 56 | C2×C42×C67 | D4 | 1 | 1 | yes | characteristic |
| 57 | #107×C67 | C23 | 1 | 2 | yes | — |
| 58 | #106×C67 | — | 2 | 1 | no | — |
| 59 | #103×C67 | — | 2 | 2 | no | — |
| 60 | #102×C67 | — | 2 | 2 | no | — |
| 61 | C4×C8×C67 | D4 | 1 | 2 | yes | — |
| 62 | C4×C8×C67 | C23 | 1 | 2 | yes | — |
| 63 | C2×Dic134 | — | 2 | 4 | no | — |
| 64 | C2×Dic134 | — | 4 | 4 | no | — |
| 65 | C2×Dic134 | — | 4 | 4 | no | — |
| 66 | C2×Dic134 | — | 4 | 2 | no | — |
| 67 | C22×Dic67 | — | 4 | 2 | no | — |
| 68 | Dic268 | — | 2 | 4 | no | — |
| 69 | Dic268 | — | 2 | 4 | no | — |
| 70 | Dic268 | — | 4 | 8 | no | — |
| 71 | Dic268 | — | 4 | 4 | no | — |
| 72 | C4⋊Dic67 | — | 4 | 4 | no | — |
| 73 | C4⋊Dic67 | — | 4 | 4 | no | — |
| 74 | C67⋊C16 | — | 2 | 4 | no | — |
| 75 | #41×C67 | — | 2 | 4 | no | — |
| 76 | #41×C67 | — | 4 | 2 | no | — |
| 77 | C2×Q8×C67 | — | 2 | 1 | no | — |
| 78 | C2×D4×C67 | — | 2 | 1 | no | — |
| 79 | C2×D4×C67 | — | 4 | 1 | no | — |
| 80 | C22×C4×C67 | — | 4 | 1 | no | — |
| 81 | C22×C4×C67 | — | 4 | 2 | no | — |
| 82 | C22×C4×C67 | D8 | 1 | 2 | yes | — |
| 83 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | Q16×C67 | — | 2 | 1 | no | — |
| 85 | Q16×C67 | C2×D4 | 1 | 2 | yes | — |
| 86 | SD16×C67 | — | 2 | 4 | no | — |
| 87 | D8×C67 | — | 2 | 1 | no | — |
| 88 | D8×C67 | C2×D4 | 1 | 2 | yes | — |
| 89 | C2×C8×C67 | — | 2 | 2 | no | — |
| 90 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 91 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 93 | C2×C8×C67 | #41 | 1 | 2 | yes | — |
| 94 | C2×C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 95 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 96 | C4⋊C4×C67 | — | 4 | 2 | no | — |
| 97 | C22⋊C4×C67 | — | 4 | 1 | no | — |
| 98 | C22⋊C4×C67 | — | 4 | 2 | no | — |
| 99 | C42×C67 | — | 4 | 1 | no | — |
| 100 | C42×C67 | — | 4 | 2 | no | — |
| 101 | C42×C67 | D8 | 1 | 1 | yes | characteristic |
| 102 | C42×C67 | D8 | 1 | 1 | yes | characteristic |
| 103 | C42×C67 | C2×D4 | 1 | 2 | yes | — |
| 104 | C23×C67 | — | 4 | 1 | no | — |
| 105 | C23×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 106 | Q8×C67 | — | 2 | 2 | no | — |
| 107 | Q8×C67 | — | 4 | 1 | no | — |
| 108 | D4×C67 | — | 2 | 2 | no | — |
| 109 | D4×C67 | — | 4 | 4 | no | — |
| 110 | D4×C67 | — | 4 | 1 | no | — |
| 111 | C2×C4×C67 | — | 2 | 4 | no | — |
| 112 | C2×C4×C67 | — | 4 | 1 | no | — |
| 113 | C2×C4×C67 | — | 4 | 4 | no | — |
| 114 | C2×C4×C67 | — | 4 | 2 | no | — |
| 115 | C2×C4×C67 | — | 8 | 4 | no | — |
| 116 | C2×C4×C67 | #121 | 1 | 2 | yes | — |
| 117 | C2×C4×C67 | C2×D8 | 1 | 2 | yes | — |
| 118 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C67 | C2×D8 | 1 | 2 | yes | — |
| 120 | C2×C4×C67 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 121 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×Dic67 | — | 4 | 4 | no | — |
| 123 | C2×Dic67 | — | 8 | 4 | no | — |
| 124 | Dic134 | — | 4 | 4 | no | — |
| 125 | Dic134 | — | 4 | 4 | no | — |
| 126 | Dic134 | — | 8 | 4 | no | — |
| 127 | C8×C67 | — | 2 | 2 | no | — |
| 128 | C8×C67 | #121 | 1 | 2 | yes | — |
| 129 | C8×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 130 | C8×C67 | C2×#41 | 1 | 1 | yes | characteristic |
| 131 | C22×C67 | — | 2 | 2 | no | — |
| 132 | C22×C67 | — | 4 | 2 | no | — |
| 133 | C22×C67 | — | 8 | 2 | no | — |
| 134 | C22×C67 | #459 | 1 | 2 | yes | — |
| 135 | C22×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 136 | C4×C67 | — | 2 | 2 | no | — |
| 137 | C4×C67 | — | 4 | 2 | no | — |
| 138 | C4×C67 | — | 8 | 2 | no | — |
| 139 | C4×C67 | #459 | 1 | 2 | yes | — |
| 140 | C4×C67 | #510 | 1 | 4 | yes | complemented |
| 141 | C4×C67 | C2×#121 | 1 | 1 | yes | characteristic |
| 142 | C4×C67 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 143 | Dic67 | — | 8 | 4 | no | — |
| 144 | #33583 | — | 67 | 1 | no | maximal, Sylow |
| 145 | C2×C67 | — | 2 | 2 | no | — |
| 146 | C2×C67 | — | 8 | 2 | no | — |
| 147 | C2×C67 | C2×#459 | 1 | 1 | yes | characteristic |
| 148 | C2×C67 | C2×#510 | 1 | 2 | yes | — |
| 149 | C2×#510 | — | 67 | 2 | no | — |
| 150 | C2×#494 | — | 67 | 1 | no | — |
| 151 | C4×#135 | D67 | 1 | 1 | yes | characteristic |
| 152 | #2209 | — | 67 | 4 | no | — |
| 153 | #2208 | — | 67 | 4 | no | — |
| 154 | #2150 | — | 67 | 1 | no | — |
| 155 | #2141 | — | 67 | 2 | no | — |
| 156 | C67 | #33583 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 157 | C2×#135 | C2×D67 | 1 | 1 | yes | characteristic |
| 158 | C22×Q16 | — | 67 | 2 | no | — |
| 159 | C22×Q16 | — | 134 | 1 | no | — |
| 160 | C2×C4⋊Q8 | — | 134 | 1 | no | — |
| 161 | C4×#41 | — | 2 | 1 | no | — |
| 162 | #510 | — | 134 | 8 | no | — |
| 163 | C2×Q32 | — | 67 | 4 | no | — |
| 164 | C2×SD32 | — | 67 | 4 | no | — |
| 165 | C2×M32 | — | 67 | 2 | no | — |
| 166 | #494 | — | 67 | 4 | no | — |
| 167 | #494 | — | 134 | 2 | no | — |
| 168 | C4×Q16 | C2×D67 | 1 | 2 | yes | — |
| 169 | C4×SD16 | — | 2 | 2 | no | — |
| 170 | C4×D8 | C2×D67 | 1 | 2 | yes | — |
| 171 | #427 | C2×D67 | 1 | 1 | yes | characteristic |
| 172 | #416 | — | 2 | 1 | no | — |
| 173 | C2×C4×C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 174 | C4⋊C16 | — | 67 | 4 | no | — |
| 175 | #358 | — | 67 | 8 | no | — |
| 176 | C2×#41 | — | 2 | 1 | no | — |
| 177 | C22×Q8 | — | 134 | 2 | no | — |
| 178 | #135 | — | 2 | 2 | no | — |
| 179 | #135 | D268 | 1 | 4 | yes | — |
| 180 | C2×Q16 | — | 268 | 4 | no | — |
| 181 | C2×Q16 | — | 67 | 4 | no | — |
| 182 | C2×Q16 | — | 134 | 2 | no | — |
| 183 | C2×Q16 | — | 134 | 4 | no | — |
| 184 | C2×Q16 | — | 134 | 4 | no | — |
| 185 | C2×Q16 | — | 134 | 4 | no | — |
| 186 | C2×Q16 | C22×D67 | 1 | 1 | yes | characteristic |
| 187 | C2×SD16 | — | 2 | 1 | no | — |
| 188 | C2×D8 | C22×D67 | 1 | 1 | yes | characteristic |
| 189 | C22×C8 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 190 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 191 | C4⋊Q8 | — | 268 | 1 | no | — |
| 192 | C4⋊Q8 | — | 268 | 1 | no | — |
| 193 | C4⋊Q8 | — | 134 | 4 | no | — |
| 194 | C4×Q8 | — | 2 | 2 | no | — |
| 195 | C4×D4 | — | 2 | 2 | no | — |
| 196 | C4×D4 | — | 4 | 2 | no | — |
| 197 | #117 | — | 2 | 1 | no | — |
| 198 | #117 | — | 4 | 1 | no | — |
| 199 | C2×C4⋊C4 | — | 268 | 2 | no | — |
| 200 | C2×C42 | — | 4 | 1 | no | — |
| 201 | C2×C42 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 202 | Q32 | — | 134 | 8 | no | — |
| 203 | SD32 | — | 134 | 8 | no | — |
| 204 | M32 | — | 134 | 4 | no | — |
| 205 | C2×C16 | — | 67 | 4 | no | — |
| 206 | #107 | C22×D67 | 1 | 2 | yes | — |
| 207 | #106 | — | 2 | 1 | no | — |
| 208 | #103 | — | 2 | 2 | no | — |
| 209 | #102 | — | 2 | 2 | no | — |
| 210 | C4×C8 | C22⋊D67 | 1 | 2 | yes | — |
| 211 | C4×C8 | C22×D67 | 1 | 2 | yes | — |
| 212 | #41 | — | 2 | 4 | no | — |
| 213 | #41 | — | 4 | 2 | no | — |
| 214 | C2×Q8 | — | 2 | 1 | no | — |
| 215 | C2×Q8 | — | 268 | 2 | no | — |
| 216 | C2×Q8 | — | 268 | 4 | no | — |
| 217 | C2×Q8 | — | 268 | 4 | no | — |
| 218 | C2×Q8 | — | 134 | 4 | no | — |
| 219 | C2×D4 | — | 2 | 1 | no | — |
| 220 | C2×D4 | — | 4 | 1 | no | — |
| 221 | C22×C4 | — | 4 | 1 | no | — |
| 222 | C22×C4 | — | 4 | 2 | no | — |
| 223 | C22×C4 | — | 268 | 2 | no | — |
| 224 | C22×C4 | #11761613 | 1 | 2 | yes | — |
| 225 | C22×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 226 | Q16 | — | 2 | 1 | no | — |
| 227 | Q16 | — | 268 | 8 | no | — |
| 228 | Q16 | — | 268 | 4 | no | — |
| 229 | Q16 | — | 134 | 4 | no | — |
| 230 | Q16 | — | 134 | 4 | no | — |
| 231 | Q16 | C2×D268 | 1 | 2 | yes | — |
| 232 | SD16 | — | 2 | 4 | no | — |
| 233 | D8 | — | 2 | 1 | no | — |
| 234 | D8 | C2×D268 | 1 | 2 | yes | — |
| 235 | C2×C8 | — | 2 | 2 | no | — |
| 236 | C2×C8 | C2×C22⋊D67 | 1 | 2 | yes | — |
| 237 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | C2×D268 | 1 | 2 | yes | — |
| 239 | C2×C8 | #11761628 | 1 | 2 | yes | — |
| 240 | C2×C8 | C23×D67 | 1 | 1 | yes | characteristic |
| 241 | C4⋊C4 | — | 2 | 2 | no | — |
| 242 | C4⋊C4 | — | 4 | 2 | no | — |
| 243 | C4⋊C4 | — | 268 | 4 | no | — |
| 244 | C4⋊C4 | — | 268 | 4 | no | — |
| 245 | C22⋊C4 | — | 4 | 1 | no | — |
| 246 | C22⋊C4 | — | 4 | 2 | no | — |
| 247 | C42 | — | 4 | 1 | no | — |
| 248 | C42 | — | 4 | 2 | no | — |
| 249 | C42 | #11761613 | 1 | 1 | yes | characteristic |
| 250 | C42 | #11761613 | 1 | 1 | yes | characteristic |
| 251 | C42 | C2×C22⋊D67 | 1 | 2 | yes | — |
| 252 | C16 | — | 134 | 4 | no | — |
| 253 | C23 | — | 4 | 1 | no | — |
| 254 | C23 | C2×#11761613 | 1 | 1 | yes | characteristic |
| 255 | Q8 | — | 2 | 2 | no | — |
| 256 | Q8 | — | 4 | 1 | no | — |
| 257 | Q8 | — | 268 | 4 | no | — |
| 258 | Q8 | — | 268 | 4 | no | — |
| 259 | Q8 | — | 536 | 4 | no | — |
| 260 | D4 | — | 2 | 2 | no | — |
| 261 | D4 | — | 4 | 4 | no | — |
| 262 | D4 | — | 4 | 1 | no | — |
| 263 | C2×C4 | — | 2 | 4 | no | — |
| 264 | C2×C4 | — | 4 | 1 | no | — |
| 265 | C2×C4 | — | 4 | 4 | no | — |
| 266 | C2×C4 | — | 4 | 2 | no | — |
| 267 | C2×C4 | — | 8 | 4 | no | — |
| 268 | C2×C4 | — | 268 | 4 | no | — |
| 269 | C2×C4 | — | 536 | 4 | no | — |
| 270 | C2×C4 | #14524017 | 1 | 2 | yes | — |
| 271 | C2×C4 | C2×#11761613 | 1 | 2 | yes | — |
| 272 | C2×C4 | C2×#11761613 | 1 | 2 | yes | — |
| 273 | C2×C4 | C2×#11761613 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | #14524068 | 1 | 2 | yes | — |
| 275 | C2×C4 | C22×C22⋊D67 | 1 | 1 | yes | characteristic |
| 276 | C8 | — | 2 | 2 | no | — |
| 277 | C8 | #14524017 | 1 | 2 | yes | — |
| 278 | C8 | C22×D268 | 1 | 1 | yes | characteristic |
| 279 | C8 | C2×#11761628 | 1 | 1 | yes | characteristic |
| 280 | C22 | — | 2 | 2 | no | — |
| 281 | C22 | — | 4 | 2 | no | — |
| 282 | C22 | — | 8 | 2 | no | — |
| 283 | C22 | #16863591 | 1 | 2 | yes | — |
| 284 | C22 | C22×#11761613 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 2 | no | — |
| 286 | C4 | — | 4 | 2 | no | — |
| 287 | C4 | — | 8 | 2 | no | — |
| 288 | C4 | — | 536 | 4 | no | — |
| 289 | C4 | #16863591 | 1 | 2 | yes | — |
| 290 | C4 | #16863836 | 1 | 4 | yes | complemented |
| 291 | C4 | C2×#14524017 | 1 | 1 | yes | characteristic |
| 292 | C4 | C2×#14524068 | 1 | 1 | yes | characteristic |
| 293 | C2 | — | 2 | 2 | no | — |
| 294 | C2 | — | 8 | 2 | no | — |
| 295 | C2 | C2×#16863591 | 1 | 1 | yes | characteristic |
| 296 | C2 | C2×#16863836 | 1 | 2 | yes | — |
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