← #30772315
Subgroups — #30772315
Order 17152.
384 conjugacy classes of subgroups.
The subgroup lattice has 316 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16863813 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16863216 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22561969 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22560044 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22558297 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22558286 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#350×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16863813 | — | 2 | 2 | no | — |
| 9 | #16863813 | — | 2 | 2 | no | — |
| 10 | C2×#14523983 | — | 2 | 1 | no | — |
| 11 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×C67⋊C16 | — | 2 | 1 | no | — |
| 13 | C22×C67⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#14523970 | C4 | 1 | 1 | yes | characteristic |
| 15 | #16863392 | C4 | 1 | 1 | yes | characteristic |
| 16 | #16863351 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×C8×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 18 | #16863216 | — | 2 | 2 | no | — |
| 19 | Dic67⋊C16 | — | 2 | 1 | no | — |
| 20 | Dic67⋊C16 | — | 2 | 1 | no | — |
| 21 | #16863160 | C4 | 1 | 1 | yes | characteristic |
| 22 | #16863157 | C4 | 1 | 1 | yes | characteristic |
| 23 | C2×#131×C67 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×M32×C67 | C22 | 1 | 1 | yes | characteristic |
| 25 | C22×C16×C67 | C22 | 1 | 1 | yes | characteristic |
| 26 | #350×C67 | — | 2 | 2 | no | — |
| 27 | C2×#11761612 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C22×C67⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #14524002 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C2×C4×Dic67 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | #14523983 | — | 2 | 4 | no | — |
| 32 | #14523983 | — | 2 | 1 | no | — |
| 33 | #14523983 | D4 | 1 | 2 | yes | — |
| 34 | C2×C67⋊C16 | — | 2 | 4 | no | — |
| 35 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 36 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 37 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 38 | C2×C67⋊C16 | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×C67⋊C16 | D4 | 1 | 1 | yes | characteristic |
| 40 | #14523970 | — | 2 | 2 | no | — |
| 41 | #14523967 | C8 | 1 | 1 | yes | characteristic |
| 42 | #14523967 | C8 | 1 | 1 | yes | characteristic |
| 43 | #14523964 | C8 | 1 | 1 | yes | characteristic |
| 44 | #14523964 | C8 | 1 | 1 | yes | characteristic |
| 45 | #14523953 | — | 2 | 1 | no | — |
| 46 | C8×Dic67 | — | 2 | 1 | no | — |
| 47 | C8×Dic67 | — | 2 | 1 | no | — |
| 48 | C8×Dic67 | — | 2 | 1 | no | — |
| 49 | C2×#41×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #131×C67 | — | 2 | 1 | no | — |
| 51 | #131×C67 | — | 2 | 1 | no | — |
| 52 | #131×C67 | D4 | 1 | 2 | yes | — |
| 53 | #131×C67 | Q8 | 1 | 2 | yes | — |
| 54 | C2×M16×C67 | — | 2 | 1 | no | — |
| 55 | C2×M16×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C22×C8×C67 | — | 2 | 1 | no | — |
| 57 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 58 | M32×C67 | — | 2 | 1 | no | — |
| 59 | M32×C67 | — | 2 | 1 | no | — |
| 60 | C2×C16×C67 | — | 2 | 1 | no | — |
| 61 | C2×C16×C67 | — | 2 | 1 | no | — |
| 62 | C2×C16×C67 | — | 2 | 1 | no | — |
| 63 | C2×C16×C67 | — | 2 | 1 | no | — |
| 64 | C22×Dic67 | — | 2 | 1 | no | — |
| 65 | #11761612 | — | 2 | 1 | no | — |
| 66 | #11761612 | — | 2 | 1 | no | — |
| 67 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 68 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 69 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 70 | C2×C67⋊C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 71 | C2×C67⋊C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 72 | C4⋊Dic67 | C2×C8 | 1 | 1 | yes | characteristic |
| 73 | C4⋊Dic67 | C2×C8 | 1 | 1 | yes | characteristic |
| 74 | C22⋊Dic67 | — | 2 | 1 | no | — |
| 75 | C4×Dic67 | — | 2 | 1 | no | — |
| 76 | C4×Dic67 | — | 2 | 1 | no | — |
| 77 | C4×Dic67 | — | 2 | 1 | no | — |
| 78 | C67⋊C16 | — | 2 | 1 | no | — |
| 79 | C67⋊C16 | — | 2 | 2 | no | — |
| 80 | C67⋊C16 | — | 2 | 1 | no | — |
| 81 | #41×C67 | — | 2 | 1 | no | — |
| 82 | #41×C67 | — | 2 | 1 | no | — |
| 83 | #41×C67 | C4⋊C4 | 1 | 2 | yes | — |
| 84 | #41×C67 | M16 | 1 | 2 | yes | — |
| 85 | C2×Q8×C67 | C2×C8 | 1 | 1 | yes | characteristic |
| 86 | C2×D4×C67 | — | 2 | 1 | no | — |
| 87 | C2×D4×C67 | C2×C8 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C67 | — | 2 | 1 | no | — |
| 89 | C22×C4×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 90 | M16×C67 | — | 2 | 1 | no | — |
| 91 | M16×C67 | — | 2 | 2 | no | — |
| 92 | M16×C67 | — | 2 | 2 | no | — |
| 93 | M16×C67 | C4⋊C4 | 1 | 2 | yes | — |
| 94 | C2×C8×C67 | — | 2 | 1 | no | — |
| 95 | C2×C8×C67 | — | 2 | 2 | no | — |
| 96 | C2×C8×C67 | — | 2 | 2 | no | — |
| 97 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C8×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 101 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 102 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 103 | C16×C67 | — | 4 | 1 | no | — |
| 104 | C16×C67 | — | 4 | 1 | no | — |
| 105 | C23×C67 | — | 2 | 1 | no | — |
| 106 | C23×C67 | #131 | 1 | 1 | yes | characteristic |
| 107 | Q8×C67 | — | 2 | 1 | no | — |
| 108 | Q8×C67 | C4⋊C8 | 1 | 2 | yes | — |
| 109 | D4×C67 | — | 2 | 2 | no | — |
| 110 | D4×C67 | — | 2 | 2 | no | — |
| 111 | D4×C67 | — | 2 | 1 | no | — |
| 112 | D4×C67 | C4⋊C8 | 1 | 2 | yes | — |
| 113 | C2×C4×C67 | — | 2 | 1 | no | — |
| 114 | C2×C4×C67 | — | 2 | 2 | no | — |
| 115 | C2×C4×C67 | — | 2 | 2 | no | — |
| 116 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C67 | C22×C8 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C67 | C2×M16 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C67 | #131 | 1 | 1 | yes | characteristic |
| 122 | C2×Dic67 | — | 2 | 2 | no | — |
| 123 | C2×Dic67 | — | 2 | 1 | no | — |
| 124 | C2×Dic67 | — | 2 | 1 | no | — |
| 125 | C2×Dic67 | — | 4 | 1 | no | — |
| 126 | C2×Dic67 | — | 4 | 1 | no | — |
| 127 | C8×C67 | — | 2 | 2 | no | — |
| 128 | C8×C67 | #120 | 1 | 1 | yes | characteristic |
| 129 | C8×C67 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 130 | C8×C67 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 131 | C8×C67 | #123 | 1 | 1 | yes | characteristic |
| 132 | C67⋊C8 | — | 2 | 1 | no | — |
| 133 | C67⋊C8 | — | 2 | 1 | no | — |
| 134 | C67⋊C8 | — | 4 | 1 | no | — |
| 135 | C22×C67 | — | 2 | 1 | no | — |
| 136 | C22×C67 | — | 2 | 1 | no | — |
| 137 | C22×C67 | — | 2 | 2 | no | — |
| 138 | C22×C67 | — | 4 | 1 | no | — |
| 139 | C22×C67 | — | 4 | 1 | no | — |
| 140 | C22×C67 | #424 | 1 | 1 | yes | characteristic |
| 141 | C22×C67 | C8×D4 | 1 | 1 | yes | characteristic |
| 142 | C22×C67 | #435 | 1 | 1 | yes | characteristic |
| 143 | C4×C67 | — | 2 | 2 | no | — |
| 144 | C4×C67 | #385 | 1 | 1 | yes | characteristic |
| 145 | C4×C67 | C2×C4⋊C8 | 1 | 1 | yes | characteristic |
| 146 | C4×C67 | C8×D4 | 1 | 1 | yes | characteristic |
| 147 | C4×C67 | #435 | 1 | 1 | yes | characteristic |
| 148 | Dic67 | — | 4 | 2 | no | — |
| 149 | Dic67 | — | 4 | 1 | no | — |
| 150 | Dic67 | — | 4 | 1 | no | — |
| 151 | #12917 | — | 67 | 1 | no | maximal, Sylow |
| 152 | C2×C67 | — | 2 | 1 | no | — |
| 153 | C2×C67 | — | 2 | 1 | no | — |
| 154 | C2×C67 | — | 4 | 2 | no | — |
| 155 | C2×C67 | #1850 | 1 | 1 | yes | characteristic |
| 156 | C2×C67 | #2170 | 1 | 1 | yes | characteristic |
| 157 | C2×C67 | #2171 | 1 | 1 | yes | characteristic |
| 158 | C2×#504 | — | 67 | 1 | no | — |
| 159 | C2×#364 | — | 67 | 1 | no | — |
| 160 | #2151 | — | 67 | 1 | no | — |
| 161 | C2×#350 | D67 | 1 | 1 | yes | characteristic |
| 162 | #1764 | — | 67 | 1 | no | — |
| 163 | #1379 | — | 67 | 1 | no | — |
| 164 | #1374 | — | 67 | 1 | no | — |
| 165 | C67 | #12917 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 166 | C2×#131 | C2×D67 | 1 | 1 | yes | characteristic |
| 167 | #504 | — | 134 | 2 | no | — |
| 168 | #504 | — | 134 | 2 | no | — |
| 169 | C2×M32 | — | 67 | 1 | no | — |
| 170 | C2×M32 | — | 134 | 1 | no | — |
| 171 | C2×M32 | C2×D67 | 1 | 1 | yes | characteristic |
| 172 | C22×C16 | — | 67 | 1 | no | — |
| 173 | C22×C16 | — | 134 | 1 | no | — |
| 174 | C22×C16 | C2×D67 | 1 | 1 | yes | characteristic |
| 175 | C2×#104 | — | 67 | 1 | no | — |
| 176 | #416 | — | 67 | 1 | no | — |
| 177 | #405 | — | 67 | 1 | no | — |
| 178 | C2×C4×C8 | — | 67 | 1 | no | — |
| 179 | #364 | — | 134 | 2 | no | — |
| 180 | C4⋊C16 | — | 134 | 1 | no | — |
| 181 | C4⋊C16 | — | 134 | 1 | no | — |
| 182 | #350 | — | 2 | 2 | no | — |
| 183 | #341 | — | 67 | 1 | no | — |
| 184 | #339 | — | 67 | 1 | no | — |
| 185 | C2×#41 | C4×D67 | 1 | 1 | yes | characteristic |
| 186 | #131 | — | 2 | 1 | no | — |
| 187 | #131 | — | 2 | 1 | no | — |
| 188 | #131 | Dic134 | 1 | 2 | yes | — |
| 189 | #131 | C22⋊D67 | 1 | 2 | yes | — |
| 190 | C2×M16 | — | 2 | 1 | no | — |
| 191 | C2×M16 | — | 67 | 1 | no | — |
| 192 | C2×M16 | C4×D67 | 1 | 1 | yes | characteristic |
| 193 | C22×C8 | — | 2 | 1 | no | — |
| 194 | C22×C8 | — | 67 | 1 | no | — |
| 195 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 196 | #117 | — | 67 | 1 | no | — |
| 197 | C2×C42 | — | 67 | 1 | no | — |
| 198 | M32 | — | 2 | 1 | no | — |
| 199 | M32 | — | 2 | 1 | no | — |
| 200 | M32 | — | 67 | 2 | no | — |
| 201 | M32 | — | 134 | 4 | no | — |
| 202 | M32 | — | 134 | 1 | no | — |
| 203 | C2×C16 | — | 2 | 1 | no | — |
| 204 | C2×C16 | — | 2 | 1 | no | — |
| 205 | C2×C16 | — | 2 | 1 | no | — |
| 206 | C2×C16 | — | 2 | 1 | no | — |
| 207 | C2×C16 | — | 67 | 1 | no | — |
| 208 | C2×C16 | — | 67 | 1 | no | — |
| 209 | C2×C16 | — | 134 | 4 | no | — |
| 210 | C2×C16 | — | 134 | 1 | no | — |
| 211 | C2×C16 | — | 134 | 1 | no | — |
| 212 | C2×C16 | — | 134 | 1 | no | — |
| 213 | #104 | — | 134 | 2 | no | — |
| 214 | #103 | — | 67 | 1 | no | — |
| 215 | #103 | — | 67 | 1 | no | — |
| 216 | #102 | — | 67 | 1 | no | — |
| 217 | #102 | — | 67 | 1 | no | — |
| 218 | #97 | — | 134 | 1 | no | — |
| 219 | C4×C8 | — | 134 | 1 | no | — |
| 220 | C4×C8 | — | 134 | 1 | no | — |
| 221 | C4×C8 | — | 134 | 1 | no | — |
| 222 | #41 | — | 2 | 1 | no | — |
| 223 | #41 | — | 2 | 1 | no | — |
| 224 | #41 | Dic67⋊C4 | 1 | 2 | yes | — |
| 225 | #41 | #11761611 | 1 | 2 | yes | — |
| 226 | C2×Q8 | C8×D67 | 1 | 1 | yes | characteristic |
| 227 | C2×D4 | — | 2 | 1 | no | — |
| 228 | C2×D4 | C8×D67 | 1 | 1 | yes | characteristic |
| 229 | C22×C4 | — | 2 | 1 | no | — |
| 230 | C22×C4 | — | 134 | 1 | no | — |
| 231 | C22×C4 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 232 | M16 | — | 2 | 1 | no | — |
| 233 | M16 | — | 2 | 2 | no | — |
| 234 | M16 | — | 2 | 2 | no | — |
| 235 | M16 | — | 134 | 1 | no | — |
| 236 | M16 | — | 134 | 1 | no | — |
| 237 | M16 | Dic67⋊C4 | 1 | 2 | yes | — |
| 238 | C2×C8 | — | 2 | 1 | no | — |
| 239 | C2×C8 | — | 2 | 2 | no | — |
| 240 | C2×C8 | — | 2 | 2 | no | — |
| 241 | C2×C8 | — | 67 | 1 | no | — |
| 242 | C2×C8 | — | 67 | 1 | no | — |
| 243 | C2×C8 | — | 134 | 1 | no | — |
| 244 | C2×C8 | — | 134 | 1 | no | — |
| 245 | C2×C8 | — | 134 | 1 | no | — |
| 246 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | C2×Dic134 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 252 | C4⋊C4 | — | 67 | 1 | no | — |
| 253 | C4⋊C4 | — | 67 | 1 | no | — |
| 254 | C22⋊C4 | — | 134 | 1 | no | — |
| 255 | C42 | — | 134 | 1 | no | — |
| 256 | C42 | — | 134 | 1 | no | — |
| 257 | C42 | — | 134 | 1 | no | — |
| 258 | C16 | — | 4 | 1 | no | — |
| 259 | C16 | — | 4 | 1 | no | — |
| 260 | C16 | — | 134 | 1 | no | — |
| 261 | C16 | — | 134 | 1 | no | — |
| 262 | C16 | — | 134 | 2 | no | — |
| 263 | C23 | — | 2 | 1 | no | — |
| 264 | C23 | #14524051 | 1 | 1 | yes | characteristic |
| 265 | Q8 | — | 2 | 1 | no | — |
| 266 | Q8 | Dic67⋊C8 | 1 | 2 | yes | — |
| 267 | D4 | — | 2 | 2 | no | — |
| 268 | D4 | — | 2 | 2 | no | — |
| 269 | D4 | — | 2 | 1 | no | — |
| 270 | D4 | Dic67⋊C8 | 1 | 2 | yes | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 2 | no | — |
| 273 | C2×C4 | — | 2 | 2 | no | — |
| 274 | C2×C4 | — | 268 | 1 | no | — |
| 275 | C2×C4 | — | 268 | 1 | no | — |
| 276 | C2×C4 | — | 134 | 2 | no | — |
| 277 | C2×C4 | — | 134 | 1 | no | — |
| 278 | C2×C4 | — | 134 | 1 | no | — |
| 279 | C2×C4 | C2×Dic67⋊C4 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | (C22⋊D67)⋊C4 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | (C22⋊D67)⋊C4 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C2×C8×D67 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | C2×#11761611 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #14524051 | 1 | 1 | yes | characteristic |
| 285 | C8 | — | 2 | 2 | no | — |
| 286 | C8 | — | 268 | 1 | no | — |
| 287 | C8 | — | 134 | 1 | no | — |
| 288 | C8 | — | 134 | 1 | no | — |
| 289 | C8 | #14524011 | 1 | 1 | yes | characteristic |
| 290 | C8 | #14524020 | 1 | 1 | yes | characteristic |
| 291 | C8 | #14524020 | 1 | 1 | yes | characteristic |
| 292 | C8 | #14524024 | 1 | 1 | yes | characteristic |
| 293 | C22 | — | 2 | 1 | no | — |
| 294 | C22 | — | 2 | 1 | no | — |
| 295 | C22 | — | 2 | 2 | no | — |
| 296 | C22 | — | 4 | 1 | no | — |
| 297 | C22 | — | 4 | 1 | no | — |
| 298 | C22 | #16863424 | 1 | 1 | yes | characteristic |
| 299 | C22 | (C22⋊D67)⋊C8 | 1 | 1 | yes | characteristic |
| 300 | C22 | #16863464 | 1 | 1 | yes | characteristic |
| 301 | C4 | — | 2 | 2 | no | — |
| 302 | C4 | — | 268 | 2 | no | — |
| 303 | C4 | — | 268 | 1 | no | — |
| 304 | C4 | — | 268 | 1 | no | — |
| 305 | C4 | #16863268 | 1 | 1 | yes | characteristic |
| 306 | C4 | C2×Dic67⋊C8 | 1 | 1 | yes | characteristic |
| 307 | C4 | (C22⋊D67)⋊C8 | 1 | 1 | yes | characteristic |
| 308 | C4 | #16863464 | 1 | 1 | yes | characteristic |
| 309 | C2 | — | 2 | 1 | no | — |
| 310 | C2 | — | 2 | 1 | no | — |
| 311 | C2 | — | 4 | 2 | no | — |
| 312 | C2 | #22560386 | 1 | 1 | yes | characteristic |
| 313 | C2 | #22562043 | 1 | 1 | yes | characteristic |
| 314 | C2 | #22562048 | 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