← #30772326
Subgroups — #30772326
Order 17152.
376 conjugacy classes of subgroups.
The subgroup lattice has 294 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22573312 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22561975 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22561965 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22561837 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22558295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22558291 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C67×#1764 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16863813 | — | 2 | 2 | no | — |
| 9 | #16863813 | — | 2 | 2 | no | — |
| 10 | C2×#14523983 | — | 2 | 2 | no | — |
| 11 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16863217 | — | 2 | 2 | no | — |
| 16 | C4⋊(C67⋊C16) | — | 2 | 2 | no | — |
| 17 | #16863180 | — | 2 | 2 | no | — |
| 18 | C2×#131×C67 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#104×C67 | C4 | 1 | 1 | yes | characteristic |
| 20 | #416×C67 | C4 | 1 | 1 | yes | characteristic |
| 21 | #405×C67 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4×C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 23 | #341×C67 | C4 | 1 | 1 | yes | characteristic |
| 24 | #339×C67 | C4 | 1 | 1 | yes | characteristic |
| 25 | #14523983 | — | 2 | 4 | no | — |
| 26 | #14523983 | — | 2 | 1 | no | — |
| 27 | #14523983 | — | 2 | 1 | no | — |
| 28 | #14523983 | — | 2 | 1 | no | — |
| 29 | #14523983 | — | 2 | 1 | no | — |
| 30 | #14523983 | — | 2 | 1 | no | — |
| 31 | #14523983 | — | 2 | 1 | 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 | 2 | yes | — |
| 39 | C2×#41×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | #131×C67 | — | 2 | 1 | no | — |
| 41 | #131×C67 | — | 2 | 1 | no | — |
| 42 | #131×C67 | D4 | 1 | 2 | yes | — |
| 43 | #131×C67 | Q8 | 1 | 2 | yes | — |
| 44 | C2×M16×C67 | — | 2 | 1 | no | — |
| 45 | C2×M16×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C2×M16×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C22×C8×C67 | — | 2 | 1 | no | — |
| 48 | C22×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 50 | #117×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C2×C42×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #104×C67 | C8 | 1 | 2 | yes | — |
| 53 | #104×C67 | C8 | 1 | 2 | yes | — |
| 54 | #103×C67 | — | 2 | 1 | no | — |
| 55 | #102×C67 | — | 2 | 1 | no | — |
| 56 | #97×C67 | — | 2 | 1 | no | — |
| 57 | C4×C8×C67 | — | 2 | 1 | no | — |
| 58 | C4×C8×C67 | — | 2 | 1 | no | — |
| 59 | C4×C8×C67 | — | 2 | 1 | no | — |
| 60 | C67⋊C16 | — | 2 | 2 | no | — |
| 61 | C67⋊C16 | — | 2 | 2 | no | — |
| 62 | C67⋊C16 | — | 4 | 1 | no | — |
| 63 | C67⋊C16 | — | 4 | 1 | no | — |
| 64 | #41×C67 | — | 2 | 1 | no | — |
| 65 | #41×C67 | — | 2 | 1 | no | — |
| 66 | #41×C67 | C4⋊C4 | 1 | 2 | yes | — |
| 67 | #41×C67 | C2×C8 | 1 | 2 | yes | — |
| 68 | C2×Q8×C67 | M16 | 1 | 1 | yes | characteristic |
| 69 | C2×D4×C67 | — | 2 | 1 | no | — |
| 70 | C2×D4×C67 | M16 | 1 | 1 | yes | characteristic |
| 71 | C22×C4×C67 | — | 2 | 1 | no | — |
| 72 | C22×C4×C67 | — | 2 | 1 | no | — |
| 73 | C22×C4×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | M16×C67 | — | 2 | 1 | no | — |
| 75 | M16×C67 | — | 2 | 1 | no | — |
| 76 | M16×C67 | — | 2 | 2 | no | — |
| 77 | M16×C67 | — | 2 | 2 | no | — |
| 78 | M16×C67 | C4⋊C4 | 1 | 2 | yes | — |
| 79 | M16×C67 | C2×C8 | 1 | 2 | yes | — |
| 80 | C2×C8×C67 | — | 2 | 1 | no | — |
| 81 | C2×C8×C67 | — | 2 | 1 | no | — |
| 82 | C2×C8×C67 | — | 2 | 2 | no | — |
| 83 | C2×C8×C67 | — | 2 | 2 | no | — |
| 84 | C2×C8×C67 | — | 2 | 1 | no | — |
| 85 | C2×C8×C67 | — | 2 | 1 | no | — |
| 86 | C2×C8×C67 | — | 2 | 1 | no | — |
| 87 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 91 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 93 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 94 | C22⋊C4×C67 | — | 2 | 1 | no | — |
| 95 | C42×C67 | — | 2 | 1 | no | — |
| 96 | C42×C67 | — | 2 | 1 | no | — |
| 97 | C42×C67 | C2×C8 | 1 | 1 | yes | characteristic |
| 98 | C42×C67 | C2×C8 | 1 | 1 | yes | characteristic |
| 99 | C23×C67 | — | 2 | 1 | no | — |
| 100 | C23×C67 | #131 | 1 | 1 | yes | characteristic |
| 101 | Q8×C67 | — | 2 | 1 | no | — |
| 102 | Q8×C67 | C4⋊C8 | 1 | 2 | yes | — |
| 103 | D4×C67 | — | 2 | 2 | no | — |
| 104 | D4×C67 | — | 2 | 2 | no | — |
| 105 | D4×C67 | — | 2 | 1 | no | — |
| 106 | D4×C67 | C4⋊C8 | 1 | 2 | yes | — |
| 107 | C2×C4×C67 | — | 2 | 1 | no | — |
| 108 | C2×C4×C67 | — | 2 | 1 | no | — |
| 109 | C2×C4×C67 | — | 2 | 2 | no | — |
| 110 | C2×C4×C67 | — | 2 | 2 | no | — |
| 111 | C2×C4×C67 | — | 2 | 1 | no | — |
| 112 | C2×C4×C67 | — | 4 | 2 | no | — |
| 113 | C2×C4×C67 | — | 4 | 1 | no | — |
| 114 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C67 | C22×C8 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C67 | C2×M16 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C67 | #131 | 1 | 1 | yes | characteristic |
| 120 | C8×C67 | — | 2 | 2 | no | — |
| 121 | C8×C67 | — | 2 | 2 | no | — |
| 122 | C8×C67 | — | 4 | 1 | no | — |
| 123 | C8×C67 | #120 | 1 | 1 | yes | characteristic |
| 124 | C8×C67 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 125 | C8×C67 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 126 | C8×C67 | #123 | 1 | 1 | yes | characteristic |
| 127 | C22×C67 | — | 2 | 2 | no | — |
| 128 | C22×C67 | — | 4 | 1 | no | — |
| 129 | C22×C67 | — | 8 | 1 | no | — |
| 130 | C22×C67 | #424 | 1 | 1 | yes | characteristic |
| 131 | C22×C67 | C8×D4 | 1 | 1 | yes | characteristic |
| 132 | C22×C67 | #435 | 1 | 1 | yes | characteristic |
| 133 | C4×C67 | — | 2 | 2 | no | — |
| 134 | C4×C67 | — | 2 | 2 | no | — |
| 135 | C4×C67 | — | 4 | 1 | no | — |
| 136 | C4×C67 | — | 8 | 1 | no | — |
| 137 | C4×C67 | #385 | 1 | 1 | yes | characteristic |
| 138 | C4×C67 | C2×C4⋊C8 | 1 | 1 | yes | characteristic |
| 139 | C4×C67 | C8×D4 | 1 | 1 | yes | characteristic |
| 140 | C4×C67 | #435 | 1 | 1 | yes | characteristic |
| 141 | #12918 | — | 67 | 1 | no | maximal, Sylow |
| 142 | C2×C67 | — | 2 | 1 | no | — |
| 143 | C2×C67 | — | 2 | 1 | no | — |
| 144 | C2×C67 | — | 2 | 1 | no | — |
| 145 | C2×C67 | — | 4 | 2 | no | — |
| 146 | C2×C67 | #1850 | 1 | 1 | yes | characteristic |
| 147 | #3407 | — | 67 | 1 | no | — |
| 148 | #2153 | — | 67 | 1 | no | — |
| 149 | #2150 | — | 67 | 1 | no | — |
| 150 | #2117 | — | 67 | 1 | no | — |
| 151 | #1764 | D67 | 1 | 1 | yes | characteristic |
| 152 | #1378 | — | 67 | 1 | no | — |
| 153 | #1376 | — | 67 | 1 | no | — |
| 154 | C67 | #12918 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 155 | C2×#131 | C2×D67 | 1 | 1 | yes | characteristic |
| 156 | #504 | — | 134 | 2 | no | — |
| 157 | #504 | — | 134 | 2 | no | — |
| 158 | C2×M32 | — | 67 | 1 | no | — |
| 159 | C2×M32 | — | 67 | 1 | no | — |
| 160 | C2×M32 | — | 67 | 1 | no | — |
| 161 | C2×M32 | — | 67 | 1 | no | — |
| 162 | C2×M32 | — | 134 | 2 | no | — |
| 163 | C2×#104 | Dic67 | 1 | 1 | yes | characteristic |
| 164 | #416 | Dic67 | 1 | 1 | yes | characteristic |
| 165 | #405 | C2×D67 | 1 | 1 | yes | characteristic |
| 166 | C2×C4×C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 167 | #364 | — | 134 | 2 | no | — |
| 168 | C4⋊C16 | — | 134 | 2 | no | — |
| 169 | #350 | — | 134 | 2 | no | — |
| 170 | #341 | Dic67 | 1 | 1 | yes | characteristic |
| 171 | #339 | Dic67 | 1 | 1 | yes | characteristic |
| 172 | C2×#41 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 173 | #131 | — | 2 | 1 | no | — |
| 174 | #131 | — | 2 | 1 | no | — |
| 175 | #131 | Dic134 | 1 | 2 | yes | — |
| 176 | #131 | D268 | 1 | 2 | yes | — |
| 177 | C2×M16 | — | 2 | 1 | no | — |
| 178 | C2×M16 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 179 | C2×M16 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 180 | C22×C8 | — | 2 | 1 | no | — |
| 181 | C22×C8 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 182 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 183 | #117 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 184 | C2×C42 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 185 | M32 | — | 67 | 2 | no | — |
| 186 | M32 | — | 134 | 1 | no | — |
| 187 | M32 | — | 134 | 1 | no | — |
| 188 | M32 | — | 134 | 1 | no | — |
| 189 | M32 | — | 134 | 1 | no | — |
| 190 | M32 | — | 134 | 4 | no | — |
| 191 | M32 | — | 134 | 1 | no | — |
| 192 | M32 | — | 134 | 1 | no | — |
| 193 | M32 | — | 134 | 1 | no | — |
| 194 | C2×C16 | — | 67 | 2 | no | — |
| 195 | C2×C16 | — | 134 | 1 | no | — |
| 196 | C2×C16 | — | 134 | 1 | no | — |
| 197 | C2×C16 | — | 134 | 4 | no | — |
| 198 | C2×C16 | — | 134 | 1 | no | — |
| 199 | #104 | C67⋊C8 | 1 | 2 | yes | — |
| 200 | #104 | C67⋊C8 | 1 | 2 | yes | — |
| 201 | #103 | — | 2 | 1 | no | — |
| 202 | #102 | — | 2 | 1 | no | — |
| 203 | #97 | — | 2 | 1 | no | — |
| 204 | C4×C8 | — | 2 | 1 | no | — |
| 205 | C4×C8 | — | 2 | 1 | no | — |
| 206 | C4×C8 | — | 2 | 1 | no | — |
| 207 | #41 | — | 2 | 1 | no | — |
| 208 | #41 | — | 2 | 1 | no | — |
| 209 | #41 | C4⋊Dic67 | 1 | 2 | yes | — |
| 210 | #41 | C2×C67⋊C8 | 1 | 2 | yes | — |
| 211 | C2×Q8 | #11761612 | 1 | 1 | yes | characteristic |
| 212 | C2×D4 | — | 2 | 1 | no | — |
| 213 | C2×D4 | #11761612 | 1 | 1 | yes | characteristic |
| 214 | C22×C4 | — | 2 | 1 | no | — |
| 215 | C22×C4 | — | 2 | 1 | no | — |
| 216 | C22×C4 | C22×Dic67 | 1 | 1 | yes | characteristic |
| 217 | M16 | — | 2 | 1 | no | — |
| 218 | M16 | — | 2 | 1 | no | — |
| 219 | M16 | — | 2 | 2 | no | — |
| 220 | M16 | — | 2 | 2 | no | — |
| 221 | M16 | C4⋊Dic67 | 1 | 2 | yes | — |
| 222 | M16 | C2×C67⋊C8 | 1 | 2 | yes | — |
| 223 | C2×C8 | — | 2 | 1 | no | — |
| 224 | C2×C8 | — | 2 | 1 | no | — |
| 225 | C2×C8 | — | 2 | 2 | no | — |
| 226 | C2×C8 | — | 2 | 2 | no | — |
| 227 | C2×C8 | — | 2 | 1 | no | — |
| 228 | C2×C8 | — | 2 | 1 | no | — |
| 229 | C2×C8 | — | 2 | 1 | no | — |
| 230 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 231 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 232 | C2×C8 | C2×D268 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | C2×Dic134 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 236 | C4⋊C4 | — | 2 | 1 | no | — |
| 237 | C22⋊C4 | — | 2 | 1 | no | — |
| 238 | C42 | — | 2 | 1 | no | — |
| 239 | C42 | — | 2 | 1 | no | — |
| 240 | C42 | C2×C67⋊C8 | 1 | 1 | yes | characteristic |
| 241 | C42 | C2×C67⋊C8 | 1 | 1 | yes | characteristic |
| 242 | C16 | — | 268 | 1 | no | — |
| 243 | C16 | — | 268 | 1 | no | — |
| 244 | C16 | — | 134 | 2 | no | — |
| 245 | C16 | — | 134 | 2 | no | — |
| 246 | C23 | — | 2 | 1 | no | — |
| 247 | C23 | #14524050 | 1 | 1 | yes | characteristic |
| 248 | Q8 | — | 2 | 1 | no | — |
| 249 | Q8 | C4⋊(C67⋊C8) | 1 | 2 | yes | — |
| 250 | D4 | — | 2 | 2 | no | — |
| 251 | D4 | — | 2 | 2 | no | — |
| 252 | D4 | — | 2 | 1 | no | — |
| 253 | D4 | C4⋊(C67⋊C8) | 1 | 2 | yes | — |
| 254 | C2×C4 | — | 2 | 1 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 2 | no | — |
| 257 | C2×C4 | — | 2 | 2 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 4 | 2 | no | — |
| 260 | C2×C4 | — | 4 | 1 | no | — |
| 261 | C2×C4 | C2×C4⋊Dic67 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 263 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | C22×C67⋊C8 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | C2×#11761612 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | #14524050 | 1 | 1 | yes | characteristic |
| 267 | C8 | — | 2 | 2 | no | — |
| 268 | C8 | — | 2 | 2 | no | — |
| 269 | C8 | — | 4 | 1 | no | — |
| 270 | C8 | #14524013 | 1 | 1 | yes | characteristic |
| 271 | C8 | #14524020 | 1 | 1 | yes | characteristic |
| 272 | C8 | #14524020 | 1 | 1 | yes | characteristic |
| 273 | C8 | #14524027 | 1 | 1 | yes | characteristic |
| 274 | C22 | — | 2 | 2 | no | — |
| 275 | C22 | — | 4 | 1 | no | — |
| 276 | C22 | — | 8 | 1 | no | — |
| 277 | C22 | #16863423 | 1 | 1 | yes | characteristic |
| 278 | C22 | D4×C67⋊C8 | 1 | 1 | yes | characteristic |
| 279 | C22 | #16863468 | 1 | 1 | yes | characteristic |
| 280 | C4 | — | 2 | 2 | no | — |
| 281 | C4 | — | 2 | 2 | no | — |
| 282 | C4 | — | 4 | 1 | no | — |
| 283 | C4 | — | 8 | 1 | no | — |
| 284 | C4 | #16863271 | 1 | 1 | yes | characteristic |
| 285 | C4 | C2×C4⋊(C67⋊C8) | 1 | 1 | yes | characteristic |
| 286 | C4 | D4×C67⋊C8 | 1 | 1 | yes | characteristic |
| 287 | C4 | #16863468 | 1 | 1 | yes | characteristic |
| 288 | C2 | — | 2 | 1 | no | — |
| 289 | C2 | — | 2 | 1 | no | — |
| 290 | C2 | — | 2 | 1 | no | — |
| 291 | C2 | — | 4 | 2 | no | — |
| 292 | C2 | #22560389 | 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