← #30772854
Subgroups — #30772854
Order 17152.
472 conjugacy classes of subgroups.
The subgroup lattice has 328 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22568820 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16863212 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#16863212 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22561837 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22558320 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22558312 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#473×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16864163 | — | 2 | 1 | no | — |
| 9 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16863445 | — | 2 | 2 | no | — |
| 12 | #16863445 | — | 2 | 1 | no | — |
| 13 | C2×#14523979 | — | 2 | 2 | no | — |
| 14 | C2×#14523979 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#14523979 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16863212 | — | 2 | 2 | no | — |
| 17 | #16863212 | — | 2 | 2 | no | — |
| 18 | #16863180 | — | 2 | 2 | no | — |
| 19 | C2×#137×C67 | — | 2 | 1 | no | — |
| 20 | C22×Q16×C67 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#131×C67 | C22 | 1 | 1 | yes | characteristic |
| 22 | #473×C67 | — | 2 | 2 | no | — |
| 23 | #473×C67 | C4 | 1 | 4 | yes | — |
| 24 | C2×#108×C67 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#101×C67 | — | 2 | 1 | no | — |
| 26 | #14524050 | — | 4 | 2 | no | — |
| 27 | #14524050 | — | 4 | 2 | no | — |
| 28 | C2×#11761612 | — | 2 | 4 | no | — |
| 29 | C2×#11761612 | — | 2 | 1 | no | — |
| 30 | C2×#11761612 | — | 2 | 1 | no | — |
| 31 | #14523983 | — | 2 | 1 | no | — |
| 32 | #14523983 | — | 2 | 1 | no | — |
| 33 | #14523983 | — | 2 | 1 | no | — |
| 34 | #14523983 | — | 2 | 1 | no | — |
| 35 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 36 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 37 | #14523979 | — | 2 | 4 | no | — |
| 38 | #14523979 | — | 2 | 1 | no | — |
| 39 | #14523979 | — | 2 | 1 | no | — |
| 40 | #14523979 | D4 | 1 | 2 | yes | — |
| 41 | #14523979 | D4 | 1 | 2 | yes | — |
| 42 | C2×#41×C67 | D4 | 1 | 1 | yes | characteristic |
| 43 | C22×Q8×C67 | — | 2 | 1 | no | — |
| 44 | #137×C67 | — | 2 | 4 | no | — |
| 45 | #137×C67 | — | 4 | 2 | no | — |
| 46 | C2×Q16×C67 | — | 2 | 1 | no | — |
| 47 | C2×Q16×C67 | — | 2 | 1 | no | — |
| 48 | C2×Q16×C67 | — | 2 | 1 | no | — |
| 49 | C2×Q16×C67 | — | 2 | 1 | no | — |
| 50 | C2×Q16×C67 | — | 2 | 1 | no | — |
| 51 | C2×Q16×C67 | — | 4 | 1 | no | — |
| 52 | C2×Q16×C67 | C2×C4 | 1 | 2 | yes | — |
| 53 | C2×SD16×C67 | — | 4 | 1 | no | — |
| 54 | #131×C67 | — | 2 | 1 | no | — |
| 55 | #131×C67 | — | 2 | 1 | no | — |
| 56 | #131×C67 | C2×C4 | 1 | 2 | yes | — |
| 57 | #131×C67 | D4 | 1 | 2 | yes | — |
| 58 | C2×M16×C67 | — | 2 | 1 | no | — |
| 59 | C2×M16×C67 | — | 2 | 1 | no | — |
| 60 | C2×M16×C67 | D4 | 1 | 1 | yes | characteristic |
| 61 | C22×C8×C67 | — | 2 | 1 | no | — |
| 62 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 63 | #108×C67 | — | 2 | 1 | no | — |
| 64 | #108×C67 | C2×C4 | 1 | 2 | yes | — |
| 65 | #101×C67 | — | 2 | 4 | no | — |
| 66 | #11761612 | — | 2 | 2 | no | — |
| 67 | #11761612 | — | 2 | 2 | no | — |
| 68 | #11761612 | — | 4 | 4 | no | — |
| 69 | #11761612 | — | 4 | 1 | no | — |
| 70 | #11761612 | — | 4 | 1 | no | — |
| 71 | C2×C67⋊C8 | — | 4 | 4 | no | — |
| 72 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 73 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 74 | C67⋊C16 | — | 4 | 1 | no | — |
| 75 | C67⋊C16 | — | 4 | 1 | no | — |
| 76 | #41×C67 | — | 2 | 1 | no | — |
| 77 | #41×C67 | — | 2 | 1 | no | — |
| 78 | #41×C67 | C22⋊C4 | 1 | 2 | yes | — |
| 79 | #41×C67 | D8 | 1 | 2 | yes | — |
| 80 | C2×Q8×C67 | — | 2 | 2 | no | — |
| 81 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 82 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 83 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 84 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 85 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 86 | C2×Q8×C67 | SD16 | 1 | 1 | yes | characteristic |
| 87 | C2×D4×C67 | — | 2 | 1 | no | — |
| 88 | C2×D4×C67 | SD16 | 1 | 1 | yes | characteristic |
| 89 | C22×C4×C67 | — | 2 | 1 | no | — |
| 90 | C22×C4×C67 | — | 4 | 1 | no | — |
| 91 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | Q16×C67 | — | 2 | 2 | no | — |
| 93 | Q16×C67 | — | 2 | 2 | no | — |
| 94 | Q16×C67 | — | 4 | 4 | no | — |
| 95 | Q16×C67 | — | 4 | 1 | no | — |
| 96 | Q16×C67 | — | 4 | 1 | no | — |
| 97 | SD16×C67 | — | 4 | 4 | no | — |
| 98 | M16×C67 | — | 2 | 1 | no | — |
| 99 | M16×C67 | — | 2 | 2 | no | — |
| 100 | M16×C67 | — | 2 | 2 | no | — |
| 101 | M16×C67 | — | 2 | 2 | no | — |
| 102 | M16×C67 | — | 4 | 1 | no | — |
| 103 | M16×C67 | C22⋊C4 | 1 | 2 | yes | — |
| 104 | C2×C8×C67 | — | 2 | 1 | no | — |
| 105 | C2×C8×C67 | — | 2 | 2 | no | — |
| 106 | C2×C8×C67 | — | 2 | 2 | no | — |
| 107 | C2×C8×C67 | — | 4 | 1 | no | — |
| 108 | C2×C8×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 113 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 114 | C23×C67 | — | 2 | 1 | no | — |
| 115 | C23×C67 | #137 | 1 | 1 | yes | characteristic |
| 116 | Q8×C67 | — | 2 | 1 | no | — |
| 117 | Q8×C67 | — | 4 | 2 | no | — |
| 118 | Q8×C67 | — | 4 | 2 | no | — |
| 119 | Q8×C67 | — | 8 | 1 | no | — |
| 120 | Q8×C67 | — | 8 | 1 | no | — |
| 121 | Q8×C67 | #102 | 1 | 2 | yes | — |
| 122 | D4×C67 | — | 2 | 2 | no | — |
| 123 | D4×C67 | — | 2 | 2 | no | — |
| 124 | D4×C67 | — | 2 | 1 | no | — |
| 125 | D4×C67 | #102 | 1 | 2 | yes | — |
| 126 | C2×C4×C67 | — | 2 | 1 | no | — |
| 127 | C2×C4×C67 | — | 2 | 2 | no | — |
| 128 | C2×C4×C67 | — | 2 | 2 | no | — |
| 129 | C2×C4×C67 | — | 4 | 2 | no | — |
| 130 | C2×C4×C67 | — | 8 | 1 | no | — |
| 131 | C2×C4×C67 | — | 8 | 1 | no | — |
| 132 | C2×C4×C67 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C67 | #120 | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C67 | #123 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 136 | C2×C4×C67 | C2×SD16 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C67 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 138 | C8×C67 | — | 2 | 2 | no | — |
| 139 | C8×C67 | — | 4 | 2 | no | — |
| 140 | C8×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 141 | C8×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 142 | C8×C67 | #121 | 1 | 1 | yes | characteristic |
| 143 | C8×C67 | #121 | 1 | 1 | yes | characteristic |
| 144 | C67⋊C8 | — | 4 | 2 | no | — |
| 145 | C67⋊C8 | — | 4 | 2 | no | — |
| 146 | C22×C67 | — | 2 | 2 | no | — |
| 147 | C22×C67 | — | 4 | 1 | no | — |
| 148 | C22×C67 | — | 8 | 1 | no | — |
| 149 | C22×C67 | #417 | 1 | 1 | yes | characteristic |
| 150 | C22×C67 | #448 | 1 | 1 | yes | characteristic |
| 151 | C22×C67 | #480 | 1 | 1 | yes | characteristic |
| 152 | C4×C67 | — | 2 | 2 | no | — |
| 153 | C4×C67 | — | 8 | 2 | no | — |
| 154 | C4×C67 | #386 | 1 | 1 | yes | characteristic |
| 155 | C4×C67 | C2×#102 | 1 | 1 | yes | characteristic |
| 156 | C4×C67 | #447 | 1 | 1 | yes | characteristic |
| 157 | C4×C67 | #481 | 1 | 1 | yes | characteristic |
| 158 | #13006 | — | 67 | 1 | no | maximal, Sylow |
| 159 | C2×C67 | — | 2 | 1 | no | — |
| 160 | C2×C67 | — | 2 | 1 | no | — |
| 161 | C2×C67 | — | 2 | 1 | no | — |
| 162 | C2×C67 | — | 4 | 2 | no | — |
| 163 | C2×C67 | #1874 | 1 | 1 | yes | characteristic |
| 164 | C2×#473 | D67 | 1 | 1 | yes | characteristic |
| 165 | #2916 | — | 67 | 1 | no | — |
| 166 | C2×#362 | — | 67 | 1 | no | — |
| 167 | C2×#362 | — | 67 | 1 | no | — |
| 168 | #2117 | — | 67 | 1 | no | — |
| 169 | #1388 | — | 67 | 1 | no | — |
| 170 | #1385 | — | 67 | 1 | no | — |
| 171 | C67 | #13006 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | C2×#137 | — | 2 | 1 | no | — |
| 173 | C22×Q16 | C2×D67 | 1 | 1 | yes | characteristic |
| 174 | #568 | — | 134 | 1 | no | — |
| 175 | C2×#131 | C2×D67 | 1 | 1 | yes | characteristic |
| 176 | C2×M32 | — | 67 | 1 | no | — |
| 177 | C2×M32 | — | 67 | 1 | no | — |
| 178 | #473 | — | 2 | 2 | no | — |
| 179 | #473 | Dic67 | 1 | 4 | yes | — |
| 180 | #430 | — | 134 | 2 | no | — |
| 181 | #430 | — | 134 | 1 | no | — |
| 182 | C2×#108 | — | 67 | 1 | no | — |
| 183 | C2×#108 | — | 67 | 1 | no | — |
| 184 | C2×#108 | — | 134 | 2 | no | — |
| 185 | C2×#108 | C2×D67 | 1 | 1 | yes | characteristic |
| 186 | C2×#101 | — | 2 | 1 | no | — |
| 187 | #362 | — | 134 | 2 | no | — |
| 188 | #362 | — | 134 | 2 | no | — |
| 189 | #350 | — | 134 | 2 | no | — |
| 190 | C2×#41 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 191 | C22×Q8 | — | 2 | 1 | no | — |
| 192 | #137 | — | 2 | 4 | no | — |
| 193 | #137 | — | 4 | 2 | no | — |
| 194 | C2×Q16 | — | 2 | 1 | no | — |
| 195 | C2×Q16 | — | 2 | 1 | no | — |
| 196 | C2×Q16 | — | 2 | 1 | no | — |
| 197 | C2×Q16 | — | 2 | 1 | no | — |
| 198 | C2×Q16 | — | 2 | 1 | no | — |
| 199 | C2×Q16 | — | 4 | 1 | no | — |
| 200 | C2×Q16 | C2×Dic67 | 1 | 2 | yes | — |
| 201 | C2×SD16 | — | 4 | 1 | no | — |
| 202 | #131 | — | 2 | 1 | no | — |
| 203 | #131 | — | 2 | 1 | no | — |
| 204 | #131 | — | 268 | 2 | no | — |
| 205 | #131 | — | 268 | 2 | no | — |
| 206 | #131 | C2×Dic67 | 1 | 2 | yes | — |
| 207 | #131 | C22⋊D67 | 1 | 2 | yes | — |
| 208 | C2×M16 | — | 2 | 1 | no | — |
| 209 | C2×M16 | — | 2 | 1 | no | — |
| 210 | C2×M16 | — | 134 | 4 | no | — |
| 211 | C2×M16 | — | 134 | 1 | no | — |
| 212 | C2×M16 | — | 134 | 1 | no | — |
| 213 | C2×M16 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 214 | C22×C8 | — | 2 | 1 | no | — |
| 215 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 216 | M32 | — | 134 | 1 | no | — |
| 217 | M32 | — | 134 | 1 | no | — |
| 218 | M32 | — | 134 | 1 | no | — |
| 219 | M32 | — | 134 | 1 | no | — |
| 220 | C2×C16 | — | 134 | 1 | no | — |
| 221 | C2×C16 | — | 134 | 1 | no | — |
| 222 | #108 | — | 2 | 1 | no | — |
| 223 | #108 | — | 67 | 2 | no | — |
| 224 | #108 | — | 67 | 2 | no | — |
| 225 | #108 | — | 134 | 4 | no | — |
| 226 | #108 | — | 134 | 1 | no | — |
| 227 | #108 | — | 134 | 1 | no | — |
| 228 | #108 | C2×Dic67 | 1 | 2 | yes | — |
| 229 | #101 | — | 2 | 4 | no | — |
| 230 | #41 | — | 2 | 1 | no | — |
| 231 | #41 | — | 2 | 1 | no | — |
| 232 | #41 | C22⋊Dic67 | 1 | 2 | yes | — |
| 233 | #41 | #11761613 | 1 | 2 | yes | — |
| 234 | C2×Q8 | — | 2 | 2 | no | — |
| 235 | C2×Q8 | — | 4 | 1 | no | — |
| 236 | C2×Q8 | — | 4 | 1 | no | — |
| 237 | C2×Q8 | — | 4 | 1 | no | — |
| 238 | C2×Q8 | — | 4 | 1 | no | — |
| 239 | C2×Q8 | — | 4 | 1 | no | — |
| 240 | C2×Q8 | #11761616 | 1 | 1 | yes | characteristic |
| 241 | C2×D4 | — | 2 | 1 | no | — |
| 242 | C2×D4 | #11761616 | 1 | 1 | yes | characteristic |
| 243 | C22×C4 | — | 2 | 1 | no | — |
| 244 | C22×C4 | — | 4 | 1 | no | — |
| 245 | C22×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 246 | Q16 | — | 2 | 2 | no | — |
| 247 | Q16 | — | 2 | 2 | no | — |
| 248 | Q16 | — | 4 | 4 | no | — |
| 249 | Q16 | — | 4 | 1 | no | — |
| 250 | Q16 | — | 4 | 1 | no | — |
| 251 | SD16 | — | 4 | 4 | no | — |
| 252 | M16 | — | 2 | 1 | no | — |
| 253 | M16 | — | 2 | 2 | no | — |
| 254 | M16 | — | 2 | 2 | no | — |
| 255 | M16 | — | 2 | 2 | no | — |
| 256 | M16 | — | 4 | 1 | no | — |
| 257 | M16 | — | 268 | 4 | no | — |
| 258 | M16 | — | 268 | 1 | no | — |
| 259 | M16 | — | 268 | 1 | no | — |
| 260 | M16 | — | 134 | 2 | no | — |
| 261 | M16 | — | 134 | 2 | no | — |
| 262 | M16 | C22⋊Dic67 | 1 | 2 | yes | — |
| 263 | C2×C8 | — | 2 | 1 | no | — |
| 264 | C2×C8 | — | 2 | 2 | no | — |
| 265 | C2×C8 | — | 2 | 2 | no | — |
| 266 | C2×C8 | — | 4 | 1 | no | — |
| 267 | C2×C8 | — | 268 | 4 | no | — |
| 268 | C2×C8 | — | 268 | 1 | no | — |
| 269 | C2×C8 | — | 268 | 1 | no | — |
| 270 | C2×C8 | C22×Dic67 | 1 | 1 | yes | characteristic |
| 271 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 272 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 273 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 274 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 275 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 276 | C16 | — | 268 | 1 | no | — |
| 277 | C16 | — | 268 | 1 | no | — |
| 278 | C23 | — | 2 | 1 | no | — |
| 279 | C23 | #14524073 | 1 | 1 | yes | characteristic |
| 280 | Q8 | — | 2 | 1 | no | — |
| 281 | Q8 | — | 4 | 2 | no | — |
| 282 | Q8 | — | 4 | 2 | no | — |
| 283 | Q8 | — | 8 | 1 | no | — |
| 284 | Q8 | — | 8 | 1 | no | — |
| 285 | Q8 | #14523964 | 1 | 2 | yes | — |
| 286 | D4 | — | 2 | 2 | no | — |
| 287 | D4 | — | 2 | 2 | no | — |
| 288 | D4 | — | 2 | 1 | no | — |
| 289 | D4 | #14523964 | 1 | 2 | yes | — |
| 290 | C2×C4 | — | 2 | 1 | no | — |
| 291 | C2×C4 | — | 2 | 2 | no | — |
| 292 | C2×C4 | — | 2 | 2 | no | — |
| 293 | C2×C4 | — | 4 | 2 | no | — |
| 294 | C2×C4 | — | 8 | 1 | no | — |
| 295 | C2×C4 | — | 8 | 1 | no | — |
| 296 | C2×C4 | C2×C22⋊Dic67 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #14524011 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #14524024 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | C2×#11761613 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | C2×#11761616 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | #14524068 | 1 | 1 | yes | characteristic |
| 302 | C8 | — | 2 | 2 | no | — |
| 303 | C8 | — | 4 | 2 | no | — |
| 304 | C8 | — | 268 | 2 | no | — |
| 305 | C8 | — | 268 | 2 | no | — |
| 306 | C8 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 307 | C8 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 308 | C8 | #14524015 | 1 | 1 | yes | characteristic |
| 309 | C8 | #14524015 | 1 | 1 | yes | characteristic |
| 310 | C22 | — | 2 | 2 | no | — |
| 311 | C22 | — | 4 | 1 | no | — |
| 312 | C22 | — | 8 | 1 | no | — |
| 313 | C22 | #16863397 | 1 | 1 | yes | characteristic |
| 314 | C22 | #16863534 | 1 | 1 | yes | characteristic |
| 315 | C22 | #16863707 | 1 | 1 | yes | characteristic |
| 316 | C4 | — | 2 | 2 | no | — |
| 317 | C4 | — | 8 | 2 | no | — |
| 318 | C4 | #16863276 | 1 | 1 | yes | characteristic |
| 319 | C4 | C2×#14523964 | 1 | 1 | yes | characteristic |
| 320 | C4 | #16863529 | 1 | 1 | yes | characteristic |
| 321 | C4 | #16863712 | 1 | 1 | yes | characteristic |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 2 | 1 | no | — |
| 324 | C2 | — | 2 | 1 | no | — |
| 325 | C2 | — | 4 | 2 | no | — |
| 326 | C2 | #22560527 | 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