← #36471013
Subgroups — #36471013
Order 22784.
394 conjugacy classes of subgroups.
The subgroup lattice has 306 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #26022024 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#19151420 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #26021880 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #26020306 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #26018359 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #26018353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C89×#2912 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#15667952 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#15667952 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#15667948 | C22 | 1 | 1 | yes | characteristic |
| 11 | #19151636 | C22 | 1 | 1 | yes | characteristic |
| 12 | #19151615 | — | 2 | 1 | no | — |
| 13 | #19151420 | — | 2 | 2 | no | — |
| 14 | #19151420 | — | 2 | 1 | no | — |
| 15 | #19151420 | — | 2 | 1 | no | — |
| 16 | #19151382 | C4 | 1 | 4 | yes | complemented |
| 17 | #19151370 | — | 2 | 1 | no | — |
| 18 | C2×#131×C89 | C22 | 1 | 1 | yes | characteristic |
| 19 | #520×C89 | — | 2 | 1 | no | — |
| 20 | #428×C89 | — | 2 | 1 | no | — |
| 21 | #428×C89 | — | 2 | 2 | no | — |
| 22 | #427×C89 | — | 2 | 1 | no | — |
| 23 | #427×C89 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#106×C89 | — | 2 | 1 | no | — |
| 25 | C2×#106×C89 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#13083558 | — | 2 | 1 | no | — |
| 27 | #15667971 | — | 2 | 1 | no | — |
| 28 | #15667952 | — | 2 | 1 | no | — |
| 29 | #15667952 | — | 2 | 1 | no | — |
| 30 | #15667952 | — | 2 | 1 | no | — |
| 31 | #15667952 | C2×C4 | 1 | 2 | yes | complemented |
| 32 | C2×#13083549 | — | 2 | 1 | no | — |
| 33 | C2×#13083549 | C2×C4 | 1 | 2 | yes | complemented |
| 34 | #15667948 | — | 2 | 1 | no | — |
| 35 | #15667948 | — | 2 | 1 | no | — |
| 36 | #15667946 | — | 2 | 1 | no | — |
| 37 | #15667944 | — | 2 | 1 | no | — |
| 38 | #15667939 | — | 4 | 2 | no | — |
| 39 | C2×#41×C89 | D4 | 1 | 1 | yes | characteristic |
| 40 | #131×C89 | — | 2 | 1 | no | — |
| 41 | #131×C89 | — | 2 | 1 | no | — |
| 42 | #131×C89 | C2×C4 | 1 | 2 | yes | — |
| 43 | #131×C89 | Q8 | 1 | 2 | yes | — |
| 44 | C2×M16×C89 | — | 2 | 1 | no | — |
| 45 | C2×M16×C89 | D4 | 1 | 1 | yes | characteristic |
| 46 | C22×C8×C89 | — | 2 | 1 | no | — |
| 47 | C22×C8×C89 | C23 | 1 | 1 | yes | characteristic |
| 48 | C4×Q8×C89 | — | 4 | 1 | no | — |
| 49 | C4×D4×C89 | — | 4 | 2 | no | — |
| 50 | C4×D4×C89 | — | 4 | 1 | no | — |
| 51 | #117×C89 | — | 2 | 2 | no | — |
| 52 | #117×C89 | — | 2 | 1 | no | — |
| 53 | C2×C4⋊C4×C89 | — | 2 | 2 | no | — |
| 54 | C2×C4⋊C4×C89 | — | 2 | 1 | no | — |
| 55 | #107×C89 | — | 2 | 2 | no | — |
| 56 | #107×C89 | — | 2 | 1 | no | — |
| 57 | #106×C89 | — | 2 | 2 | no | — |
| 58 | #106×C89 | — | 2 | 1 | no | — |
| 59 | #106×C89 | D4 | 1 | 1 | yes | characteristic |
| 60 | #106×C89 | D4 | 1 | 2 | yes | — |
| 61 | #106×C89 | D4 | 1 | 1 | yes | characteristic |
| 62 | #13083558 | — | 4 | 1 | no | — |
| 63 | #13083558 | — | 4 | 1 | no | — |
| 64 | C2×#10623494 | — | 4 | 1 | no | — |
| 65 | C4⋊Dic89 | — | 4 | 1 | no | — |
| 66 | C22⋊Dic89 | — | 4 | 1 | no | — |
| 67 | C4×Dic89 | — | 4 | 1 | no | — |
| 68 | #13083549 | — | 2 | 2 | no | — |
| 69 | #13083549 | — | 2 | 2 | no | — |
| 70 | #41×C89 | — | 2 | 1 | no | — |
| 71 | #41×C89 | — | 2 | 1 | no | — |
| 72 | #41×C89 | C4⋊C4 | 1 | 2 | yes | — |
| 73 | #41×C89 | SD16 | 1 | 2 | yes | — |
| 74 | C2×Q8×C89 | SD16 | 1 | 1 | yes | characteristic |
| 75 | C2×D4×C89 | — | 2 | 1 | no | — |
| 76 | C2×D4×C89 | SD16 | 1 | 1 | yes | characteristic |
| 77 | C22×C4×C89 | — | 2 | 1 | no | — |
| 78 | C22×C4×C89 | — | 4 | 2 | no | — |
| 79 | C22×C4×C89 | — | 4 | 1 | no | — |
| 80 | C22×C4×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | M16×C89 | — | 2 | 1 | no | — |
| 82 | M16×C89 | — | 2 | 2 | no | — |
| 83 | M16×C89 | — | 2 | 2 | no | — |
| 84 | M16×C89 | C4⋊C4 | 1 | 2 | yes | — |
| 85 | C2×C8×C89 | — | 2 | 1 | no | — |
| 86 | C2×C8×C89 | — | 2 | 2 | no | — |
| 87 | C2×C8×C89 | — | 2 | 2 | no | — |
| 88 | C2×C8×C89 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C8×C89 | C2×Q8 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C89 | #41 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C89 | #41 | 1 | 1 | yes | characteristic |
| 94 | C4⋊C4×C89 | — | 2 | 1 | no | — |
| 95 | C4⋊C4×C89 | — | 2 | 2 | no | — |
| 96 | C4⋊C4×C89 | — | 2 | 1 | no | — |
| 97 | C4⋊C4×C89 | — | 4 | 2 | no | — |
| 98 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 99 | C22⋊C4×C89 | — | 4 | 2 | no | — |
| 100 | C22⋊C4×C89 | — | 4 | 1 | no | — |
| 101 | C42×C89 | — | 4 | 2 | no | — |
| 102 | C42×C89 | — | 4 | 1 | no | — |
| 103 | C23×C89 | — | 2 | 1 | no | — |
| 104 | C23×C89 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 105 | Q8×C89 | — | 2 | 1 | no | — |
| 106 | Q8×C89 | #106 | 1 | 2 | yes | — |
| 107 | D4×C89 | — | 2 | 2 | no | — |
| 108 | D4×C89 | — | 2 | 2 | no | — |
| 109 | D4×C89 | — | 2 | 1 | no | — |
| 110 | D4×C89 | #106 | 1 | 2 | yes | — |
| 111 | C2×C4×C89 | — | 2 | 1 | no | — |
| 112 | C2×C4×C89 | — | 2 | 2 | no | — |
| 113 | C2×C4×C89 | — | 2 | 2 | no | — |
| 114 | C2×C4×C89 | — | 4 | 1 | no | — |
| 115 | C2×C4×C89 | — | 4 | 2 | no | — |
| 116 | C2×C4×C89 | — | 4 | 1 | no | — |
| 117 | C2×C4×C89 | — | 16 | 1 | no | — |
| 118 | C2×C4×C89 | — | 16 | 2 | no | — |
| 119 | C2×C4×C89 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C89 | #120 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C89 | #123 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C89 | C2×SD16 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C89 | C2×SD16 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C89 | #137 | 1 | 1 | yes | characteristic |
| 125 | C2×Dic89 | — | 8 | 1 | no | — |
| 126 | C8×C89 | — | 2 | 2 | no | — |
| 127 | C8×C89 | C4×D4 | 1 | 1 | yes | characteristic |
| 128 | C8×C89 | C4×D4 | 1 | 1 | yes | characteristic |
| 129 | C8×C89 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 130 | C8×C89 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 131 | #10623494 | — | 8 | 1 | no | — |
| 132 | C22×C89 | — | 2 | 2 | no | — |
| 133 | C22×C89 | — | 4 | 1 | no | — |
| 134 | C22×C89 | — | 4 | 2 | no | — |
| 135 | C22×C89 | #428 | 1 | 1 | yes | characteristic |
| 136 | C22×C89 | #450 | 1 | 1 | yes | characteristic |
| 137 | C22×C89 | #481 | 1 | 1 | yes | characteristic |
| 138 | C4×C89 | — | 2 | 2 | no | — |
| 139 | C4×C89 | — | 8 | 1 | no | — |
| 140 | C4×C89 | — | 8 | 2 | no | — |
| 141 | C4×C89 | — | 8 | 1 | no | — |
| 142 | C4×C89 | #385 | 1 | 1 | yes | characteristic |
| 143 | C4×C89 | C2×#106 | 1 | 1 | yes | characteristic |
| 144 | C4×C89 | #448 | 1 | 1 | yes | characteristic |
| 145 | C4×C89 | #483 | 1 | 1 | yes | characteristic |
| 146 | Dic89 | — | 16 | 1 | no | — |
| 147 | #12934 | — | 89 | 1 | no | maximal, Sylow |
| 148 | C2×C89 | — | 2 | 1 | no | — |
| 149 | C2×C89 | — | 2 | 1 | no | — |
| 150 | C2×C89 | — | 2 | 1 | no | — |
| 151 | C2×C89 | — | 4 | 2 | no | — |
| 152 | C2×C89 | #1853 | 1 | 1 | yes | characteristic |
| 153 | #2912 | D89 | 1 | 1 | yes | characteristic |
| 154 | #2155 | — | 89 | 1 | no | — |
| 155 | C2×#365 | — | 89 | 1 | no | — |
| 156 | #2117 | — | 89 | 1 | no | — |
| 157 | #1814 | — | 89 | 1 | no | — |
| 158 | #1386 | — | 89 | 1 | no | — |
| 159 | #1384 | — | 89 | 1 | no | — |
| 160 | C89 | #12934 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 161 | C2×#131 | C2×D89 | 1 | 1 | yes | characteristic |
| 162 | #520 | — | 2 | 1 | no | — |
| 163 | C2×M32 | — | 89 | 1 | no | — |
| 164 | C2×M32 | — | 89 | 1 | no | — |
| 165 | C2×#108 | — | 89 | 1 | no | — |
| 166 | #428 | — | 2 | 1 | no | — |
| 167 | #428 | — | 2 | 2 | no | — |
| 168 | #427 | — | 2 | 1 | no | — |
| 169 | #427 | — | 89 | 1 | no | — |
| 170 | #427 | C2×D89 | 1 | 1 | yes | characteristic |
| 171 | C2×#106 | — | 2 | 1 | no | — |
| 172 | C2×#106 | C2×D89 | 1 | 1 | yes | characteristic |
| 173 | #421 | — | 178 | 1 | no | — |
| 174 | #365 | — | 178 | 2 | no | — |
| 175 | #365 | — | 178 | 1 | no | — |
| 176 | #365 | — | 178 | 1 | no | — |
| 177 | #350 | — | 89 | 4 | no | — |
| 178 | #344 | — | 178 | 1 | no | — |
| 179 | C2×#41 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 180 | #131 | — | 2 | 1 | no | — |
| 181 | #131 | — | 2 | 1 | no | — |
| 182 | #131 | Dic178 | 1 | 2 | yes | — |
| 183 | #131 | C4×D89 | 1 | 2 | yes | — |
| 184 | C2×M16 | — | 2 | 1 | no | — |
| 185 | C2×M16 | — | 178 | 1 | no | — |
| 186 | C2×M16 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 187 | C22×C8 | — | 2 | 1 | no | — |
| 188 | C22×C8 | C22×D89 | 1 | 1 | yes | characteristic |
| 189 | C4×Q8 | — | 4 | 1 | no | — |
| 190 | C4×D4 | — | 4 | 2 | no | — |
| 191 | C4×D4 | — | 4 | 1 | no | — |
| 192 | #117 | — | 2 | 2 | no | — |
| 193 | #117 | — | 2 | 1 | no | — |
| 194 | #117 | — | 178 | 1 | no | — |
| 195 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 196 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 197 | M32 | — | 89 | 2 | no | — |
| 198 | M32 | — | 178 | 1 | no | — |
| 199 | M32 | — | 178 | 1 | no | — |
| 200 | M32 | — | 178 | 1 | no | — |
| 201 | C2×C16 | — | 89 | 2 | no | — |
| 202 | C2×C16 | — | 178 | 1 | no | — |
| 203 | #108 | — | 178 | 1 | no | — |
| 204 | #108 | — | 178 | 1 | no | — |
| 205 | #107 | — | 2 | 2 | no | — |
| 206 | #107 | — | 2 | 1 | no | — |
| 207 | #107 | — | 178 | 1 | no | — |
| 208 | #106 | — | 2 | 2 | no | — |
| 209 | #106 | — | 2 | 1 | no | — |
| 210 | #106 | — | 178 | 1 | no | — |
| 211 | #106 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 212 | #106 | C22⋊D89 | 1 | 2 | yes | — |
| 213 | #106 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 214 | #104 | — | 356 | 2 | no | — |
| 215 | #41 | — | 2 | 1 | no | — |
| 216 | #41 | — | 2 | 1 | no | — |
| 217 | #41 | #13083553 | 1 | 2 | yes | — |
| 218 | #41 | #13083561 | 1 | 2 | yes | — |
| 219 | C2×Q8 | #13083562 | 1 | 1 | yes | characteristic |
| 220 | C2×D4 | — | 2 | 1 | no | — |
| 221 | C2×D4 | #13083562 | 1 | 1 | yes | characteristic |
| 222 | C22×C4 | — | 2 | 1 | no | — |
| 223 | C22×C4 | — | 4 | 2 | no | — |
| 224 | C22×C4 | — | 4 | 1 | no | — |
| 225 | C22×C4 | C2×C22⋊D89 | 1 | 1 | yes | characteristic |
| 226 | M16 | — | 2 | 1 | no | — |
| 227 | M16 | — | 2 | 2 | no | — |
| 228 | M16 | — | 2 | 2 | no | — |
| 229 | M16 | — | 356 | 1 | no | — |
| 230 | M16 | — | 356 | 1 | no | — |
| 231 | M16 | #13083553 | 1 | 2 | yes | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 2 | no | — |
| 234 | C2×C8 | — | 2 | 2 | no | — |
| 235 | C2×C8 | — | 356 | 1 | no | — |
| 236 | C2×C8 | C2×C4×D89 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | C2×C22⋊D89 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | C2×C22⋊D89 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | C2×Dic178 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | #13083574 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | #13083574 | 1 | 1 | yes | characteristic |
| 242 | C4⋊C4 | — | 2 | 1 | no | — |
| 243 | C4⋊C4 | — | 2 | 2 | no | — |
| 244 | C4⋊C4 | — | 2 | 1 | no | — |
| 245 | C4⋊C4 | — | 4 | 2 | no | — |
| 246 | C4⋊C4 | — | 4 | 1 | no | — |
| 247 | C4⋊C4 | — | 356 | 1 | no | — |
| 248 | C22⋊C4 | — | 4 | 2 | no | — |
| 249 | C22⋊C4 | — | 4 | 1 | no | — |
| 250 | C22⋊C4 | — | 356 | 1 | no | — |
| 251 | C42 | — | 4 | 2 | no | — |
| 252 | C42 | — | 4 | 1 | no | — |
| 253 | C42 | — | 356 | 1 | no | — |
| 254 | C16 | — | 178 | 2 | no | — |
| 255 | C16 | — | 178 | 2 | no | — |
| 256 | C23 | — | 2 | 1 | no | — |
| 257 | C23 | #15668038 | 1 | 1 | yes | characteristic |
| 258 | Q8 | — | 2 | 1 | no | — |
| 259 | Q8 | #15667943 | 1 | 2 | yes | — |
| 260 | D4 | — | 2 | 2 | no | — |
| 261 | D4 | — | 2 | 2 | no | — |
| 262 | D4 | — | 2 | 1 | no | — |
| 263 | D4 | #15667943 | 1 | 2 | yes | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 2 | 2 | no | — |
| 267 | C2×C4 | — | 4 | 1 | no | — |
| 268 | C2×C4 | — | 4 | 2 | no | — |
| 269 | C2×C4 | — | 4 | 1 | no | — |
| 270 | C2×C4 | — | 16 | 1 | no | — |
| 271 | C2×C4 | — | 16 | 2 | no | — |
| 272 | C2×C4 | — | 712 | 1 | no | — |
| 273 | C2×C4 | C2×#13083553 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | #15667981 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | #15667995 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | C2×#13083561 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C2×#13083562 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | #15668043 | 1 | 1 | yes | characteristic |
| 279 | C8 | — | 2 | 2 | no | — |
| 280 | C8 | — | 712 | 1 | no | — |
| 281 | C8 | C4×C22⋊D89 | 1 | 1 | yes | characteristic |
| 282 | C8 | C4×C22⋊D89 | 1 | 1 | yes | characteristic |
| 283 | C8 | #15667991 | 1 | 1 | yes | characteristic |
| 284 | C8 | #15667991 | 1 | 1 | yes | characteristic |
| 285 | C22 | — | 2 | 2 | no | — |
| 286 | C22 | — | 4 | 1 | no | — |
| 287 | C22 | — | 4 | 2 | no | — |
| 288 | C22 | #19151638 | 1 | 1 | yes | characteristic |
| 289 | C22 | #19151749 | 1 | 1 | yes | characteristic |
| 290 | C22 | #19151916 | 1 | 1 | yes | characteristic |
| 291 | C4 | — | 2 | 2 | no | — |
| 292 | C4 | — | 8 | 2 | no | — |
| 293 | C4 | — | 8 | 1 | no | — |
| 294 | C4 | — | 8 | 1 | no | — |
| 295 | C4 | — | 1424 | 1 | no | — |
| 296 | C4 | #19151471 | 1 | 1 | yes | characteristic |
| 297 | C4 | C2×#15667943 | 1 | 1 | yes | characteristic |
| 298 | C4 | #19151737 | 1 | 1 | yes | characteristic |
| 299 | C4 | #19151928 | 1 | 1 | yes | characteristic |
| 300 | C2 | — | 2 | 1 | no | — |
| 301 | C2 | — | 2 | 1 | no | — |
| 302 | C2 | — | 2 | 1 | no | — |
| 303 | C2 | — | 4 | 2 | no | — |
| 304 | C2 | #26020445 | 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