← #34204885
Subgroups — #34204885
Order 20224.
426 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4×Dic316 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#19098391 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#19098368 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#19098368 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #23764133 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #23761492 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C79×#1374 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C22×Dic316 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×Dic158 | — | 2 | 1 | no | — |
| 10 | C4×Dic316 | — | 2 | 2 | no | — |
| 11 | C4×Dic316 | — | 2 | 2 | no | — |
| 12 | C2×#14579095 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#14579095 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#14579095 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#14579086 | — | 2 | 1 | no | — |
| 16 | #19098461 | — | 2 | 1 | no | — |
| 17 | #19098391 | — | 2 | 2 | no | — |
| 18 | #19098368 | — | 2 | 2 | no | — |
| 19 | #19098368 | C4 | 1 | 4 | yes | complemented |
| 20 | #19098326 | — | 2 | 1 | no | — |
| 21 | C22×C16×C79 | C22 | 1 | 1 | yes | characteristic |
| 22 | C22×C16×C79 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×C4×C8×C79 | C22 | 1 | 1 | yes | characteristic |
| 24 | C22×Dic158 | — | 2 | 1 | no | — |
| 25 | C2×Dic316 | — | 2 | 1 | no | — |
| 26 | C2×Dic316 | — | 2 | 1 | no | — |
| 27 | C2×Dic316 | — | 2 | 1 | no | — |
| 28 | C2×Dic316 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×Dic316 | D4 | 1 | 2 | yes | — |
| 30 | C2×Dic316 | Q8 | 1 | 2 | yes | — |
| 31 | C4×Dic158 | — | 2 | 4 | no | — |
| 32 | C4×Dic158 | — | 4 | 2 | no | — |
| 33 | C2×C4⋊Dic79 | — | 2 | 1 | no | — |
| 34 | C2×C4⋊Dic79 | — | 2 | 1 | no | — |
| 35 | C2×C4⋊Dic79 | — | 2 | 1 | no | — |
| 36 | C2×Dic79⋊C4 | — | 4 | 1 | no | — |
| 37 | C2×Dic79⋊C4 | — | 4 | 1 | no | — |
| 38 | C2×C4×Dic79 | — | 4 | 1 | no | — |
| 39 | #14579095 | — | 2 | 1 | no | — |
| 40 | #14579095 | — | 2 | 1 | no | — |
| 41 | #14579095 | — | 2 | 1 | no | — |
| 42 | #14579095 | — | 2 | 1 | no | — |
| 43 | #14579095 | C2×C4 | 1 | 2 | yes | — |
| 44 | #14579095 | D4 | 1 | 2 | yes | — |
| 45 | #14579086 | — | 2 | 4 | no | — |
| 46 | C79⋊#95 | — | 4 | 1 | no | — |
| 47 | C22×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 48 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C42×C79 | D4 | 1 | 1 | yes | characteristic |
| 50 | C2×C16×C79 | — | 2 | 1 | no | — |
| 51 | C2×C16×C79 | — | 2 | 1 | no | — |
| 52 | C2×C16×C79 | — | 2 | 1 | no | — |
| 53 | C2×C16×C79 | — | 2 | 1 | no | — |
| 54 | C2×C16×C79 | — | 2 | 1 | no | — |
| 55 | C2×C16×C79 | C2×C4 | 1 | 2 | yes | — |
| 56 | C4×C8×C79 | — | 2 | 1 | no | — |
| 57 | C4×C8×C79 | — | 2 | 1 | no | — |
| 58 | C2×Dic158 | — | 2 | 2 | no | — |
| 59 | C2×Dic158 | — | 2 | 2 | no | — |
| 60 | C2×Dic158 | — | 2 | 2 | no | — |
| 61 | C2×Dic158 | — | 4 | 1 | no | — |
| 62 | C2×Dic158 | — | 4 | 1 | no | — |
| 63 | C2×Dic158 | — | 4 | 1 | no | — |
| 64 | C22×Dic79 | — | 4 | 1 | no | — |
| 65 | C22×Dic79 | — | 4 | 1 | no | — |
| 66 | C22×Dic79 | — | 4 | 1 | no | — |
| 67 | C22×Dic79 | — | 4 | 1 | no | — |
| 68 | Dic316 | — | 2 | 2 | no | — |
| 69 | Dic316 | — | 2 | 2 | no | — |
| 70 | Dic316 | — | 2 | 2 | no | — |
| 71 | Dic316 | C4⋊C4 | 1 | 4 | yes | complemented |
| 72 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 73 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 74 | C4⋊Dic79 | — | 4 | 1 | no | — |
| 75 | C4⋊Dic79 | — | 4 | 1 | no | — |
| 76 | C4⋊Dic79 | — | 4 | 1 | no | — |
| 77 | C4⋊Dic79 | — | 4 | 1 | no | — |
| 78 | Dic79⋊C4 | — | 4 | 4 | no | — |
| 79 | Dic79⋊C4 | — | 8 | 2 | no | — |
| 80 | C4×Dic79 | — | 4 | 4 | no | — |
| 81 | C22×C4×C79 | D8 | 1 | 1 | yes | characteristic |
| 82 | C22×C4×C79 | D8 | 1 | 1 | yes | characteristic |
| 83 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C79 | — | 2 | 1 | no | — |
| 85 | C2×C8×C79 | — | 2 | 1 | no | — |
| 86 | C2×C8×C79 | — | 2 | 1 | no | — |
| 87 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C79 | C2×Q8 | 1 | 1 | yes | characteristic |
| 91 | C2×C8×C79 | #41 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C79 | #41 | 1 | 1 | yes | characteristic |
| 93 | C42×C79 | — | 2 | 1 | no | — |
| 94 | C42×C79 | — | 2 | 1 | no | — |
| 95 | C16×C79 | — | 2 | 2 | no | — |
| 96 | C16×C79 | — | 4 | 1 | no | — |
| 97 | C23×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 98 | C2×C4×C79 | — | 2 | 1 | no | — |
| 99 | C2×C4×C79 | — | 2 | 1 | no | — |
| 100 | C2×C4×C79 | — | 2 | 1 | no | — |
| 101 | C2×C4×C79 | — | 2 | 1 | no | — |
| 102 | C2×C4×C79 | — | 2 | 1 | no | — |
| 103 | C2×C4×C79 | SD32 | 1 | 2 | yes | — |
| 104 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C79 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C79 | #135 | 1 | 1 | yes | characteristic |
| 108 | C2×C4×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 109 | C2×C4×C79 | #137 | 1 | 1 | yes | characteristic |
| 110 | C2×Dic79 | — | 4 | 2 | no | — |
| 111 | C2×Dic79 | — | 4 | 2 | no | — |
| 112 | C2×Dic79 | — | 4 | 2 | no | — |
| 113 | C2×Dic79 | — | 4 | 2 | no | — |
| 114 | C2×Dic79 | — | 4 | 2 | no | — |
| 115 | C2×Dic79 | — | 8 | 1 | no | — |
| 116 | C2×Dic79 | — | 8 | 1 | no | — |
| 117 | C2×Dic79 | — | 8 | 1 | no | — |
| 118 | C2×Dic79 | — | 8 | 1 | no | — |
| 119 | C2×Dic79 | — | 8 | 1 | no | — |
| 120 | C2×Dic79 | — | 8 | 1 | no | — |
| 121 | C2×Dic79 | — | 8 | 1 | no | — |
| 122 | Dic158 | — | 2 | 4 | no | — |
| 123 | Dic158 | — | 4 | 2 | no | — |
| 124 | Dic158 | — | 4 | 2 | no | — |
| 125 | Dic158 | — | 4 | 2 | no | — |
| 126 | C8×C79 | — | 4 | 1 | no | — |
| 127 | C8×C79 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 128 | C8×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 129 | C8×C79 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 130 | C8×C79 | #123 | 1 | 1 | yes | characteristic |
| 131 | C22×C79 | C4×D8 | 1 | 1 | yes | characteristic |
| 132 | C22×C79 | #447 | 1 | 1 | yes | characteristic |
| 133 | C22×C79 | #474 | 1 | 1 | yes | characteristic |
| 134 | C22×C79 | C2×SD32 | 1 | 1 | yes | characteristic |
| 135 | C22×C79 | #508 | 1 | 1 | yes | characteristic |
| 136 | C22×C79 | #510 | 1 | 1 | yes | characteristic |
| 137 | C22×C79 | #510 | 1 | 1 | yes | characteristic |
| 138 | C4×C79 | — | 2 | 2 | no | — |
| 139 | C4×C79 | — | 4 | 1 | no | — |
| 140 | C4×C79 | #385 | 1 | 1 | yes | characteristic |
| 141 | C4×C79 | #440 | 1 | 1 | yes | characteristic |
| 142 | C4×C79 | #452 | 1 | 1 | yes | characteristic |
| 143 | C4×C79 | #478 | 1 | 1 | yes | characteristic |
| 144 | Dic79 | — | 4 | 4 | no | — |
| 145 | Dic79 | — | 8 | 2 | no | — |
| 146 | Dic79 | — | 8 | 2 | no | — |
| 147 | Dic79 | — | 16 | 1 | no | — |
| 148 | #12938 | — | 79 | 1 | no | maximal, Sylow |
| 149 | C2×C79 | #1862 | 1 | 1 | yes | characteristic |
| 150 | C2×C79 | C4×SD32 | 1 | 1 | yes | characteristic |
| 151 | C2×C79 | #2176 | 1 | 1 | yes | characteristic |
| 152 | C2×C79 | #2185 | 1 | 1 | yes | characteristic |
| 153 | C2×C79 | #2189 | 1 | 1 | yes | characteristic |
| 154 | C2×C79 | #2225 | 1 | 1 | yes | characteristic |
| 155 | C2×C79 | #2229 | 1 | 1 | yes | characteristic |
| 156 | C2×C4×Q16 | — | 79 | 1 | no | — |
| 157 | C2×#367 | — | 79 | 1 | no | — |
| 158 | C2×#358 | — | 79 | 1 | no | — |
| 159 | C2×#358 | — | 79 | 1 | no | — |
| 160 | #1942 | — | 79 | 1 | no | — |
| 161 | #1380 | — | 79 | 1 | no | — |
| 162 | #1374 | D79 | 1 | 1 | yes | characteristic |
| 163 | C79 | #12938 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 164 | C22×Q16 | — | 79 | 1 | no | — |
| 165 | C2×C4×Q8 | — | 158 | 1 | no | — |
| 166 | C22×C16 | C2×D79 | 1 | 1 | yes | characteristic |
| 167 | C22×C16 | C2×D79 | 1 | 1 | yes | characteristic |
| 168 | C4×Q16 | — | 158 | 2 | no | — |
| 169 | C4×Q16 | — | 158 | 2 | no | — |
| 170 | C2×#107 | — | 79 | 1 | no | — |
| 171 | C2×#107 | — | 79 | 1 | no | — |
| 172 | C2×#107 | — | 79 | 1 | no | — |
| 173 | C2×#103 | — | 158 | 1 | no | — |
| 174 | C2×C4×C8 | C2×D79 | 1 | 1 | yes | characteristic |
| 175 | #389 | — | 158 | 1 | no | — |
| 176 | #367 | — | 158 | 2 | no | — |
| 177 | #358 | — | 79 | 4 | no | — |
| 178 | #358 | — | 158 | 2 | no | — |
| 179 | #340 | — | 158 | 1 | no | — |
| 180 | C22×Q8 | — | 158 | 1 | no | — |
| 181 | C2×Q16 | — | 79 | 2 | no | — |
| 182 | C2×Q16 | — | 79 | 2 | no | — |
| 183 | C2×Q16 | — | 79 | 2 | no | — |
| 184 | C2×Q16 | — | 158 | 1 | no | — |
| 185 | C2×Q16 | — | 158 | 1 | no | — |
| 186 | C2×Q16 | — | 158 | 1 | no | — |
| 187 | C22×C8 | D316 | 1 | 1 | yes | characteristic |
| 188 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 189 | C4×Q8 | — | 316 | 2 | no | — |
| 190 | C4×Q8 | — | 158 | 4 | no | — |
| 191 | C2×C4⋊C4 | — | 316 | 1 | no | — |
| 192 | C2×C4⋊C4 | — | 316 | 1 | no | — |
| 193 | C2×C4⋊C4 | — | 158 | 1 | no | — |
| 194 | C2×C4⋊C4 | — | 158 | 1 | no | — |
| 195 | C2×C4⋊C4 | — | 158 | 1 | no | — |
| 196 | C2×C42 | — | 316 | 1 | no | — |
| 197 | C2×C42 | D316 | 1 | 1 | yes | characteristic |
| 198 | C2×C16 | — | 2 | 1 | no | — |
| 199 | C2×C16 | — | 2 | 1 | no | — |
| 200 | C2×C16 | — | 2 | 1 | no | — |
| 201 | C2×C16 | — | 2 | 1 | no | — |
| 202 | C2×C16 | — | 2 | 1 | no | — |
| 203 | C2×C16 | C4×D79 | 1 | 2 | yes | — |
| 204 | #107 | — | 79 | 2 | no | — |
| 205 | #107 | — | 79 | 2 | no | — |
| 206 | #107 | — | 158 | 1 | no | — |
| 207 | #107 | — | 158 | 1 | no | — |
| 208 | #107 | — | 158 | 1 | no | — |
| 209 | #107 | — | 158 | 1 | no | — |
| 210 | #103 | — | 158 | 4 | no | — |
| 211 | C4×C8 | — | 2 | 1 | no | — |
| 212 | C4×C8 | — | 2 | 1 | no | — |
| 213 | #95 | — | 316 | 1 | no | — |
| 214 | C2×Q8 | — | 316 | 1 | no | — |
| 215 | C2×Q8 | — | 316 | 1 | no | — |
| 216 | C2×Q8 | — | 316 | 1 | no | — |
| 217 | C2×Q8 | — | 158 | 2 | no | — |
| 218 | C2×Q8 | — | 158 | 2 | no | — |
| 219 | C2×Q8 | — | 158 | 2 | no | — |
| 220 | C22×C4 | — | 316 | 1 | no | — |
| 221 | C22×C4 | — | 316 | 1 | no | — |
| 222 | C22×C4 | — | 316 | 1 | no | — |
| 223 | C22×C4 | — | 316 | 1 | no | — |
| 224 | C22×C4 | D632 | 1 | 1 | yes | characteristic |
| 225 | C22×C4 | D632 | 1 | 1 | yes | characteristic |
| 226 | C22×C4 | C2×D316 | 1 | 1 | yes | characteristic |
| 227 | Q16 | — | 79 | 4 | no | — |
| 228 | Q16 | — | 158 | 2 | no | — |
| 229 | Q16 | — | 158 | 2 | no | — |
| 230 | Q16 | — | 158 | 2 | no | — |
| 231 | C2×C8 | — | 2 | 1 | no | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 1 | no | — |
| 234 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | D4×D79 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | D4×D79 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | Q8×D79 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | #11929623 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | #11929624 | 1 | 1 | yes | characteristic |
| 240 | C4⋊C4 | — | 316 | 4 | no | — |
| 241 | C4⋊C4 | — | 316 | 1 | no | — |
| 242 | C4⋊C4 | — | 316 | 1 | no | — |
| 243 | C4⋊C4 | — | 316 | 1 | no | — |
| 244 | C4⋊C4 | — | 316 | 1 | no | — |
| 245 | C4⋊C4 | — | 632 | 2 | no | — |
| 246 | C4⋊C4 | — | 158 | 2 | no | — |
| 247 | C4⋊C4 | — | 158 | 2 | no | — |
| 248 | C42 | — | 2 | 1 | no | — |
| 249 | C42 | — | 2 | 1 | no | — |
| 250 | C42 | — | 316 | 4 | no | — |
| 251 | C16 | — | 2 | 2 | no | — |
| 252 | C16 | — | 4 | 1 | no | — |
| 253 | C23 | C2×D632 | 1 | 1 | yes | characteristic |
| 254 | Q8 | — | 316 | 2 | no | — |
| 255 | Q8 | — | 316 | 2 | no | — |
| 256 | Q8 | — | 316 | 2 | no | — |
| 257 | Q8 | — | 158 | 4 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 2 | 1 | no | — |
| 260 | C2×C4 | — | 2 | 1 | no | — |
| 261 | C2×C4 | — | 2 | 1 | no | — |
| 262 | C2×C4 | — | 2 | 1 | no | — |
| 263 | C2×C4 | — | 316 | 2 | no | — |
| 264 | C2×C4 | — | 316 | 2 | no | — |
| 265 | C2×C4 | — | 316 | 2 | no | — |
| 266 | C2×C4 | — | 316 | 2 | no | — |
| 267 | C2×C4 | — | 316 | 2 | no | — |
| 268 | C2×C4 | — | 632 | 1 | no | — |
| 269 | C2×C4 | — | 632 | 1 | no | — |
| 270 | C2×C4 | — | 632 | 1 | no | — |
| 271 | C2×C4 | — | 632 | 1 | no | — |
| 272 | C2×C4 | — | 632 | 1 | no | — |
| 273 | C2×C4 | — | 632 | 1 | no | — |
| 274 | C2×C4 | — | 632 | 1 | no | — |
| 275 | C2×C4 | #14579106 | 1 | 2 | yes | — |
| 276 | C2×C4 | C4×D316 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | #14579131 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | #14579141 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | #14579183 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | #14579189 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #14579194 | 1 | 1 | yes | characteristic |
| 282 | C8 | — | 4 | 1 | no | — |
| 283 | C8 | C4⋊C4×D79 | 1 | 1 | yes | characteristic |
| 284 | C8 | (C22⋊D79)⋊C4 | 1 | 1 | yes | characteristic |
| 285 | C8 | #14579139 | 1 | 1 | yes | characteristic |
| 286 | C8 | #14579143 | 1 | 1 | yes | characteristic |
| 287 | C22 | C4×D632 | 1 | 1 | yes | characteristic |
| 288 | C22 | #19098699 | 1 | 1 | yes | characteristic |
| 289 | C22 | #19098844 | 1 | 1 | yes | characteristic |
| 290 | C22 | C2×#14579106 | 1 | 1 | yes | characteristic |
| 291 | C22 | #19098995 | 1 | 1 | yes | characteristic |
| 292 | C22 | #19099006 | 1 | 1 | yes | characteristic |
| 293 | C22 | #19099006 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 2 | no | — |
| 295 | C4 | — | 4 | 1 | no | — |
| 296 | C4 | — | 316 | 4 | no | — |
| 297 | C4 | — | 632 | 2 | no | — |
| 298 | C4 | — | 632 | 2 | no | — |
| 299 | C4 | — | 1264 | 1 | no | — |
| 300 | C4 | #19098437 | 1 | 1 | yes | characteristic |
| 301 | C4 | #19098665 | 1 | 1 | yes | characteristic |
| 302 | C4 | #19098727 | 1 | 1 | yes | characteristic |
| 303 | C4 | #19098864 | 1 | 1 | yes | characteristic |
| 304 | C2 | #23763650 | 1 | 1 | yes | characteristic |
| 305 | C2 | C4×#14579106 | 1 | 1 | yes | characteristic |
| 306 | C2 | #23765272 | 1 | 1 | yes | characteristic |
| 307 | C2 | #23765319 | 1 | 1 | yes | characteristic |
| 308 | C2 | #23765342 | 1 | 1 | yes | characteristic |
| 309 | C2 | #23765543 | 1 | 1 | yes | characteristic |
| 310 | C2 | #23765565 | 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