Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
1002 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #23777058 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23763318 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #23761957 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C79×#1780 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#14579209 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #19099232 | — | 2 | 4 | no | — |
| 7 | #19099220 | — | 2 | 4 | no | — |
| 8 | #19099130 | — | 2 | 2 | no | — |
| 9 | #19099130 | C4 | 1 | 2 | yes | complemented |
| 10 | #19099074 | — | 2 | 2 | no | — |
| 11 | #19099074 | C22 | 1 | 2 | yes | — |
| 12 | #19099041 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#14579076 | — | 2 | 2 | no | — |
| 14 | #19098426 | C22 | 1 | 2 | yes | — |
| 15 | #19098301 | — | 2 | 4 | no | — |
| 16 | #19098298 | — | 2 | 4 | no | — |
| 17 | C22×#41×C79 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×C22⋊C8×C79 | — | 2 | 2 | no | — |
| 19 | C2×C22⋊C8×C79 | C22 | 1 | 2 | yes | — |
| 20 | #14579209 | — | 4 | 4 | no | — |
| 21 | C2×#11929625 | — | 2 | 2 | no | — |
| 22 | C2×#11929624 | — | 2 | 2 | no | — |
| 23 | C2×#11929624 | C2×C4 | 1 | 2 | yes | — |
| 24 | C2×Q8×D79 | — | 2 | 2 | no | — |
| 25 | #14579157 | — | 2 | 4 | no | — |
| 26 | #14579157 | — | 2 | 2 | no | — |
| 27 | #14579148 | — | 2 | 4 | no | — |
| 28 | #14579148 | — | 2 | 2 | no | — |
| 29 | #14579144 | — | 2 | 2 | no | — |
| 30 | #14579140 | — | 2 | 4 | no | — |
| 31 | #14579140 | D4 | 1 | 4 | yes | — |
| 32 | #14579135 | — | 2 | 4 | no | — |
| 33 | #14579135 | D4 | 1 | 4 | yes | — |
| 34 | #14579132 | — | 2 | 8 | no | — |
| 35 | #14579132 | — | 2 | 4 | no | — |
| 36 | #14579130 | — | 2 | 4 | no | — |
| 37 | #14579130 | — | 2 | 2 | no | — |
| 38 | Q8×Dic79 | — | 2 | 4 | no | — |
| 39 | C4×C22⋊D79 | — | 2 | 8 | no | — |
| 40 | C4×C22⋊D79 | — | 2 | 4 | no | — |
| 41 | #14579120 | — | 2 | 2 | no | — |
| 42 | #14579120 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×C22⋊Dic79 | — | 2 | 2 | no | — |
| 44 | #14579076 | — | 4 | 4 | no | — |
| 45 | C79⋊#95 | — | 2 | 4 | no | — |
| 46 | C2×#41×C79 | — | 2 | 4 | no | — |
| 47 | C2×#41×C79 | — | 4 | 4 | no | — |
| 48 | C22×Q8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C22×D4×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C22×D4×C79 | D4 | 1 | 2 | yes | — |
| 51 | C23×C4×C79 | D4 | 1 | 2 | yes | — |
| 52 | C23×C4×C79 | C23 | 1 | 1 | yes | characteristic |
| 53 | C22×C8×C79 | — | 2 | 2 | no | — |
| 54 | C22⋊C8×C79 | — | 2 | 4 | no | — |
| 55 | C22⋊C8×C79 | — | 4 | 4 | no | — |
| 56 | #11929625 | — | 4 | 8 | no | — |
| 57 | #11929624 | — | 4 | 4 | no | — |
| 58 | #11929624 | — | 4 | 4 | no | — |
| 59 | #11929624 | — | 4 | 4 | no | — |
| 60 | C2×Dic158 | — | 2 | 2 | no | — |
| 61 | C2×Dic158 | C22⋊C4 | 1 | 2 | yes | — |
| 62 | Q8×D79 | — | 4 | 8 | no | — |
| 63 | C2×D316 | — | 2 | 2 | no | — |
| 64 | C2×D316 | C22⋊C4 | 1 | 2 | yes | — |
| 65 | C2×C22⋊D79 | — | 2 | 2 | no | — |
| 66 | C2×C22⋊D79 | — | 2 | 8 | no | — |
| 67 | C2×C22⋊D79 | — | 2 | 4 | no | — |
| 68 | C22×Dic79 | — | 4 | 2 | no | — |
| 69 | C2×C4×D79 | — | 2 | 4 | no | — |
| 70 | C2×C4×D79 | — | 2 | 2 | no | — |
| 71 | C4⋊Dic79 | — | 2 | 4 | no | — |
| 72 | C4⋊Dic79 | C22⋊C4 | 1 | 4 | yes | — |
| 73 | Dic79⋊C4 | — | 2 | 4 | no | — |
| 74 | C22⋊Dic79 | — | 2 | 8 | no | — |
| 75 | C22⋊Dic79 | — | 2 | 2 | no | — |
| 76 | C22⋊Dic79 | — | 2 | 4 | no | — |
| 77 | C22⋊Dic79 | — | 4 | 4 | no | — |
| 78 | (C2×D79)⋊C4 | — | 2 | 4 | no | — |
| 79 | C4×Dic79 | — | 2 | 4 | no | — |
| 80 | C4×Dic79 | — | 2 | 2 | no | — |
| 81 | C24×C79 | C22⋊C4 | 1 | 2 | yes | — |
| 82 | C24×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | #41×C79 | — | 4 | 16 | no | — |
| 84 | C2×Q8×C79 | — | 2 | 4 | no | — |
| 85 | C2×Q8×C79 | — | 2 | 2 | no | — |
| 86 | C2×D4×C79 | — | 2 | 4 | no | — |
| 87 | C2×D4×C79 | — | 2 | 4 | no | — |
| 88 | C2×D4×C79 | — | 2 | 2 | no | — |
| 89 | C2×D4×C79 | — | 4 | 2 | no | — |
| 90 | C2×D4×C79 | — | 4 | 2 | no | — |
| 91 | C22×C4×C79 | — | 2 | 4 | no | — |
| 92 | C22×C4×C79 | — | 2 | 4 | no | — |
| 93 | C22×C4×C79 | — | 4 | 2 | no | — |
| 94 | C22×C4×C79 | — | 4 | 2 | no | — |
| 95 | C22×C4×C79 | C22⋊C4 | 1 | 2 | yes | — |
| 96 | C22×C4×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 97 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C22×C4×C79 | C2×D4 | 1 | 4 | yes | — |
| 99 | C2×C8×C79 | — | 4 | 4 | no | — |
| 100 | C2×C8×C79 | — | 4 | 2 | no | — |
| 101 | C23×C79 | — | 2 | 2 | no | — |
| 102 | C23×C79 | — | 2 | 2 | no | — |
| 103 | C23×C79 | — | 2 | 2 | no | — |
| 104 | C23×C79 | — | 2 | 2 | no | — |
| 105 | C23×C79 | — | 4 | 2 | no | — |
| 106 | C23×C79 | — | 4 | 4 | no | — |
| 107 | C23×C79 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 108 | C23×C79 | #120 | 1 | 2 | yes | — |
| 109 | C22×D79 | — | 2 | 2 | no | — |
| 110 | Q8×C79 | — | 2 | 8 | no | — |
| 111 | D4×C79 | — | 4 | 8 | no | — |
| 112 | D4×C79 | — | 4 | 4 | no | — |
| 113 | C2×C4×C79 | — | 2 | 2 | no | — |
| 114 | C2×C4×C79 | — | 2 | 2 | no | — |
| 115 | C2×C4×C79 | — | 2 | 2 | no | — |
| 116 | C2×C4×C79 | — | 2 | 2 | no | — |
| 117 | C2×C4×C79 | — | 4 | 4 | no | — |
| 118 | C2×C4×C79 | — | 4 | 8 | no | — |
| 119 | C2×C4×C79 | #99 | 1 | 4 | yes | — |
| 120 | C2×C4×C79 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 121 | C2×C4×C79 | #120 | 1 | 2 | yes | — |
| 122 | C22⋊D79 | — | 4 | 4 | no | — |
| 123 | C22⋊D79 | — | 8 | 4 | no | — |
| 124 | C22⋊D79 | — | 8 | 8 | no | — |
| 125 | C2×Dic79 | — | 2 | 4 | no | — |
| 126 | C2×Dic79 | — | 2 | 2 | no | — |
| 127 | C2×Dic79 | — | 8 | 2 | no | — |
| 128 | C2×Dic79 | — | 8 | 4 | no | — |
| 129 | D316 | — | 4 | 2 | no | — |
| 130 | D316 | — | 4 | 4 | no | — |
| 131 | C4×D79 | — | 4 | 8 | no | — |
| 132 | C4×D79 | — | 4 | 4 | no | — |
| 133 | Dic158 | — | 4 | 4 | no | — |
| 134 | Dic158 | — | 4 | 2 | no | — |
| 135 | C8×C79 | — | 8 | 2 | no | — |
| 136 | C22×C79 | — | 2 | 2 | no | — |
| 137 | C22×C79 | — | 2 | 4 | no | — |
| 138 | C22×C79 | — | 2 | 2 | no | — |
| 139 | C22×C79 | — | 4 | 4 | no | — |
| 140 | C22×C79 | — | 4 | 4 | no | — |
| 141 | C22×C79 | — | 4 | 2 | no | — |
| 142 | C22×C79 | — | 8 | 2 | no | — |
| 143 | C22×C79 | — | 8 | 2 | no | — |
| 144 | C22×C79 | #379 | 1 | 1 | yes | characteristic |
| 145 | C22×C79 | C2×#99 | 1 | 2 | yes | — |
| 146 | C2×D79 | — | 4 | 4 | no | — |
| 147 | C2×D79 | — | 4 | 2 | no | — |
| 148 | C4×C79 | — | 2 | 2 | no | — |
| 149 | C4×C79 | — | 2 | 4 | no | — |
| 150 | C4×C79 | — | 2 | 2 | no | — |
| 151 | Dic79 | — | 4 | 4 | no | — |
| 152 | Dic79 | — | 4 | 2 | no | — |
| 153 | Dic79 | — | 16 | 2 | no | — |
| 154 | #8464 | — | 79 | 1 | no | maximal, Sylow |
| 155 | C2×C79 | — | 2 | 2 | no | — |
| 156 | C2×C79 | — | 4 | 2 | no | — |
| 157 | C2×C79 | — | 8 | 2 | no | — |
| 158 | C2×C79 | #1781 | 1 | 1 | yes | characteristic |
| 159 | C2×C79 | #1794 | 1 | 2 | yes | — |
| 160 | D79 | — | 8 | 2 | no | — |
| 161 | #3487 | — | 79 | 1 | no | — |
| 162 | #1782 | — | 79 | 1 | no | — |
| 163 | #1780 | D79 | 1 | 1 | yes | characteristic, complemented |
| 164 | #1516 | — | 79 | 4 | no | — |
| 165 | C79 | #8464 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 166 | C2×2−1+4 | — | 79 | 1 | no | — |
| 167 | C22×#41 | C2×D79 | 1 | 1 | yes | characteristic |
| 168 | #550 | — | 158 | 4 | no | — |
| 169 | #548 | — | 158 | 4 | no | — |
| 170 | #535 | — | 79 | 2 | no | — |
| 171 | #535 | — | 158 | 2 | no | — |
| 172 | #525 | — | 79 | 2 | no | — |
| 173 | #525 | — | 158 | 2 | no | — |
| 174 | #519 | — | 79 | 1 | no | — |
| 175 | C2×#99 | — | 158 | 2 | no | — |
| 176 | C2×C22⋊C8 | — | 2 | 2 | no | — |
| 177 | C2×C22⋊C8 | C2×D79 | 1 | 2 | yes | — |
| 178 | #381 | — | 79 | 2 | no | — |
| 179 | #328 | — | 158 | 4 | no | — |
| 180 | #327 | — | 158 | 4 | no | — |
| 181 | 2−1+4 | — | 316 | 4 | no | — |
| 182 | C2×#41 | — | 2 | 4 | no | — |
| 183 | C2×#41 | — | 4 | 4 | no | — |
| 184 | C2×#41 | — | 79 | 2 | no | — |
| 185 | C2×#41 | — | 158 | 2 | no | — |
| 186 | C2×#41 | — | 158 | 2 | no | — |
| 187 | C22×Q8 | — | 158 | 2 | no | — |
| 188 | C22×Q8 | C4×D79 | 1 | 1 | yes | characteristic |
| 189 | C22×D4 | C4×D79 | 1 | 1 | yes | characteristic |
| 190 | C22×D4 | D316 | 1 | 2 | yes | — |
| 191 | C23×C4 | C22⋊D79 | 1 | 2 | yes | — |
| 192 | C23×C4 | C22×D79 | 1 | 1 | yes | characteristic |
| 193 | C22×C8 | — | 2 | 2 | no | — |
| 194 | #127 | — | 158 | 2 | no | — |
| 195 | #127 | — | 158 | 4 | no | — |
| 196 | #124 | — | 158 | 2 | no | — |
| 197 | #124 | — | 158 | 4 | no | — |
| 198 | #123 | — | 158 | 2 | no | — |
| 199 | C22⋊Q8 | — | 79 | 4 | no | — |
| 200 | C22⋊Q8 | — | 158 | 4 | no | — |
| 201 | #121 | — | 79 | 4 | no | — |
| 202 | #121 | — | 158 | 4 | no | — |
| 203 | #121 | — | 158 | 4 | no | — |
| 204 | #121 | — | 158 | 8 | no | — |
| 205 | #120 | — | 158 | 4 | no | — |
| 206 | #120 | — | 158 | 2 | no | — |
| 207 | C4×Q8 | — | 158 | 4 | no | — |
| 208 | C4×D4 | — | 158 | 8 | no | — |
| 209 | C4×D4 | — | 158 | 4 | no | — |
| 210 | #117 | — | 79 | 2 | no | — |
| 211 | #117 | — | 158 | 2 | no | — |
| 212 | C2×C22⋊C4 | — | 158 | 2 | no | — |
| 213 | #99 | — | 316 | 4 | no | — |
| 214 | C22⋊C8 | — | 2 | 4 | no | — |
| 215 | C22⋊C8 | — | 4 | 4 | no | — |
| 216 | #95 | — | 158 | 4 | no | — |
| 217 | C24 | (C2×D79)⋊C4 | 1 | 2 | yes | — |
| 218 | C24 | C2×D316 | 1 | 1 | yes | characteristic |
| 219 | #41 | — | 4 | 16 | no | — |
| 220 | #41 | — | 316 | 4 | no | — |
| 221 | #41 | — | 316 | 4 | no | — |
| 222 | #41 | — | 316 | 8 | no | — |
| 223 | #41 | — | 316 | 4 | no | — |
| 224 | C2×Q8 | — | 2 | 4 | no | — |
| 225 | C2×Q8 | — | 2 | 2 | no | — |
| 226 | C2×Q8 | — | 316 | 8 | no | — |
| 227 | C2×Q8 | — | 79 | 2 | no | — |
| 228 | C2×Q8 | — | 158 | 2 | no | — |
| 229 | C2×D4 | — | 2 | 4 | no | — |
| 230 | C2×D4 | — | 2 | 4 | no | — |
| 231 | C2×D4 | — | 2 | 2 | no | — |
| 232 | C2×D4 | — | 4 | 2 | no | — |
| 233 | C2×D4 | — | 4 | 2 | no | — |
| 234 | C2×D4 | — | 79 | 2 | no | — |
| 235 | C2×D4 | — | 158 | 2 | no | — |
| 236 | C2×D4 | — | 158 | 2 | no | — |
| 237 | C2×D4 | — | 158 | 8 | no | — |
| 238 | C2×D4 | — | 158 | 4 | no | — |
| 239 | C22×C4 | — | 2 | 4 | no | — |
| 240 | C22×C4 | — | 2 | 4 | no | — |
| 241 | C22×C4 | — | 4 | 2 | no | — |
| 242 | C22×C4 | — | 4 | 2 | no | — |
| 243 | C22×C4 | — | 316 | 2 | no | — |
| 244 | C22×C4 | — | 158 | 4 | no | — |
| 245 | C22×C4 | — | 158 | 2 | no | — |
| 246 | C22×C4 | (C2×D79)⋊C4 | 1 | 2 | yes | — |
| 247 | C22×C4 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 248 | C22×C4 | D4×D79 | 1 | 4 | yes | — |
| 249 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | — | 4 | 4 | no | — |
| 251 | C2×C8 | — | 4 | 2 | no | — |
| 252 | C4⋊C4 | — | 79 | 4 | no | — |
| 253 | C4⋊C4 | — | 158 | 4 | no | — |
| 254 | C4⋊C4 | — | 158 | 4 | no | — |
| 255 | C22⋊C4 | — | 316 | 4 | no | — |
| 256 | C22⋊C4 | — | 158 | 8 | no | — |
| 257 | C22⋊C4 | — | 158 | 2 | no | — |
| 258 | C22⋊C4 | — | 158 | 4 | no | — |
| 259 | C22⋊C4 | — | 158 | 4 | no | — |
| 260 | C42 | — | 158 | 4 | no | — |
| 261 | C42 | — | 158 | 2 | no | — |
| 262 | C23 | — | 2 | 2 | no | — |
| 263 | C23 | — | 2 | 2 | no | — |
| 264 | C23 | — | 2 | 2 | no | — |
| 265 | C23 | — | 2 | 2 | no | — |
| 266 | C23 | — | 4 | 2 | no | — |
| 267 | C23 | — | 4 | 4 | no | — |
| 268 | C23 | — | 158 | 2 | no | — |
| 269 | C23 | C2×(C2×D79)⋊C4 | 1 | 1 | yes | characteristic |
| 270 | C23 | #14579131 | 1 | 2 | yes | — |
| 271 | Q8 | — | 2 | 8 | no | — |
| 272 | Q8 | — | 316 | 2 | no | — |
| 273 | Q8 | — | 316 | 4 | no | — |
| 274 | D4 | — | 4 | 8 | no | — |
| 275 | D4 | — | 4 | 4 | no | — |
| 276 | D4 | — | 316 | 2 | no | — |
| 277 | D4 | — | 316 | 4 | no | — |
| 278 | D4 | — | 316 | 4 | no | — |
| 279 | D4 | — | 632 | 4 | no | — |
| 280 | D4 | — | 632 | 8 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 2 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 2 | 2 | no | — |
| 285 | C2×C4 | — | 4 | 4 | no | — |
| 286 | C2×C4 | — | 4 | 8 | no | — |
| 287 | C2×C4 | — | 316 | 8 | no | — |
| 288 | C2×C4 | — | 316 | 4 | no | — |
| 289 | C2×C4 | — | 632 | 2 | no | — |
| 290 | C2×C4 | — | 632 | 4 | no | — |
| 291 | C2×C4 | — | 158 | 4 | no | — |
| 292 | C2×C4 | — | 158 | 2 | no | — |
| 293 | C2×C4 | #14579075 | 1 | 4 | yes | — |
| 294 | C2×C4 | C22⋊C4×D79 | 1 | 2 | yes | — |
| 295 | C2×C4 | #14579129 | 1 | 2 | yes | — |
| 296 | C8 | — | 8 | 2 | no | — |
| 297 | C22 | — | 2 | 2 | no | — |
| 298 | C22 | — | 2 | 2 | no | — |
| 299 | C22 | — | 2 | 4 | no | — |
| 300 | C22 | — | 4 | 4 | no | — |
| 301 | C22 | — | 4 | 2 | no | — |
| 302 | C22 | — | 4 | 4 | no | — |
| 303 | C22 | — | 8 | 2 | no | — |
| 304 | C22 | — | 8 | 2 | no | — |
| 305 | C22 | — | 316 | 2 | no | — |
| 306 | C22 | — | 316 | 4 | no | — |
| 307 | C22 | #19098418 | 1 | 1 | yes | characteristic |
| 308 | C22 | C2×#14579075 | 1 | 2 | yes | — |
| 309 | C4 | — | 2 | 2 | no | — |
| 310 | C4 | — | 2 | 4 | no | — |
| 311 | C4 | — | 2 | 2 | no | — |
| 312 | C4 | — | 316 | 2 | no | — |
| 313 | C4 | — | 316 | 4 | no | — |
| 314 | C4 | — | 1264 | 2 | no | — |
| 315 | C2 | — | 2 | 2 | no | — |
| 316 | C2 | — | 4 | 2 | no | — |
| 317 | C2 | — | 8 | 2 | no | — |
| 318 | C2 | — | 632 | 2 | no | — |
| 319 | C2 | #23763314 | 1 | 1 | yes | characteristic |
| 320 | C2 | #23763363 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23763318 | C2 | semidirect |
| 3. #23761957 | C2 | semidirect |
| 4. C79×#1780 | C2 | semidirect |
| 5. C2×#14579209 | C4 | semidirect |
| 9. #19099130 | C4 | semidirect |
| 12. #19099041 | C4 | semidirect |
| 163. #1780 | D79 | semidirect |
| 165. C79 | #8464 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23777058 | 2 |
| #23763318 | 2 |
| #23761957 | 2 |
| C79×#1780 | 2 |
| #8464 | 79 |