Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
822 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 822 conjugacy classes of subgroups — too many to draw. Only its 81 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 | #23772650 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #23763317 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #23761929 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23761308 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C79×#2815 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#11929624 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#14579187 | — | 2 | 2 | no | — |
| 8 | #19098706 | — | 2 | 4 | no | — |
| 9 | #19098701 | — | 2 | 2 | no | — |
| 10 | #19098696 | — | 2 | 2 | no | — |
| 11 | #19098562 | — | 2 | 2 | no | — |
| 12 | C2×#14579075 | — | 2 | 2 | no | — |
| 13 | #19098527 | C22 | 1 | 2 | yes | — |
| 14 | #19098426 | C22 | 1 | 1 | yes | characteristic |
| 15 | #19098297 | — | 2 | 4 | no | — |
| 16 | C2×#121×C79 | C22 | 1 | 2 | yes | — |
| 17 | #397×C79 | — | 2 | 2 | no | — |
| 18 | #392×C79 | — | 2 | 2 | no | — |
| 19 | #392×C79 | — | 2 | 2 | no | — |
| 20 | #381×C79 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#11929624 | — | 2 | 2 | no | — |
| 22 | C2×#11929624 | — | 2 | 2 | no | — |
| 23 | C2×#11929624 | — | 4 | 4 | no | — |
| 24 | C22×Dic158 | D4 | 1 | 1 | yes | characteristic, complemented |
| 25 | C22×D316 | D4 | 1 | 1 | yes | characteristic, complemented |
| 26 | C22×C22⋊D79 | — | 2 | 1 | no | — |
| 27 | C22×C4×D79 | — | 2 | 1 | no | — |
| 28 | #14579187 | — | 4 | 8 | no | — |
| 29 | C2×#11929612 | — | 2 | 4 | no | — |
| 30 | C2×#11929609 | — | 2 | 4 | no | — |
| 31 | C2×#11929608 | — | 2 | 2 | no | — |
| 32 | C2×C22⋊Dic79 | — | 2 | 1 | no | — |
| 33 | #14579084 | — | 2 | 4 | no | — |
| 34 | #14579081 | — | 2 | 4 | no | — |
| 35 | #14579075 | — | 4 | 4 | no | — |
| 36 | C22⋊(C79⋊C8) | — | 2 | 2 | no | — |
| 37 | C22⋊(C79⋊C8) | D4 | 1 | 4 | yes | complemented |
| 38 | C79⋊#95 | — | 2 | 2 | no | — |
| 39 | C22×D4×C79 | — | 2 | 2 | no | — |
| 40 | C22×D4×C79 | — | 2 | 2 | no | — |
| 41 | C22×D4×C79 | D4 | 1 | 2 | yes | — |
| 42 | C23×C4×C79 | C23 | 1 | 1 | yes | characteristic |
| 43 | #121×C79 | — | 2 | 2 | no | — |
| 44 | #121×C79 | — | 2 | 2 | no | — |
| 45 | #121×C79 | — | 2 | 2 | no | — |
| 46 | #121×C79 | — | 2 | 2 | no | — |
| 47 | C2×C4⋊C4×C79 | D4 | 1 | 2 | yes | — |
| 48 | C2×C22⋊C4×C79 | — | 2 | 2 | no | — |
| 49 | C2×C22⋊C4×C79 | — | 2 | 4 | no | — |
| 50 | C2×C22⋊C4×C79 | D4 | 1 | 2 | yes | — |
| 51 | #95×C79 | — | 2 | 2 | no | — |
| 52 | #95×C79 | — | 2 | 2 | no | — |
| 53 | C23×D79 | — | 2 | 1 | no | — |
| 54 | #11929624 | — | 2 | 8 | no | — |
| 55 | #11929624 | — | 4 | 4 | no | — |
| 56 | #11929624 | — | 4 | 4 | no | — |
| 57 | #11929624 | — | 4 | 4 | no | — |
| 58 | C2×Dic158 | — | 2 | 2 | no | — |
| 59 | C2×Dic158 | — | 4 | 2 | no | — |
| 60 | C2×D316 | — | 2 | 2 | no | — |
| 61 | C2×D316 | — | 4 | 2 | no | — |
| 62 | C2×C22⋊D79 | — | 2 | 4 | no | — |
| 63 | C2×C22⋊D79 | — | 4 | 2 | no | — |
| 64 | C2×C22⋊D79 | — | 4 | 1 | no | — |
| 65 | C2×C22⋊D79 | — | 4 | 1 | no | — |
| 66 | C22×Dic79 | — | 2 | 1 | no | — |
| 67 | C22×Dic79 | — | 4 | 1 | no | — |
| 68 | C2×C4×D79 | — | 4 | 2 | no | — |
| 69 | C2×C4×D79 | — | 4 | 2 | no | — |
| 70 | C2×C4×D79 | — | 4 | 2 | no | — |
| 71 | #11929612 | — | 4 | 8 | no | — |
| 72 | #11929609 | — | 4 | 8 | no | — |
| 73 | #11929608 | — | 4 | 4 | no | — |
| 74 | C2×C79⋊C8 | — | 2 | 4 | no | — |
| 75 | C22⋊Dic79 | — | 4 | 2 | no | — |
| 76 | C24×C79 | — | 2 | 2 | no | — |
| 77 | C24×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C2×D4×C79 | — | 2 | 2 | no | — |
| 79 | C2×D4×C79 | — | 2 | 2 | no | — |
| 80 | C2×D4×C79 | — | 4 | 4 | no | — |
| 81 | C2×D4×C79 | — | 4 | 2 | no | — |
| 82 | C2×D4×C79 | — | 4 | 2 | no | — |
| 83 | C2×D4×C79 | — | 8 | 4 | no | — |
| 84 | C2×D4×C79 | — | 8 | 4 | no | — |
| 85 | C2×D4×C79 | — | 8 | 2 | no | — |
| 86 | C22×C4×C79 | — | 2 | 2 | no | — |
| 87 | C22×C4×C79 | — | 2 | 2 | no | — |
| 88 | C22×C4×C79 | — | 4 | 2 | no | — |
| 89 | C22×C4×C79 | C2×D4 | 1 | 2 | yes | — |
| 90 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C22×C4×C79 | C2×D4 | 1 | 2 | yes | — |
| 92 | C22×C4×C79 | #41 | 1 | 1 | yes | characteristic |
| 93 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 94 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 95 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 96 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 97 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 98 | C22⋊C4×C79 | — | 8 | 4 | no | — |
| 99 | C23×C79 | — | 2 | 2 | no | — |
| 100 | C23×C79 | — | 4 | 2 | no | — |
| 101 | C23×C79 | — | 4 | 2 | no | — |
| 102 | C23×C79 | — | 4 | 2 | no | — |
| 103 | C23×C79 | — | 4 | 4 | no | — |
| 104 | C23×C79 | — | 4 | 2 | no | — |
| 105 | C23×C79 | — | 4 | 4 | no | — |
| 106 | C23×C79 | #120 | 1 | 1 | yes | characteristic |
| 107 | C23×C79 | #121 | 1 | 1 | yes | characteristic |
| 108 | C23×C79 | #127 | 1 | 1 | yes | characteristic |
| 109 | C22×D79 | — | 2 | 2 | no | — |
| 110 | C22×D79 | — | 4 | 1 | no | — |
| 111 | C22×D79 | — | 4 | 1 | no | — |
| 112 | C22×D79 | — | 8 | 2 | no | — |
| 113 | D4×C79 | — | 4 | 4 | no | — |
| 114 | D4×C79 | — | 8 | 2 | no | — |
| 115 | D4×C79 | — | 8 | 4 | no | — |
| 116 | D4×C79 | — | 8 | 4 | no | — |
| 117 | D4×C79 | — | 8 | 8 | no | — |
| 118 | C2×C4×C79 | — | 2 | 4 | no | — |
| 119 | C2×C4×C79 | — | 4 | 1 | no | — |
| 120 | C2×C4×C79 | — | 4 | 1 | no | — |
| 121 | C2×C4×C79 | — | 4 | 2 | no | — |
| 122 | C2×C4×C79 | — | 4 | 4 | no | — |
| 123 | C2×C4×C79 | — | 4 | 4 | no | — |
| 124 | C2×C4×C79 | — | 4 | 2 | no | — |
| 125 | C2×C4×C79 | — | 4 | 2 | no | — |
| 126 | C2×C4×C79 | #120 | 1 | 2 | yes | — |
| 127 | C2×C4×C79 | #121 | 1 | 2 | yes | — |
| 128 | C22⋊D79 | — | 4 | 4 | no | — |
| 129 | C22⋊D79 | — | 4 | 4 | no | — |
| 130 | C2×Dic79 | — | 2 | 2 | no | — |
| 131 | C2×Dic79 | — | 4 | 1 | no | — |
| 132 | C2×Dic79 | — | 4 | 1 | no | — |
| 133 | C2×Dic79 | — | 8 | 1 | no | — |
| 134 | C2×Dic79 | — | 8 | 2 | no | — |
| 135 | D316 | — | 2 | 4 | no | — |
| 136 | D316 | — | 4 | 2 | no | — |
| 137 | C4×D79 | — | 4 | 8 | no | — |
| 138 | Dic158 | — | 2 | 4 | no | — |
| 139 | Dic158 | — | 4 | 2 | no | — |
| 140 | C79⋊C8 | — | 4 | 4 | no | — |
| 141 | C22×C79 | — | 2 | 2 | no | — |
| 142 | C22×C79 | — | 2 | 2 | no | — |
| 143 | C22×C79 | — | 2 | 2 | no | — |
| 144 | C22×C79 | — | 2 | 1 | no | — |
| 145 | C22×C79 | — | 2 | 1 | no | — |
| 146 | C22×C79 | — | 8 | 2 | no | — |
| 147 | C22×C79 | — | 8 | 4 | no | — |
| 148 | C22×C79 | — | 8 | 4 | no | — |
| 149 | C22×C79 | — | 8 | 2 | no | — |
| 150 | C22×C79 | — | 8 | 2 | no | — |
| 151 | C22×C79 | — | 8 | 2 | no | — |
| 152 | C22×C79 | #392 | 1 | 1 | yes | characteristic |
| 153 | C22×C79 | B(4,𝔽2) | 1 | 2 | yes | — |
| 154 | C2×D79 | — | 4 | 2 | no | — |
| 155 | C2×D79 | — | 8 | 2 | no | — |
| 156 | C2×D79 | — | 8 | 2 | no | — |
| 157 | C2×D79 | — | 8 | 2 | no | — |
| 158 | C4×C79 | — | 2 | 2 | no | — |
| 159 | C4×C79 | — | 2 | 2 | no | — |
| 160 | C4×C79 | — | 8 | 2 | no | — |
| 161 | C4×C79 | — | 8 | 2 | no | — |
| 162 | Dic79 | — | 4 | 2 | no | — |
| 163 | Dic79 | — | 16 | 1 | no | — |
| 164 | #9980 | — | 79 | 1 | no | maximal, Sylow |
| 165 | C2×C79 | — | 2 | 2 | no | — |
| 166 | C2×C79 | — | 4 | 1 | no | — |
| 167 | C2×C79 | — | 4 | 1 | no | — |
| 168 | C2×C79 | — | 16 | 2 | no | — |
| 169 | C2×C79 | #2008 | 1 | 2 | yes | — |
| 170 | C2×C79 | #2021 | 1 | 1 | yes | characteristic |
| 171 | D79 | — | 8 | 2 | no | — |
| 172 | #3006 | — | 79 | 2 | no | — |
| 173 | #2815 | D79 | 1 | 1 | yes | characteristic, complemented |
| 174 | #1782 | — | 79 | 1 | no | — |
| 175 | #1510 | — | 79 | 2 | no | — |
| 176 | #1306 | — | 79 | 1 | no | — |
| 177 | C79 | #9980 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 178 | C22×#41 | — | 79 | 1 | no | — |
| 179 | C2×AGL(1,ℤ/8) | — | 158 | 2 | no | — |
| 180 | C2×#121 | C2×D79 | 1 | 2 | yes | — |
| 181 | #449 | — | 158 | 4 | no | — |
| 182 | #448 | — | 158 | 2 | no | — |
| 183 | #447 | — | 158 | 2 | no | — |
| 184 | #417 | — | 158 | 2 | no | — |
| 185 | C2×#99 | — | 158 | 2 | no | — |
| 186 | #407 | — | 79 | 2 | no | — |
| 187 | #397 | — | 2 | 2 | no | — |
| 188 | #392 | — | 2 | 2 | no | — |
| 189 | #392 | — | 2 | 2 | no | — |
| 190 | #381 | — | 79 | 1 | no | — |
| 191 | #381 | C2×D79 | 1 | 1 | yes | characteristic |
| 192 | #327 | — | 158 | 4 | no | — |
| 193 | C2×#41 | — | 316 | 4 | no | — |
| 194 | C2×#41 | — | 158 | 2 | no | — |
| 195 | C2×#41 | — | 158 | 2 | no | — |
| 196 | C22×Q8 | — | 79 | 1 | no | — |
| 197 | C22×D4 | — | 2 | 2 | no | — |
| 198 | C22×D4 | — | 2 | 2 | no | — |
| 199 | C22×D4 | — | 79 | 1 | no | — |
| 200 | C22×D4 | — | 158 | 1 | no | — |
| 201 | C22×D4 | C22⋊D79 | 1 | 2 | yes | — |
| 202 | C23×C4 | — | 158 | 1 | no | — |
| 203 | C23×C4 | C22×D79 | 1 | 1 | yes | characteristic |
| 204 | AGL(1,ℤ/8) | — | 316 | 8 | no | — |
| 205 | C2×SD16 | — | 158 | 4 | no | — |
| 206 | C2×D8 | — | 158 | 4 | no | — |
| 207 | C2×M16 | — | 158 | 2 | no | — |
| 208 | #121 | — | 2 | 2 | no | — |
| 209 | #121 | — | 2 | 2 | no | — |
| 210 | #121 | — | 2 | 2 | no | — |
| 211 | #121 | — | 2 | 2 | no | — |
| 212 | C2×C4⋊C4 | C22⋊D79 | 1 | 2 | yes | — |
| 213 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 214 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 215 | C2×C22⋊C4 | — | 158 | 1 | no | — |
| 216 | C2×C22⋊C4 | C22⋊D79 | 1 | 2 | yes | — |
| 217 | #103 | — | 158 | 4 | no | — |
| 218 | #102 | — | 158 | 4 | no | — |
| 219 | #99 | — | 316 | 4 | no | — |
| 220 | C22⋊C8 | — | 79 | 4 | no | — |
| 221 | C22⋊C8 | — | 158 | 2 | no | — |
| 222 | #95 | — | 2 | 2 | no | — |
| 223 | #95 | — | 2 | 2 | no | — |
| 224 | #95 | — | 158 | 2 | no | — |
| 225 | C24 | — | 2 | 2 | no | — |
| 226 | C24 | — | 158 | 1 | no | — |
| 227 | C24 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 228 | #41 | — | 316 | 4 | no | — |
| 229 | #41 | — | 316 | 4 | no | — |
| 230 | #41 | — | 316 | 4 | no | — |
| 231 | #41 | — | 158 | 8 | no | — |
| 232 | C2×Q8 | — | 316 | 2 | no | — |
| 233 | C2×Q8 | — | 158 | 2 | no | — |
| 234 | C2×D4 | — | 2 | 2 | no | — |
| 235 | C2×D4 | — | 2 | 2 | no | — |
| 236 | C2×D4 | — | 4 | 4 | no | — |
| 237 | C2×D4 | — | 4 | 2 | no | — |
| 238 | C2×D4 | — | 4 | 2 | no | — |
| 239 | C2×D4 | — | 8 | 4 | no | — |
| 240 | C2×D4 | — | 8 | 4 | no | — |
| 241 | C2×D4 | — | 8 | 2 | no | — |
| 242 | C2×D4 | — | 316 | 2 | no | — |
| 243 | C2×D4 | — | 316 | 2 | no | — |
| 244 | C2×D4 | — | 316 | 1 | no | — |
| 245 | C2×D4 | — | 316 | 1 | no | — |
| 246 | C2×D4 | — | 158 | 4 | no | — |
| 247 | C2×D4 | — | 158 | 2 | no | — |
| 248 | C22×C4 | — | 2 | 2 | no | — |
| 249 | C22×C4 | — | 2 | 2 | no | — |
| 250 | C22×C4 | — | 4 | 2 | no | — |
| 251 | C22×C4 | — | 316 | 2 | no | — |
| 252 | C22×C4 | — | 316 | 2 | no | — |
| 253 | C22×C4 | — | 316 | 2 | no | — |
| 254 | C22×C4 | — | 316 | 1 | no | — |
| 255 | C22×C4 | — | 158 | 1 | no | — |
| 256 | C22×C4 | D4×D79 | 1 | 1 | yes | characteristic |
| 257 | C22×C4 | D4×D79 | 1 | 2 | yes | — |
| 258 | C22×C4 | C2×C22⋊D79 | 1 | 2 | yes | — |
| 259 | C22×C4 | #11929623 | 1 | 1 | yes | characteristic |
| 260 | SD16 | — | 316 | 8 | no | — |
| 261 | D8 | — | 316 | 8 | no | — |
| 262 | M16 | — | 316 | 4 | no | — |
| 263 | C2×C8 | — | 158 | 4 | no | — |
| 264 | C4⋊C4 | — | 2 | 2 | no | — |
| 265 | C4⋊C4 | — | 2 | 2 | no | — |
| 266 | C22⋊C4 | — | 4 | 2 | no | — |
| 267 | C22⋊C4 | — | 4 | 2 | no | — |
| 268 | C22⋊C4 | — | 4 | 2 | no | — |
| 269 | C22⋊C4 | — | 8 | 4 | no | — |
| 270 | C22⋊C4 | — | 316 | 2 | no | — |
| 271 | C23 | — | 2 | 2 | no | — |
| 272 | C23 | — | 4 | 2 | no | — |
| 273 | C23 | — | 4 | 2 | no | — |
| 274 | C23 | — | 4 | 2 | no | — |
| 275 | C23 | — | 4 | 4 | no | — |
| 276 | C23 | — | 4 | 2 | no | — |
| 277 | C23 | — | 4 | 4 | no | — |
| 278 | C23 | — | 316 | 1 | no | — |
| 279 | C23 | — | 316 | 1 | no | — |
| 280 | C23 | — | 632 | 2 | no | — |
| 281 | C23 | — | 158 | 2 | no | — |
| 282 | C23 | #14579130 | 1 | 1 | yes | characteristic |
| 283 | C23 | #14579132 | 1 | 1 | yes | characteristic |
| 284 | C23 | #14579157 | 1 | 1 | yes | characteristic |
| 285 | Q8 | — | 316 | 2 | no | — |
| 286 | Q8 | — | 158 | 4 | no | — |
| 287 | D4 | — | 4 | 4 | no | — |
| 288 | D4 | — | 8 | 2 | no | — |
| 289 | D4 | — | 8 | 4 | no | — |
| 290 | D4 | — | 8 | 4 | no | — |
| 291 | D4 | — | 8 | 8 | no | — |
| 292 | D4 | — | 316 | 2 | no | — |
| 293 | D4 | — | 316 | 4 | no | — |
| 294 | D4 | — | 316 | 4 | no | — |
| 295 | D4 | — | 158 | 4 | no | — |
| 296 | C2×C4 | — | 2 | 4 | no | — |
| 297 | C2×C4 | — | 4 | 1 | no | — |
| 298 | C2×C4 | — | 4 | 1 | no | — |
| 299 | C2×C4 | — | 4 | 2 | no | — |
| 300 | C2×C4 | — | 4 | 4 | no | — |
| 301 | C2×C4 | — | 4 | 4 | no | — |
| 302 | C2×C4 | — | 4 | 2 | no | — |
| 303 | C2×C4 | — | 4 | 2 | no | — |
| 304 | C2×C4 | — | 316 | 8 | no | — |
| 305 | C2×C4 | — | 316 | 1 | no | — |
| 306 | C2×C4 | — | 316 | 1 | no | — |
| 307 | C2×C4 | — | 632 | 1 | no | — |
| 308 | C2×C4 | — | 632 | 2 | no | — |
| 309 | C2×C4 | — | 158 | 2 | no | — |
| 310 | C2×C4 | #14579129 | 1 | 2 | yes | — |
| 311 | C2×C4 | #14579133 | 1 | 2 | yes | — |
| 312 | C8 | — | 316 | 4 | no | — |
| 313 | C22 | — | 2 | 2 | no | — |
| 314 | C22 | — | 2 | 1 | no | — |
| 315 | C22 | — | 2 | 1 | no | — |
| 316 | C22 | — | 2 | 2 | no | — |
| 317 | C22 | — | 2 | 2 | no | — |
| 318 | C22 | — | 8 | 2 | no | — |
| 319 | C22 | — | 8 | 4 | no | — |
| 320 | C22 | — | 8 | 4 | no | — |
| 321 | C22 | — | 8 | 2 | no | — |
| 322 | C22 | — | 8 | 2 | no | — |
| 323 | C22 | — | 8 | 2 | no | — |
| 324 | C22 | — | 316 | 2 | no | — |
| 325 | C22 | — | 632 | 2 | no | — |
| 326 | C22 | — | 632 | 2 | no | — |
| 327 | C22 | — | 632 | 2 | no | — |
| 328 | C22 | #19098471 | 1 | 1 | yes | characteristic |
| 329 | C22 | #19098750 | 1 | 2 | yes | — |
| 330 | C4 | — | 2 | 2 | no | — |
| 331 | C4 | — | 2 | 2 | no | — |
| 332 | C4 | — | 8 | 2 | no | — |
| 333 | C4 | — | 8 | 2 | no | — |
| 334 | C4 | — | 316 | 2 | no | — |
| 335 | C4 | — | 1264 | 1 | no | — |
| 336 | C2 | — | 2 | 2 | no | — |
| 337 | C2 | — | 4 | 1 | no | — |
| 338 | C2 | — | 4 | 1 | no | — |
| 339 | C2 | — | 16 | 2 | no | — |
| 340 | C2 | — | 632 | 2 | no | — |
| 341 | C2 | #23764453 | 1 | 2 | yes | — |
| 342 | C2 | #23764524 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23772650 | C2 | semidirect |
| 2. #23763317 | C2 | semidirect |
| 3. #23761929 | C2 | semidirect |
| 4. #23761308 | C2 | semidirect |
| 5. C79×#2815 | C2 | semidirect |
| 24. C22×Dic158 | D4 | semidirect |
| 25. C22×D316 | D4 | semidirect |
| 37. C22⋊(C79⋊C8) | D4 | semidirect |
| 173. #2815 | D79 | semidirect |
| 177. C79 | #9980 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23772650 | 2 |
| #23763317 | 2 |
| #23761929 | 2 |
| #23761308 | 2 |
| C79×#2815 | 2 |
| #9980 | 79 |