Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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 | #23765311 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #23764509 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #23761415 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23761326 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C79×#3199 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14579186 | — | 2 | 1 | no | — |
| 7 | C22×D632 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×#14579102 | — | 2 | 2 | no | — |
| 9 | C2×#14579077 | — | 2 | 1 | no | — |
| 10 | #19098363 | — | 2 | 2 | no | — |
| 11 | C22⋊(C79⋊C16) | C22 | 1 | 2 | yes | complemented |
| 12 | #19098328 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | #535×C79 | — | 2 | 1 | no | — |
| 14 | #482×C79 | — | 2 | 2 | no | — |
| 15 | #469×C79 | — | 2 | 1 | no | — |
| 16 | #466×C79 | C22 | 1 | 2 | yes | — |
| 17 | #447×C79 | — | 2 | 2 | no | — |
| 18 | #408×C79 | C22 | 1 | 1 | yes | characteristic |
| 19 | C22×D316 | — | 2 | 1 | no | — |
| 20 | #14579186 | — | 4 | 2 | no | — |
| 21 | #14579186 | — | 4 | 2 | no | — |
| 22 | C2×#11929611 | — | 4 | 1 | no | — |
| 23 | C2×D632 | — | 2 | 2 | no | — |
| 24 | C2×D632 | — | 4 | 2 | no | — |
| 25 | C2×#11929609 | — | 4 | 1 | no | — |
| 26 | C2×#11929608 | — | 2 | 1 | no | — |
| 27 | #14579102 | — | 4 | 4 | no | — |
| 28 | C2×C79⋊C16 | — | 2 | 2 | no | — |
| 29 | #14579077 | — | 4 | 2 | no | — |
| 30 | C2×#41×C79 | D4 | 1 | 1 | yes | characteristic |
| 31 | C22×D4×C79 | — | 2 | 1 | no | — |
| 32 | C2×D8×C79 | — | 2 | 1 | no | — |
| 33 | C2×D8×C79 | D4 | 1 | 2 | yes | — |
| 34 | C2×M16×C79 | D4 | 1 | 1 | yes | characteristic |
| 35 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 36 | #127×C79 | — | 4 | 1 | no | — |
| 37 | #124×C79 | — | 4 | 1 | no | — |
| 38 | #121×C79 | — | 2 | 2 | no | — |
| 39 | #121×C79 | — | 2 | 2 | no | — |
| 40 | #120×C79 | — | 4 | 1 | no | — |
| 41 | #120×C79 | — | 4 | 1 | no | — |
| 42 | #117×C79 | — | 2 | 1 | no | — |
| 43 | #107×C79 | D4 | 1 | 2 | yes | — |
| 44 | #106×C79 | — | 2 | 1 | no | — |
| 45 | #102×C79 | — | 2 | 2 | no | — |
| 46 | #102×C79 | — | 2 | 2 | no | — |
| 47 | C22⋊C8×C79 | — | 2 | 2 | no | — |
| 48 | C23×D79 | — | 4 | 1 | no | — |
| 49 | C2×D316 | — | 2 | 2 | no | — |
| 50 | C2×D316 | — | 4 | 1 | no | — |
| 51 | C2×D316 | — | 4 | 1 | no | — |
| 52 | C2×D316 | — | 4 | 1 | no | — |
| 53 | C2×D316 | — | 4 | 1 | no | — |
| 54 | C2×D316 | — | 4 | 1 | no | — |
| 55 | #11929611 | — | 4 | 4 | no | — |
| 56 | D632 | — | 2 | 4 | no | — |
| 57 | D632 | — | 4 | 2 | no | — |
| 58 | #11929609 | — | 4 | 4 | no | — |
| 59 | #11929608 | — | 2 | 2 | no | — |
| 60 | #11929608 | — | 4 | 1 | no | — |
| 61 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 62 | C79⋊C16 | — | 4 | 2 | no | — |
| 63 | C24×C79 | — | 4 | 1 | no | — |
| 64 | #41×C79 | — | 4 | 1 | no | — |
| 65 | #41×C79 | — | 4 | 1 | no | — |
| 66 | C2×Q8×C79 | D8 | 1 | 1 | yes | characteristic |
| 67 | C2×D4×C79 | — | 2 | 1 | no | — |
| 68 | C2×D4×C79 | — | 2 | 2 | no | — |
| 69 | C2×D4×C79 | — | 4 | 1 | no | — |
| 70 | C2×D4×C79 | — | 4 | 2 | no | — |
| 71 | C2×D4×C79 | — | 4 | 2 | no | — |
| 72 | C2×D4×C79 | — | 8 | 1 | no | — |
| 73 | C2×D4×C79 | — | 8 | 1 | no | — |
| 74 | C2×D4×C79 | D8 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C79 | — | 2 | 1 | no | — |
| 76 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | D8×C79 | — | 4 | 2 | no | — |
| 78 | D8×C79 | — | 8 | 1 | no | — |
| 79 | M16×C79 | — | 4 | 1 | no | — |
| 80 | C2×C8×C79 | — | 2 | 1 | no | — |
| 81 | C2×C8×C79 | — | 4 | 1 | no | — |
| 82 | C2×C8×C79 | C2×D4 | 1 | 2 | yes | — |
| 83 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 84 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 85 | C22⋊C4×C79 | — | 4 | 1 | no | — |
| 86 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 87 | C42×C79 | — | 4 | 1 | no | — |
| 88 | C23×C79 | — | 2 | 1 | no | — |
| 89 | C23×C79 | — | 4 | 2 | no | — |
| 90 | C23×C79 | — | 8 | 2 | no | — |
| 91 | C23×C79 | — | 16 | 2 | no | — |
| 92 | C23×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 93 | C22×D79 | — | 4 | 2 | no | — |
| 94 | C22×D79 | — | 8 | 1 | no | — |
| 95 | C22×D79 | — | 8 | 1 | no | — |
| 96 | C22×D79 | — | 16 | 2 | no | — |
| 97 | Q8×C79 | — | 4 | 1 | no | — |
| 98 | D4×C79 | — | 4 | 1 | no | — |
| 99 | D4×C79 | — | 4 | 2 | no | — |
| 100 | D4×C79 | — | 4 | 2 | no | — |
| 101 | D4×C79 | — | 8 | 1 | no | — |
| 102 | D4×C79 | — | 8 | 2 | no | — |
| 103 | D4×C79 | — | 16 | 2 | no | — |
| 104 | D4×C79 | — | 16 | 2 | no | — |
| 105 | C2×C4×C79 | — | 2 | 1 | no | — |
| 106 | C2×C4×C79 | — | 2 | 1 | no | — |
| 107 | C2×C4×C79 | — | 2 | 1 | no | — |
| 108 | C2×C4×C79 | — | 4 | 1 | no | — |
| 109 | C2×C4×C79 | — | 4 | 1 | no | — |
| 110 | C2×C4×C79 | — | 4 | 2 | no | — |
| 111 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 113 | D316 | — | 4 | 2 | no | — |
| 114 | D316 | — | 4 | 2 | no | — |
| 115 | D316 | — | 8 | 1 | no | — |
| 116 | D316 | — | 8 | 1 | no | — |
| 117 | C8×C79 | — | 2 | 2 | no | — |
| 118 | C8×C79 | — | 4 | 1 | no | — |
| 119 | C79⋊C8 | — | 4 | 2 | no | — |
| 120 | C22×C79 | — | 2 | 1 | no | — |
| 121 | C22×C79 | — | 2 | 2 | no | — |
| 122 | C22×C79 | — | 4 | 1 | no | — |
| 123 | C22×C79 | — | 4 | 2 | no | — |
| 124 | C22×C79 | — | 8 | 2 | no | — |
| 125 | C22×C79 | — | 8 | 2 | no | — |
| 126 | C22×C79 | — | 8 | 2 | no | — |
| 127 | C22×C79 | — | 16 | 4 | no | — |
| 128 | C22×C79 | #447 | 1 | 1 | yes | characteristic |
| 129 | C2×D79 | — | 8 | 2 | no | — |
| 130 | C2×D79 | — | 16 | 2 | no | — |
| 131 | C2×D79 | — | 16 | 2 | no | — |
| 132 | C2×D79 | — | 16 | 2 | no | — |
| 133 | C4×C79 | — | 2 | 1 | no | — |
| 134 | C4×C79 | — | 4 | 1 | no | — |
| 135 | C4×C79 | — | 8 | 2 | no | — |
| 136 | C4×C79 | #453 | 1 | 1 | yes | characteristic |
| 137 | C4×C79 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 138 | #12108 | — | 79 | 1 | no | maximal, Sylow |
| 139 | C2×C79 | — | 4 | 1 | no | — |
| 140 | C2×C79 | — | 8 | 1 | no | — |
| 141 | C2×C79 | — | 16 | 2 | no | — |
| 142 | C2×C79 | #1595 | 1 | 1 | yes | characteristic |
| 143 | C2×C79 | #2192 | 1 | 2 | yes | — |
| 144 | D79 | — | 16 | 2 | no | — |
| 145 | #3199 | D79 | 1 | 1 | yes | characteristic, complemented |
| 146 | #2184 | — | 79 | 2 | no | — |
| 147 | #2018 | — | 79 | 1 | no | — |
| 148 | #1349 | — | 79 | 2 | no | — |
| 149 | #1315 | — | 79 | 1 | no | — |
| 150 | C79 | #12108 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 151 | C2×AGL(1,ℤ/8) | — | 158 | 1 | no | — |
| 152 | C22×D8 | — | 79 | 1 | no | — |
| 153 | #535 | — | 2 | 1 | no | — |
| 154 | C2×D16 | — | 158 | 2 | no | — |
| 155 | #482 | — | 2 | 2 | no | — |
| 156 | #469 | — | 2 | 1 | no | — |
| 157 | #466 | C2×D79 | 1 | 2 | yes | — |
| 158 | #447 | — | 2 | 2 | no | — |
| 159 | C2×#100 | — | 158 | 1 | no | — |
| 160 | #408 | C2×D79 | 1 | 1 | yes | characteristic |
| 161 | #357 | — | 158 | 2 | no | — |
| 162 | C22⋊C16 | — | 79 | 2 | no | — |
| 163 | #341 | — | 79 | 1 | no | — |
| 164 | C2×#41 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 165 | C22×D4 | — | 2 | 1 | no | — |
| 166 | C22×D4 | — | 158 | 1 | no | — |
| 167 | AGL(1,ℤ/8) | — | 316 | 2 | no | — |
| 168 | AGL(1,ℤ/8) | — | 316 | 2 | no | — |
| 169 | C2×SD16 | — | 316 | 1 | no | — |
| 170 | C2×D8 | — | 2 | 1 | no | — |
| 171 | C2×D8 | — | 316 | 2 | no | — |
| 172 | C2×D8 | — | 316 | 1 | no | — |
| 173 | C2×D8 | — | 158 | 2 | no | — |
| 174 | C2×D8 | C22⋊D79 | 1 | 2 | yes | — |
| 175 | C2×M16 | — | 158 | 1 | no | — |
| 176 | C2×M16 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 177 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 178 | #127 | — | 4 | 1 | no | — |
| 179 | #124 | — | 4 | 1 | no | — |
| 180 | #121 | — | 2 | 2 | no | — |
| 181 | #121 | — | 2 | 2 | no | — |
| 182 | #120 | — | 4 | 1 | no | — |
| 183 | #120 | — | 4 | 1 | no | — |
| 184 | #117 | — | 2 | 1 | no | — |
| 185 | D16 | — | 316 | 4 | no | — |
| 186 | C2×C16 | — | 158 | 2 | no | — |
| 187 | #107 | C22⋊D79 | 1 | 2 | yes | — |
| 188 | #106 | — | 2 | 1 | no | — |
| 189 | #102 | — | 2 | 2 | no | — |
| 190 | #102 | — | 2 | 2 | no | — |
| 191 | #100 | — | 316 | 2 | no | — |
| 192 | C22⋊C8 | — | 2 | 2 | no | — |
| 193 | C24 | — | 4 | 1 | no | — |
| 194 | C24 | — | 316 | 1 | no | — |
| 195 | #41 | — | 4 | 1 | no | — |
| 196 | #41 | — | 4 | 1 | no | — |
| 197 | C2×Q8 | #11929609 | 1 | 1 | yes | characteristic |
| 198 | C2×D4 | — | 2 | 1 | no | — |
| 199 | C2×D4 | — | 2 | 2 | no | — |
| 200 | C2×D4 | — | 4 | 1 | no | — |
| 201 | C2×D4 | — | 4 | 2 | no | — |
| 202 | C2×D4 | — | 4 | 2 | no | — |
| 203 | C2×D4 | — | 8 | 1 | no | — |
| 204 | C2×D4 | — | 8 | 1 | no | — |
| 205 | C2×D4 | — | 316 | 1 | no | — |
| 206 | C2×D4 | — | 316 | 1 | no | — |
| 207 | C2×D4 | — | 316 | 1 | no | — |
| 208 | C2×D4 | — | 316 | 1 | no | — |
| 209 | C2×D4 | — | 316 | 1 | no | — |
| 210 | C2×D4 | — | 158 | 2 | no | — |
| 211 | C2×D4 | #11929609 | 1 | 1 | yes | characteristic |
| 212 | C22×C4 | — | 2 | 1 | no | — |
| 213 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 214 | SD16 | — | 316 | 4 | no | — |
| 215 | D8 | — | 4 | 2 | no | — |
| 216 | D8 | — | 8 | 1 | no | — |
| 217 | D8 | — | 316 | 2 | no | — |
| 218 | D8 | — | 316 | 4 | no | — |
| 219 | D8 | — | 158 | 4 | no | — |
| 220 | M16 | — | 4 | 1 | no | — |
| 221 | M16 | — | 316 | 1 | no | — |
| 222 | M16 | — | 158 | 2 | no | — |
| 223 | C2×C8 | — | 2 | 1 | no | — |
| 224 | C2×C8 | — | 4 | 1 | no | — |
| 225 | C2×C8 | — | 316 | 1 | no | — |
| 226 | C2×C8 | C2×C22⋊D79 | 1 | 2 | yes | — |
| 227 | C4⋊C4 | — | 2 | 2 | no | — |
| 228 | C22⋊C4 | — | 4 | 2 | no | — |
| 229 | C22⋊C4 | — | 4 | 1 | no | — |
| 230 | C22⋊C4 | — | 4 | 2 | no | — |
| 231 | C42 | — | 4 | 1 | no | — |
| 232 | C16 | — | 316 | 2 | no | — |
| 233 | C23 | — | 2 | 1 | no | — |
| 234 | C23 | — | 4 | 2 | no | — |
| 235 | C23 | — | 8 | 2 | no | — |
| 236 | C23 | — | 16 | 2 | no | — |
| 237 | C23 | — | 316 | 2 | no | — |
| 238 | C23 | — | 632 | 1 | no | — |
| 239 | C23 | — | 632 | 1 | no | — |
| 240 | C23 | — | 1264 | 2 | no | — |
| 241 | C23 | #14579187 | 1 | 1 | yes | characteristic |
| 242 | Q8 | — | 4 | 1 | no | — |
| 243 | D4 | — | 4 | 1 | no | — |
| 244 | D4 | — | 4 | 2 | no | — |
| 245 | D4 | — | 4 | 2 | no | — |
| 246 | D4 | — | 8 | 1 | no | — |
| 247 | D4 | — | 8 | 2 | no | — |
| 248 | D4 | — | 16 | 2 | no | — |
| 249 | D4 | — | 16 | 2 | no | — |
| 250 | D4 | — | 316 | 2 | no | — |
| 251 | D4 | — | 316 | 2 | no | — |
| 252 | D4 | — | 632 | 1 | no | — |
| 253 | D4 | — | 632 | 1 | no | — |
| 254 | C2×C4 | — | 2 | 1 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 4 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 1 | no | — |
| 259 | C2×C4 | — | 4 | 2 | no | — |
| 260 | C2×C4 | #14579130 | 1 | 1 | yes | characteristic |
| 261 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 262 | C8 | — | 2 | 2 | no | — |
| 263 | C8 | — | 4 | 1 | no | — |
| 264 | C8 | — | 316 | 2 | no | — |
| 265 | C22 | — | 2 | 2 | no | — |
| 266 | C22 | — | 2 | 1 | no | — |
| 267 | C22 | — | 4 | 2 | no | — |
| 268 | C22 | — | 4 | 1 | no | — |
| 269 | C22 | — | 8 | 2 | no | — |
| 270 | C22 | — | 8 | 2 | no | — |
| 271 | C22 | — | 8 | 2 | no | — |
| 272 | C22 | — | 16 | 4 | no | — |
| 273 | C22 | — | 632 | 2 | no | — |
| 274 | C22 | — | 1264 | 2 | no | — |
| 275 | C22 | — | 1264 | 2 | no | — |
| 276 | C22 | — | 1264 | 2 | no | — |
| 277 | C22 | #19098697 | 1 | 1 | yes | characteristic |
| 278 | C4 | — | 2 | 1 | no | — |
| 279 | C4 | — | 4 | 1 | no | — |
| 280 | C4 | — | 8 | 2 | no | — |
| 281 | C4 | #19098731 | 1 | 1 | yes | characteristic |
| 282 | C4 | #19098750 | 1 | 1 | yes | characteristic |
| 283 | C2 | — | 4 | 1 | no | — |
| 284 | C2 | — | 8 | 1 | no | — |
| 285 | C2 | — | 16 | 2 | no | — |
| 286 | C2 | — | 1264 | 2 | no | — |
| 287 | C2 | #23762318 | 1 | 1 | yes | characteristic |
| 288 | C2 | #23765357 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23765311 | C2 | semidirect |
| 2. #23764509 | C2 | semidirect |
| 3. #23761415 | C2 | semidirect |
| 4. #23761326 | C2 | semidirect |
| 5. C79×#3199 | C2 | semidirect |
| 7. C22×D632 | C22 | semidirect |
| 11. C22⋊(C79⋊C16) | C22 | semidirect |
| 12. #19098328 | C22 | semidirect |
| 145. #3199 | D79 | semidirect |
| 150. C79 | #12108 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23765311 | 2 |
| #23764509 | 2 |
| #23761415 | 2 |
| #23761326 | 2 |
| C79×#3199 | 2 |
| #12108 | 79 |