Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
422 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #23774265 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23765319 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23764521 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #23761416 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C79×#1315 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14579191 | — | 2 | 1 | no | — |
| 7 | C22×Dic316 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | #19099127 | — | 2 | 1 | no | — |
| 9 | C2×#14579106 | — | 2 | 2 | no | — |
| 10 | #19098888 | — | 2 | 2 | no | — |
| 11 | #19098816 | — | 2 | 1 | no | — |
| 12 | #19098799 | C22 | 1 | 2 | yes | — |
| 13 | #19098699 | — | 2 | 2 | no | — |
| 14 | C2×#14579079 | — | 2 | 1 | no | — |
| 15 | #19098368 | — | 2 | 2 | no | — |
| 16 | #19098328 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #408×C79 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22⋊C16×C79 | C22 | 1 | 2 | yes | complemented |
| 19 | C22×Dic158 | — | 2 | 1 | no | — |
| 20 | C22×D316 | — | 2 | 1 | no | — |
| 21 | #14579191 | — | 4 | 2 | no | — |
| 22 | #14579191 | — | 4 | 2 | no | — |
| 23 | C2×Dic316 | — | 2 | 2 | no | — |
| 24 | C2×Dic316 | — | 4 | 2 | no | — |
| 25 | C2×#11929614 | — | 4 | 1 | no | — |
| 26 | C2×#11929612 | — | 4 | 1 | no | — |
| 27 | C2×D632 | — | 2 | 1 | no | — |
| 28 | C2×D632 | D4 | 1 | 2 | yes | — |
| 29 | C2×#11929608 | — | 2 | 1 | no | — |
| 30 | #14579157 | — | 4 | 1 | no | — |
| 31 | #14579150 | — | 4 | 1 | no | — |
| 32 | #14579135 | — | 2 | 2 | no | — |
| 33 | #14579135 | — | 2 | 2 | no | — |
| 34 | #14579129 | — | 4 | 1 | no | — |
| 35 | #14579129 | — | 4 | 1 | no | — |
| 36 | #14579120 | — | 2 | 1 | no | — |
| 37 | #14579106 | — | 4 | 4 | no | — |
| 38 | #14579095 | D4 | 1 | 2 | yes | — |
| 39 | #14579093 | — | 2 | 1 | no | — |
| 40 | #14579083 | — | 2 | 2 | no | — |
| 41 | #14579083 | — | 2 | 2 | no | — |
| 42 | #14579079 | — | 4 | 2 | no | — |
| 43 | C2×#41×C79 | D4 | 1 | 1 | yes | characteristic |
| 44 | C2×M16×C79 | D4 | 1 | 1 | yes | characteristic |
| 45 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C16×C79 | — | 2 | 2 | no | — |
| 47 | C22⋊C8×C79 | — | 2 | 2 | no | — |
| 48 | C23×D79 | — | 4 | 1 | no | — |
| 49 | C2×Dic158 | — | 2 | 2 | no | — |
| 50 | C2×Dic158 | — | 4 | 1 | no | — |
| 51 | C2×Dic158 | — | 4 | 1 | no | — |
| 52 | C2×Dic158 | — | 4 | 1 | no | — |
| 53 | C2×Dic158 | — | 4 | 1 | no | — |
| 54 | C2×Dic158 | — | 4 | 1 | no | — |
| 55 | C2×D316 | — | 2 | 2 | no | — |
| 56 | C2×D316 | — | 4 | 1 | no | — |
| 57 | C2×D316 | — | 8 | 1 | no | — |
| 58 | C2×D316 | — | 8 | 1 | no | — |
| 59 | C2×C22⋊D79 | — | 4 | 2 | no | — |
| 60 | C2×C22⋊D79 | — | 4 | 2 | no | — |
| 61 | C22×Dic79 | — | 4 | 1 | no | — |
| 62 | Dic316 | — | 2 | 4 | no | — |
| 63 | Dic316 | — | 4 | 2 | no | — |
| 64 | #11929614 | — | 4 | 4 | no | — |
| 65 | #11929612 | — | 4 | 4 | no | — |
| 66 | D632 | — | 4 | 2 | no | — |
| 67 | D632 | — | 8 | 1 | no | — |
| 68 | #11929608 | — | 2 | 2 | no | — |
| 69 | #11929608 | — | 4 | 1 | no | — |
| 70 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 71 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 72 | C22⋊Dic79 | — | 4 | 1 | no | — |
| 73 | (C2×D79)⋊C4 | — | 4 | 2 | no | — |
| 74 | (C2×D79)⋊C4 | — | 4 | 2 | no | — |
| 75 | C4×Dic79 | — | 4 | 1 | no | — |
| 76 | #41×C79 | — | 4 | 1 | no | — |
| 77 | #41×C79 | — | 4 | 1 | no | — |
| 78 | C2×Q8×C79 | D8 | 1 | 1 | yes | characteristic |
| 79 | C2×D4×C79 | — | 2 | 1 | no | — |
| 80 | C2×D4×C79 | D8 | 1 | 1 | yes | characteristic |
| 81 | C22×C4×C79 | — | 2 | 1 | no | — |
| 82 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | M16×C79 | — | 4 | 1 | no | — |
| 84 | C2×C8×C79 | — | 2 | 1 | no | — |
| 85 | C2×C8×C79 | — | 4 | 1 | no | — |
| 86 | C2×C8×C79 | C2×D4 | 1 | 2 | yes | — |
| 87 | C16×C79 | — | 4 | 2 | no | — |
| 88 | C23×C79 | — | 2 | 1 | no | — |
| 89 | C23×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 90 | C22×D79 | — | 4 | 2 | no | — |
| 91 | C22×D79 | — | 8 | 2 | no | — |
| 92 | C22×D79 | — | 16 | 2 | no | — |
| 93 | Q8×C79 | — | 4 | 1 | no | — |
| 94 | D4×C79 | — | 4 | 1 | no | — |
| 95 | D4×C79 | — | 8 | 1 | no | — |
| 96 | C2×C4×C79 | — | 2 | 1 | no | — |
| 97 | C2×C4×C79 | — | 2 | 1 | no | — |
| 98 | C2×C4×C79 | — | 2 | 1 | no | — |
| 99 | C2×C4×C79 | — | 4 | 1 | no | — |
| 100 | C2×C4×C79 | — | 4 | 1 | no | — |
| 101 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 103 | C22⋊D79 | — | 16 | 2 | no | — |
| 104 | C22⋊D79 | — | 16 | 2 | no | — |
| 105 | C2×Dic79 | — | 4 | 2 | no | — |
| 106 | C2×Dic79 | — | 4 | 2 | no | — |
| 107 | C2×Dic79 | — | 8 | 1 | no | — |
| 108 | C2×Dic79 | — | 8 | 1 | no | — |
| 109 | D316 | — | 4 | 2 | no | — |
| 110 | D316 | — | 4 | 2 | no | — |
| 111 | D316 | — | 8 | 2 | no | — |
| 112 | Dic158 | — | 4 | 2 | no | — |
| 113 | Dic158 | — | 4 | 2 | no | — |
| 114 | Dic158 | — | 8 | 1 | no | — |
| 115 | Dic158 | — | 8 | 1 | no | — |
| 116 | C8×C79 | — | 2 | 2 | no | — |
| 117 | C8×C79 | — | 4 | 1 | no | — |
| 118 | C79⋊C8 | — | 4 | 2 | no | — |
| 119 | C22×C79 | — | 2 | 1 | no | — |
| 120 | C22×C79 | — | 2 | 2 | no | — |
| 121 | C22×C79 | — | 4 | 1 | no | — |
| 122 | C22×C79 | — | 4 | 2 | no | — |
| 123 | C22×C79 | #447 | 1 | 1 | yes | characteristic |
| 124 | C2×D79 | — | 8 | 2 | no | — |
| 125 | C2×D79 | — | 8 | 2 | no | — |
| 126 | C2×D79 | — | 8 | 2 | no | — |
| 127 | C2×D79 | — | 16 | 4 | no | — |
| 128 | C4×C79 | — | 2 | 1 | no | — |
| 129 | C4×C79 | — | 4 | 1 | no | — |
| 130 | C4×C79 | #453 | 1 | 1 | yes | characteristic |
| 131 | C4×C79 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 132 | Dic79 | — | 8 | 2 | no | — |
| 133 | Dic79 | — | 8 | 2 | no | — |
| 134 | #12111 | — | 79 | 1 | no | maximal, Sylow |
| 135 | C2×C79 | — | 4 | 1 | no | — |
| 136 | C2×C79 | — | 8 | 1 | no | — |
| 137 | C2×C79 | #1595 | 1 | 1 | yes | characteristic |
| 138 | C2×C79 | #2194 | 1 | 2 | yes | — |
| 139 | D79 | — | 16 | 2 | no | — |
| 140 | #3199 | — | 79 | 1 | no | — |
| 141 | #2185 | — | 79 | 2 | no | — |
| 142 | #2020 | — | 79 | 1 | no | — |
| 143 | #1349 | — | 79 | 2 | no | — |
| 144 | #1315 | D79 | 1 | 1 | yes | characteristic, complemented |
| 145 | C79 | #12111 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 146 | C2×#137 | — | 158 | 1 | no | — |
| 147 | C22×Q16 | — | 79 | 1 | no | — |
| 148 | #535 | — | 158 | 1 | no | — |
| 149 | C2×SD32 | — | 158 | 2 | no | — |
| 150 | #482 | — | 158 | 2 | no | — |
| 151 | #469 | — | 158 | 1 | no | — |
| 152 | #466 | — | 79 | 2 | no | — |
| 153 | #447 | — | 158 | 2 | no | — |
| 154 | C2×#101 | — | 158 | 1 | no | — |
| 155 | #408 | C2×D79 | 1 | 1 | yes | characteristic |
| 156 | #358 | — | 158 | 2 | no | — |
| 157 | C22⋊C16 | C2×D79 | 1 | 2 | yes | complemented |
| 158 | #341 | — | 79 | 1 | no | — |
| 159 | C2×#41 | D316 | 1 | 1 | yes | characteristic |
| 160 | C22×Q8 | — | 158 | 1 | no | — |
| 161 | C22×D4 | — | 158 | 1 | no | — |
| 162 | #137 | — | 316 | 2 | no | — |
| 163 | #137 | — | 316 | 2 | no | — |
| 164 | C2×Q16 | — | 316 | 2 | no | — |
| 165 | C2×Q16 | — | 316 | 1 | no | — |
| 166 | C2×Q16 | — | 158 | 2 | no | — |
| 167 | C2×SD16 | — | 316 | 1 | no | — |
| 168 | C2×D8 | — | 79 | 2 | no | — |
| 169 | C2×D8 | — | 158 | 1 | no | — |
| 170 | C2×M16 | — | 158 | 1 | no | — |
| 171 | C2×M16 | D316 | 1 | 1 | yes | characteristic |
| 172 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 173 | #127 | — | 316 | 1 | no | — |
| 174 | #124 | — | 316 | 1 | no | — |
| 175 | #121 | — | 158 | 2 | no | — |
| 176 | #121 | — | 158 | 2 | no | — |
| 177 | #120 | — | 316 | 1 | no | — |
| 178 | #120 | — | 316 | 1 | no | — |
| 179 | #117 | — | 158 | 1 | no | — |
| 180 | SD32 | — | 316 | 4 | no | — |
| 181 | C2×C16 | — | 2 | 2 | no | — |
| 182 | #107 | — | 79 | 2 | no | — |
| 183 | #106 | — | 158 | 1 | no | — |
| 184 | #102 | — | 158 | 2 | no | — |
| 185 | #102 | — | 158 | 2 | no | — |
| 186 | #101 | — | 316 | 2 | no | — |
| 187 | C22⋊C8 | — | 2 | 2 | no | — |
| 188 | C24 | — | 316 | 1 | no | — |
| 189 | #41 | — | 4 | 1 | no | — |
| 190 | #41 | — | 4 | 1 | no | — |
| 191 | C2×Q8 | — | 316 | 1 | no | — |
| 192 | C2×Q8 | — | 316 | 1 | no | — |
| 193 | C2×Q8 | — | 316 | 1 | no | — |
| 194 | C2×Q8 | — | 316 | 1 | no | — |
| 195 | C2×Q8 | — | 316 | 1 | no | — |
| 196 | C2×Q8 | — | 158 | 2 | no | — |
| 197 | C2×Q8 | D632 | 1 | 1 | yes | characteristic |
| 198 | C2×D4 | — | 2 | 1 | no | — |
| 199 | C2×D4 | — | 316 | 1 | no | — |
| 200 | C2×D4 | — | 316 | 2 | no | — |
| 201 | C2×D4 | — | 316 | 2 | no | — |
| 202 | C2×D4 | — | 632 | 1 | no | — |
| 203 | C2×D4 | — | 632 | 1 | no | — |
| 204 | C2×D4 | — | 158 | 2 | no | — |
| 205 | C2×D4 | D632 | 1 | 1 | yes | characteristic |
| 206 | C22×C4 | — | 2 | 1 | no | — |
| 207 | C22×C4 | — | 316 | 1 | no | — |
| 208 | C22×C4 | C2×D316 | 1 | 1 | yes | characteristic |
| 209 | Q16 | — | 316 | 2 | no | — |
| 210 | Q16 | — | 316 | 4 | no | — |
| 211 | Q16 | — | 158 | 4 | no | — |
| 212 | SD16 | — | 316 | 4 | no | — |
| 213 | D8 | — | 316 | 2 | no | — |
| 214 | D8 | — | 632 | 1 | no | — |
| 215 | M16 | — | 4 | 1 | no | — |
| 216 | M16 | — | 316 | 1 | no | — |
| 217 | M16 | — | 158 | 2 | no | — |
| 218 | C2×C8 | — | 2 | 1 | no | — |
| 219 | C2×C8 | — | 4 | 1 | no | — |
| 220 | C2×C8 | — | 316 | 1 | no | — |
| 221 | C2×C8 | D4×D79 | 1 | 2 | yes | — |
| 222 | C4⋊C4 | — | 158 | 2 | no | — |
| 223 | C22⋊C4 | — | 316 | 2 | no | — |
| 224 | C22⋊C4 | — | 316 | 1 | no | — |
| 225 | C22⋊C4 | — | 316 | 2 | no | — |
| 226 | C42 | — | 316 | 1 | no | — |
| 227 | C16 | — | 4 | 2 | no | — |
| 228 | C23 | — | 2 | 1 | no | — |
| 229 | C23 | — | 316 | 2 | no | — |
| 230 | C23 | — | 632 | 2 | no | — |
| 231 | C23 | — | 1264 | 2 | no | — |
| 232 | C23 | #14579189 | 1 | 1 | yes | characteristic |
| 233 | Q8 | — | 4 | 1 | no | — |
| 234 | Q8 | — | 316 | 2 | no | — |
| 235 | Q8 | — | 316 | 2 | no | — |
| 236 | Q8 | — | 632 | 1 | no | — |
| 237 | Q8 | — | 632 | 1 | no | — |
| 238 | D4 | — | 4 | 1 | no | — |
| 239 | D4 | — | 8 | 1 | no | — |
| 240 | D4 | — | 316 | 2 | no | — |
| 241 | D4 | — | 316 | 2 | no | — |
| 242 | D4 | — | 632 | 2 | no | — |
| 243 | D4 | — | 1264 | 2 | no | — |
| 244 | D4 | — | 1264 | 2 | no | — |
| 245 | C2×C4 | — | 2 | 1 | no | — |
| 246 | C2×C4 | — | 2 | 1 | no | — |
| 247 | C2×C4 | — | 2 | 1 | no | — |
| 248 | C2×C4 | — | 4 | 1 | no | — |
| 249 | C2×C4 | — | 4 | 1 | no | — |
| 250 | C2×C4 | — | 316 | 2 | no | — |
| 251 | C2×C4 | — | 316 | 2 | no | — |
| 252 | C2×C4 | — | 632 | 1 | no | — |
| 253 | C2×C4 | — | 632 | 1 | no | — |
| 254 | C2×C4 | #14579131 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | C2×D632 | 1 | 1 | yes | characteristic |
| 256 | C8 | — | 2 | 2 | no | — |
| 257 | C8 | — | 4 | 1 | no | — |
| 258 | C8 | — | 316 | 2 | no | — |
| 259 | C22 | — | 2 | 2 | no | — |
| 260 | C22 | — | 2 | 1 | no | — |
| 261 | C22 | — | 4 | 2 | no | — |
| 262 | C22 | — | 4 | 1 | no | — |
| 263 | C22 | — | 632 | 2 | no | — |
| 264 | C22 | — | 632 | 2 | no | — |
| 265 | C22 | — | 632 | 2 | no | — |
| 266 | C22 | — | 1264 | 4 | no | — |
| 267 | C22 | #19098699 | 1 | 1 | yes | characteristic |
| 268 | C4 | — | 2 | 1 | no | — |
| 269 | C4 | — | 4 | 1 | no | — |
| 270 | C4 | — | 632 | 2 | no | — |
| 271 | C4 | — | 632 | 2 | no | — |
| 272 | C4 | #19098733 | 1 | 1 | yes | characteristic |
| 273 | C4 | #19098751 | 1 | 1 | yes | characteristic |
| 274 | C2 | — | 4 | 1 | no | — |
| 275 | C2 | — | 8 | 1 | no | — |
| 276 | C2 | — | 1264 | 2 | no | — |
| 277 | C2 | #23762320 | 1 | 1 | yes | characteristic |
| 278 | C2 | #23765373 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23765319 | C2 | semidirect |
| 3. #23764521 | C2 | semidirect |
| 4. #23761416 | C2 | semidirect |
| 5. C79×#1315 | C2 | semidirect |
| 7. C22×Dic316 | C22 | semidirect |
| 16. #19098328 | C22 | semidirect |
| 18. C22⋊C16×C79 | C22 | semidirect |
| 144. #1315 | D79 | semidirect |
| 145. C79 | #12111 | semidirect |
| 157. C22⋊C16 | C2×D79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23774265 | 2 |
| #23765319 | 2 |
| #23764521 | 2 |
| #23761416 | 2 |
| C79×#1315 | 2 |
| #12111 | 79 |