Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
384 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 384 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 | #23765345 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #23764687 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #23761391 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23761351 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C79×#3253 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19099363 | C22 | 1 | 1 | yes | characteristic |
| 7 | #19098999 | — | 2 | 2 | no | — |
| 8 | C2×#14579089 | — | 2 | 1 | no | — |
| 9 | #19098545 | — | 2 | 1 | no | — |
| 10 | #19098372 | — | 2 | 2 | no | — |
| 11 | #19098345 | C22 | 1 | 2 | yes | — |
| 12 | C79⋊#338 | C22 | 1 | 1 | yes | characteristic |
| 13 | #532×C79 | — | 2 | 1 | no | — |
| 14 | #476×C79 | — | 2 | 2 | no | — |
| 15 | #469×C79 | — | 2 | 1 | no | — |
| 16 | #469×C79 | C22 | 1 | 2 | yes | — |
| 17 | #459×C79 | — | 2 | 2 | no | — |
| 18 | #423×C79 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#11929624 | — | 2 | 1 | no | — |
| 20 | #14579194 | — | 4 | 1 | no | — |
| 21 | #14579189 | — | 4 | 1 | no | — |
| 22 | #14579183 | — | 2 | 2 | no | — |
| 23 | C2×#11929608 | — | 2 | 1 | no | — |
| 24 | #14579105 | — | 4 | 2 | no | — |
| 25 | #14579102 | — | 4 | 2 | no | — |
| 26 | #14579101 | — | 2 | 2 | no | — |
| 27 | #14579089 | — | 4 | 1 | no | — |
| 28 | #14579089 | — | 4 | 1 | no | — |
| 29 | #14579079 | — | 4 | 1 | no | — |
| 30 | #14579077 | — | 4 | 1 | no | — |
| 31 | C2×#41×C79 | — | 4 | 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 | C2×M16×C79 | C23 | 1 | 1 | yes | characteristic |
| 36 | C4⋊Q8×C79 | — | 2 | 1 | no | — |
| 37 | #127×C79 | — | 2 | 1 | no | — |
| 38 | #124×C79 | — | 4 | 1 | no | — |
| 39 | #121×C79 | — | 2 | 2 | no | — |
| 40 | #121×C79 | — | 4 | 1 | no | — |
| 41 | C4×D4×C79 | — | 2 | 2 | no | — |
| 42 | C4×D4×C79 | — | 4 | 1 | no | — |
| 43 | C2×C42×C79 | D4 | 1 | 1 | yes | characteristic |
| 44 | #106×C79 | — | 2 | 1 | no | — |
| 45 | #106×C79 | D4 | 1 | 2 | yes | — |
| 46 | C4⋊C8×C79 | — | 2 | 2 | no | — |
| 47 | #102×C79 | — | 2 | 2 | no | — |
| 48 | #102×C79 | — | 2 | 2 | no | — |
| 49 | #11929624 | — | 4 | 1 | no | — |
| 50 | #11929624 | — | 4 | 1 | no | — |
| 51 | #11929624 | — | 4 | 1 | no | — |
| 52 | #11929624 | — | 4 | 1 | no | — |
| 53 | C2×Dic158 | — | 2 | 1 | no | — |
| 54 | C2×D316 | — | 2 | 1 | no | — |
| 55 | C2×C22⋊D79 | — | 4 | 1 | no | — |
| 56 | C2×C4×D79 | — | 4 | 1 | no | — |
| 57 | Dic316 | — | 2 | 2 | no | — |
| 58 | #11929613 | — | 4 | 2 | no | — |
| 59 | D632 | — | 2 | 2 | no | — |
| 60 | #11929608 | — | 4 | 1 | no | — |
| 61 | #11929608 | — | 4 | 1 | no | — |
| 62 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 63 | C79⋊C16 | — | 4 | 2 | no | — |
| 64 | #41×C79 | — | 8 | 4 | no | — |
| 65 | C2×Q8×C79 | — | 4 | 1 | no | — |
| 66 | C2×D4×C79 | — | 2 | 2 | no | — |
| 67 | C2×D4×C79 | — | 4 | 1 | no | — |
| 68 | C2×D4×C79 | — | 4 | 1 | no | — |
| 69 | C2×D4×C79 | — | 4 | 2 | no | — |
| 70 | C22×C4×C79 | — | 2 | 1 | no | — |
| 71 | C22×C4×C79 | — | 4 | 2 | no | — |
| 72 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | D8×C79 | — | 4 | 2 | no | — |
| 74 | D8×C79 | — | 8 | 1 | no | — |
| 75 | M16×C79 | — | 4 | 1 | no | — |
| 76 | M16×C79 | — | 4 | 1 | no | — |
| 77 | C2×C8×C79 | — | 2 | 1 | no | — |
| 78 | C2×C8×C79 | C2×D4 | 1 | 2 | yes | — |
| 79 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 80 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 81 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 82 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 83 | C42×C79 | — | 2 | 1 | no | — |
| 84 | C42×C79 | D8 | 1 | 1 | yes | characteristic |
| 85 | C42×C79 | D8 | 1 | 1 | yes | characteristic |
| 86 | C23×C79 | — | 4 | 2 | no | — |
| 87 | C23×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 88 | C22×D79 | — | 4 | 1 | no | — |
| 89 | Q8×C79 | — | 8 | 2 | no | — |
| 90 | D4×C79 | — | 4 | 2 | no | — |
| 91 | D4×C79 | — | 4 | 2 | no | — |
| 92 | D4×C79 | — | 8 | 2 | no | — |
| 93 | D4×C79 | — | 8 | 2 | no | — |
| 94 | D4×C79 | — | 8 | 4 | no | — |
| 95 | C2×C4×C79 | — | 2 | 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 | 2 | no | — |
| 100 | C2×C4×C79 | — | 4 | 2 | no | — |
| 101 | C2×C4×C79 | — | 8 | 4 | no | — |
| 102 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 103 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 104 | C22⋊D79 | — | 8 | 1 | no | — |
| 105 | C22⋊D79 | — | 8 | 1 | no | — |
| 106 | C2×Dic79 | — | 4 | 1 | no | — |
| 107 | D316 | — | 4 | 1 | no | — |
| 108 | D316 | — | 4 | 1 | no | — |
| 109 | C4×D79 | — | 8 | 1 | no | — |
| 110 | C4×D79 | — | 8 | 1 | no | — |
| 111 | Dic158 | — | 4 | 1 | no | — |
| 112 | Dic158 | — | 4 | 1 | no | — |
| 113 | C8×C79 | — | 2 | 2 | no | — |
| 114 | C8×C79 | — | 4 | 1 | no | — |
| 115 | C79⋊C8 | — | 8 | 1 | no | — |
| 116 | C22×C79 | — | 2 | 1 | no | — |
| 117 | C22×C79 | — | 4 | 1 | no | — |
| 118 | C22×C79 | — | 8 | 2 | no | — |
| 119 | C22×C79 | — | 8 | 2 | no | — |
| 120 | C22×C79 | — | 8 | 2 | no | — |
| 121 | C22×C79 | #447 | 1 | 1 | yes | characteristic |
| 122 | C2×D79 | — | 8 | 1 | no | — |
| 123 | C2×D79 | — | 16 | 1 | no | — |
| 124 | C4×C79 | — | 2 | 2 | no | — |
| 125 | C4×C79 | — | 2 | 1 | no | — |
| 126 | C4×C79 | — | 4 | 1 | no | — |
| 127 | C4×C79 | — | 8 | 2 | no | — |
| 128 | C4×C79 | #454 | 1 | 1 | yes | characteristic |
| 129 | C4×C79 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 130 | Dic79 | — | 8 | 1 | no | — |
| 131 | #12173 | — | 79 | 1 | no | maximal, Sylow |
| 132 | C2×C79 | — | 2 | 1 | no | — |
| 133 | C2×C79 | — | 4 | 1 | no | — |
| 134 | C2×C79 | — | 16 | 2 | no | — |
| 135 | C2×C79 | #1598 | 1 | 1 | yes | characteristic |
| 136 | D79 | — | 16 | 1 | no | — |
| 137 | #3253 | D79 | 1 | 1 | yes | characteristic, complemented |
| 138 | #2190 | — | 79 | 2 | no | — |
| 139 | #2050 | — | 79 | 1 | no | — |
| 140 | #1341 | — | 79 | 2 | no | — |
| 141 | #1327 | — | 79 | 1 | no | — |
| 142 | C79 | #12173 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 143 | #575 | — | 79 | 1 | no | — |
| 144 | #532 | — | 2 | 1 | no | — |
| 145 | #509 | — | 158 | 2 | no | — |
| 146 | #476 | — | 2 | 2 | no | — |
| 147 | #469 | — | 2 | 1 | no | — |
| 148 | #469 | C2×D79 | 1 | 2 | yes | — |
| 149 | #459 | — | 2 | 2 | no | — |
| 150 | #423 | C2×D79 | 1 | 1 | yes | characteristic |
| 151 | C2×#104 | — | 158 | 1 | no | — |
| 152 | #413 | — | 158 | 1 | no | — |
| 153 | #360 | — | 158 | 2 | no | — |
| 154 | #349 | — | 79 | 2 | no | — |
| 155 | #338 | — | 79 | 1 | no | — |
| 156 | C2×#41 | — | 4 | 1 | no | — |
| 157 | C2×#41 | — | 158 | 1 | no | — |
| 158 | #137 | — | 316 | 1 | no | — |
| 159 | AGL(1,ℤ/8) | — | 316 | 1 | no | — |
| 160 | #135 | — | 158 | 2 | no | — |
| 161 | C2×D8 | — | 2 | 1 | no | — |
| 162 | C2×D8 | C22⋊D79 | 1 | 2 | yes | — |
| 163 | C2×M16 | — | 158 | 1 | no | — |
| 164 | C2×M16 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 165 | C2×M16 | C22×D79 | 1 | 1 | yes | characteristic |
| 166 | C4⋊Q8 | — | 2 | 1 | no | — |
| 167 | #127 | — | 2 | 1 | no | — |
| 168 | #124 | — | 4 | 1 | no | — |
| 169 | #121 | — | 2 | 2 | no | — |
| 170 | #121 | — | 4 | 1 | no | — |
| 171 | C4×D4 | — | 2 | 2 | no | — |
| 172 | C4×D4 | — | 4 | 1 | no | — |
| 173 | C2×C42 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 174 | SD32 | — | 316 | 2 | no | — |
| 175 | D16 | — | 316 | 2 | no | — |
| 176 | M32 | — | 158 | 2 | no | — |
| 177 | #106 | — | 2 | 1 | no | — |
| 178 | #106 | C22⋊D79 | 1 | 2 | yes | — |
| 179 | C4⋊C8 | — | 2 | 2 | no | — |
| 180 | #104 | — | 316 | 1 | no | — |
| 181 | #104 | — | 316 | 1 | no | — |
| 182 | #102 | — | 2 | 2 | no | — |
| 183 | #102 | — | 2 | 2 | no | — |
| 184 | #101 | — | 316 | 1 | no | — |
| 185 | #100 | — | 316 | 1 | no | — |
| 186 | #41 | — | 8 | 4 | no | — |
| 187 | #41 | — | 316 | 1 | no | — |
| 188 | #41 | — | 316 | 1 | no | — |
| 189 | #41 | — | 316 | 1 | no | — |
| 190 | #41 | — | 316 | 1 | no | — |
| 191 | C2×Q8 | — | 4 | 1 | no | — |
| 192 | C2×Q8 | — | 158 | 1 | no | — |
| 193 | C2×D4 | — | 2 | 2 | no | — |
| 194 | C2×D4 | — | 4 | 1 | no | — |
| 195 | C2×D4 | — | 4 | 1 | no | — |
| 196 | C2×D4 | — | 4 | 2 | no | — |
| 197 | C2×D4 | — | 316 | 1 | no | — |
| 198 | C2×D4 | — | 158 | 1 | no | — |
| 199 | C22×C4 | — | 2 | 1 | no | — |
| 200 | C22×C4 | — | 4 | 2 | no | — |
| 201 | C22×C4 | — | 316 | 1 | no | — |
| 202 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 203 | Q16 | — | 158 | 2 | no | — |
| 204 | SD16 | — | 316 | 2 | no | — |
| 205 | D8 | — | 4 | 2 | no | — |
| 206 | D8 | — | 8 | 1 | no | — |
| 207 | D8 | — | 158 | 2 | no | — |
| 208 | M16 | — | 4 | 1 | no | — |
| 209 | M16 | — | 4 | 1 | no | — |
| 210 | M16 | — | 316 | 1 | no | — |
| 211 | M16 | — | 316 | 1 | no | — |
| 212 | C2×C8 | — | 2 | 1 | no | — |
| 213 | C2×C8 | — | 316 | 1 | no | — |
| 214 | C2×C8 | C2×C22⋊D79 | 1 | 2 | yes | — |
| 215 | C4⋊C4 | — | 2 | 2 | no | — |
| 216 | C4⋊C4 | — | 4 | 1 | no | — |
| 217 | C22⋊C4 | — | 4 | 2 | no | — |
| 218 | C22⋊C4 | — | 4 | 2 | no | — |
| 219 | C42 | — | 2 | 1 | no | — |
| 220 | C42 | #11929609 | 1 | 1 | yes | characteristic |
| 221 | C42 | #11929609 | 1 | 1 | yes | characteristic |
| 222 | C16 | — | 316 | 2 | no | — |
| 223 | C23 | — | 4 | 2 | no | — |
| 224 | C23 | — | 316 | 1 | no | — |
| 225 | C23 | #14579187 | 1 | 1 | yes | characteristic |
| 226 | Q8 | — | 8 | 2 | no | — |
| 227 | Q8 | — | 316 | 1 | no | — |
| 228 | Q8 | — | 316 | 1 | no | — |
| 229 | D4 | — | 4 | 2 | no | — |
| 230 | D4 | — | 4 | 2 | no | — |
| 231 | D4 | — | 8 | 2 | no | — |
| 232 | D4 | — | 8 | 2 | no | — |
| 233 | D4 | — | 8 | 4 | no | — |
| 234 | D4 | — | 316 | 1 | no | — |
| 235 | D4 | — | 316 | 1 | no | — |
| 236 | D4 | — | 632 | 1 | no | — |
| 237 | D4 | — | 632 | 1 | no | — |
| 238 | C2×C4 | — | 2 | 1 | no | — |
| 239 | C2×C4 | — | 2 | 1 | no | — |
| 240 | C2×C4 | — | 2 | 1 | no | — |
| 241 | C2×C4 | — | 2 | 1 | no | — |
| 242 | C2×C4 | — | 4 | 2 | no | — |
| 243 | C2×C4 | — | 4 | 2 | no | — |
| 244 | C2×C4 | — | 8 | 4 | no | — |
| 245 | C2×C4 | — | 316 | 1 | no | — |
| 246 | C2×C4 | — | 632 | 1 | no | — |
| 247 | C2×C4 | — | 632 | 1 | no | — |
| 248 | C2×C4 | #14579130 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 250 | C8 | — | 2 | 2 | no | — |
| 251 | C8 | — | 4 | 1 | no | — |
| 252 | C8 | — | 632 | 1 | no | — |
| 253 | C22 | — | 2 | 1 | no | — |
| 254 | C22 | — | 4 | 1 | no | — |
| 255 | C22 | — | 8 | 2 | no | — |
| 256 | C22 | — | 8 | 2 | no | — |
| 257 | C22 | — | 8 | 2 | no | — |
| 258 | C22 | — | 632 | 1 | no | — |
| 259 | C22 | — | 1264 | 1 | no | — |
| 260 | C22 | #19098697 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 2 | 2 | no | — |
| 262 | C4 | — | 2 | 1 | no | — |
| 263 | C4 | — | 4 | 1 | no | — |
| 264 | C4 | — | 8 | 2 | no | — |
| 265 | C4 | — | 632 | 1 | no | — |
| 266 | C4 | #19098736 | 1 | 1 | yes | characteristic |
| 267 | C4 | #19098750 | 1 | 1 | yes | characteristic |
| 268 | C2 | — | 2 | 1 | no | — |
| 269 | C2 | — | 4 | 1 | no | — |
| 270 | C2 | — | 16 | 2 | no | — |
| 271 | C2 | — | 1264 | 1 | no | — |
| 272 | C2 | #23762335 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23765345 | C2 | semidirect |
| 2. #23764687 | C2 | semidirect |
| 3. #23761391 | C2 | semidirect |
| 4. #23761351 | C2 | semidirect |
| 5. C79×#3253 | C2 | semidirect |
| 137. #3253 | D79 | semidirect |
| 142. C79 | #12173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23765345 | 2 |
| #23764687 | 2 |
| #23761391 | 2 |
| #23761351 | 2 |
| C79×#3253 | 2 |
| #12173 | 79 |