Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
2080 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 2080 conjugacy classes of subgroups — too many to draw. Only its 219 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 | C2×#19099127 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#19098699 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #23764398 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #23763720 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C79×#1780 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C23×D316 | C22 | 1 | 2 | yes | complemented |
| 7 | C22×D632 | C22 | 1 | 4 | yes | — |
| 8 | #19099127 | — | 2 | 16 | no | — |
| 9 | C2×#14579157 | — | 2 | 2 | no | — |
| 10 | C2×#14579150 | — | 2 | 2 | no | — |
| 11 | C2×#14579135 | C22 | 1 | 4 | yes | — |
| 12 | C2×#14579135 | C22 | 1 | 4 | yes | complemented |
| 13 | C2×#14579129 | — | 2 | 2 | no | — |
| 14 | C2×#14579129 | — | 2 | 2 | no | — |
| 15 | C2×#14579120 | C22 | 1 | 2 | yes | complemented |
| 16 | #19098699 | — | 2 | 16 | no | — |
| 17 | C2×#14579083 | C22 | 1 | 8 | yes | complemented |
| 18 | #19098326 | C22 | 1 | 4 | yes | complemented |
| 19 | C22×#41×C79 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C22⋊C8×C79 | C22 | 1 | 4 | yes | complemented |
| 21 | C2×C22⋊C8×C79 | C22 | 1 | 2 | yes | complemented |
| 22 | C24×D79 | — | 2 | 2 | no | — |
| 23 | C22×D316 | — | 2 | 2 | no | — |
| 24 | C22×D316 | — | 2 | 8 | no | — |
| 25 | C22×D316 | — | 2 | 8 | no | — |
| 26 | C22×D316 | — | 4 | 4 | no | — |
| 27 | C22×D316 | C23 | 1 | 4 | yes | — |
| 28 | C22×C22⋊D79 | — | 2 | 4 | no | — |
| 29 | C22×C22⋊D79 | — | 2 | 4 | no | — |
| 30 | C2×D632 | — | 2 | 16 | no | — |
| 31 | C2×D632 | — | 4 | 4 | no | — |
| 32 | #14579157 | — | 2 | 16 | no | — |
| 33 | #14579150 | — | 2 | 16 | no | — |
| 34 | #14579135 | — | 2 | 8 | no | — |
| 35 | #14579135 | — | 2 | 8 | no | — |
| 36 | #14579135 | — | 2 | 8 | no | — |
| 37 | #14579135 | D4 | 1 | 16 | yes | complemented |
| 38 | #14579129 | — | 2 | 16 | no | — |
| 39 | #14579129 | — | 2 | 16 | no | — |
| 40 | #14579120 | — | 2 | 8 | no | — |
| 41 | C2×C4⋊Dic79 | C23 | 1 | 4 | yes | — |
| 42 | C2×C22⋊Dic79 | — | 2 | 2 | no | — |
| 43 | C2×(C2×D79)⋊C4 | — | 2 | 4 | no | — |
| 44 | C2×(C2×D79)⋊C4 | — | 2 | 4 | no | — |
| 45 | C2×C4×Dic79 | — | 2 | 2 | no | — |
| 46 | #14579083 | — | 2 | 16 | no | — |
| 47 | C2×#41×C79 | — | 2 | 8 | no | — |
| 48 | C2×#41×C79 | — | 4 | 2 | no | — |
| 49 | C22×Q8×C79 | D4 | 1 | 1 | yes | characteristic |
| 50 | C22×D4×C79 | D4 | 1 | 1 | yes | characteristic |
| 51 | C22×D4×C79 | D4 | 1 | 2 | yes | — |
| 52 | C23×C4×C79 | C23 | 1 | 1 | yes | characteristic |
| 53 | C23×C4×C79 | C23 | 1 | 2 | yes | — |
| 54 | C22×C8×C79 | C23 | 1 | 4 | yes | — |
| 55 | C22⋊C8×C79 | — | 2 | 8 | no | — |
| 56 | C22⋊C8×C79 | — | 4 | 2 | no | — |
| 57 | C23×D79 | — | 2 | 4 | no | — |
| 58 | C23×D79 | — | 2 | 8 | no | — |
| 59 | C23×D79 | — | 4 | 4 | no | — |
| 60 | C23×D79 | — | 4 | 4 | no | — |
| 61 | C23×D79 | — | 8 | 8 | no | — |
| 62 | C2×D316 | — | 2 | 8 | no | — |
| 63 | C2×D316 | — | 2 | 4 | no | — |
| 64 | C2×D316 | — | 2 | 8 | no | — |
| 65 | C2×D316 | — | 2 | 8 | no | — |
| 66 | C2×D316 | — | 4 | 32 | no | — |
| 67 | C2×D316 | — | 4 | 4 | no | — |
| 68 | C2×D316 | — | 4 | 4 | no | — |
| 69 | C2×D316 | C2×D4 | 1 | 8 | yes | — |
| 70 | C2×C22⋊D79 | — | 2 | 16 | no | — |
| 71 | C2×C22⋊D79 | — | 2 | 16 | no | — |
| 72 | C2×C22⋊D79 | — | 8 | 8 | no | — |
| 73 | C2×C22⋊D79 | — | 8 | 8 | no | — |
| 74 | C22×Dic79 | — | 2 | 4 | no | — |
| 75 | D632 | — | 4 | 16 | no | — |
| 76 | C4⋊Dic79 | — | 2 | 4 | no | — |
| 77 | C4⋊Dic79 | C2×D4 | 1 | 8 | yes | — |
| 78 | C22⋊Dic79 | — | 2 | 8 | no | — |
| 79 | (C2×D79)⋊C4 | — | 2 | 16 | no | — |
| 80 | (C2×D79)⋊C4 | — | 2 | 16 | no | — |
| 81 | C4×Dic79 | — | 2 | 8 | no | — |
| 82 | C24×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C24×C79 | C2×D4 | 1 | 2 | yes | — |
| 84 | #41×C79 | — | 4 | 16 | no | — |
| 85 | C2×Q8×C79 | — | 2 | 4 | no | — |
| 86 | C2×Q8×C79 | — | 4 | 1 | no | — |
| 87 | C2×D4×C79 | — | 2 | 4 | no | — |
| 88 | C2×D4×C79 | — | 2 | 4 | no | — |
| 89 | C2×D4×C79 | — | 2 | 4 | no | — |
| 90 | C2×D4×C79 | — | 4 | 2 | no | — |
| 91 | C2×D4×C79 | — | 4 | 1 | no | — |
| 92 | C22×C4×C79 | — | 2 | 4 | no | — |
| 93 | C22×C4×C79 | — | 2 | 4 | no | — |
| 94 | C22×C4×C79 | — | 2 | 4 | no | — |
| 95 | C22×C4×C79 | — | 2 | 2 | no | — |
| 96 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C22×C4×C79 | C2×D4 | 1 | 2 | yes | — |
| 98 | C22×C4×C79 | C2×D4 | 1 | 8 | yes | — |
| 99 | C22×C4×C79 | C24 | 1 | 1 | yes | characteristic |
| 100 | C2×C8×C79 | — | 2 | 8 | no | — |
| 101 | C2×C8×C79 | — | 2 | 4 | no | — |
| 102 | C23×C79 | — | 2 | 2 | no | — |
| 103 | C23×C79 | — | 2 | 2 | no | — |
| 104 | C23×C79 | — | 2 | 8 | no | — |
| 105 | C23×C79 | — | 2 | 2 | no | — |
| 106 | C23×C79 | — | 2 | 1 | no | — |
| 107 | C23×C79 | — | 2 | 4 | no | — |
| 108 | C23×C79 | #120 | 1 | 4 | yes | — |
| 109 | C23×C79 | C22×D4 | 1 | 1 | yes | characteristic |
| 110 | C22×D79 | — | 2 | 8 | no | — |
| 111 | C22×D79 | — | 4 | 16 | no | — |
| 112 | C22×D79 | — | 4 | 4 | no | — |
| 113 | C22×D79 | — | 4 | 4 | no | — |
| 114 | C22×D79 | — | 4 | 8 | no | — |
| 115 | C22×D79 | — | 4 | 8 | no | — |
| 116 | C22×D79 | — | 8 | 16 | no | — |
| 117 | C22×D79 | — | 8 | 16 | no | — |
| 118 | C22×D79 | — | 8 | 16 | no | — |
| 119 | Q8×C79 | — | 4 | 4 | no | — |
| 120 | D4×C79 | — | 4 | 8 | no | — |
| 121 | D4×C79 | — | 4 | 4 | no | — |
| 122 | C2×C4×C79 | — | 2 | 2 | no | — |
| 123 | C2×C4×C79 | — | 2 | 2 | no | — |
| 124 | C2×C4×C79 | — | 2 | 16 | no | — |
| 125 | C2×C4×C79 | — | 2 | 2 | no | — |
| 126 | C2×C4×C79 | — | 2 | 1 | no | — |
| 127 | C2×C4×C79 | — | 4 | 4 | no | — |
| 128 | C2×C4×C79 | #120 | 1 | 4 | yes | — |
| 129 | C2×C4×C79 | C22×D4 | 1 | 4 | yes | — |
| 130 | C2×C4×C79 | H(5,𝔽2) | 1 | 2 | yes | — |
| 131 | C22⋊D79 | — | 8 | 16 | no | — |
| 132 | C22⋊D79 | — | 8 | 16 | no | — |
| 133 | C2×Dic79 | — | 2 | 8 | no | — |
| 134 | C2×Dic79 | — | 4 | 8 | no | — |
| 135 | D316 | — | 2 | 16 | no | — |
| 136 | D316 | — | 4 | 16 | no | — |
| 137 | D316 | — | 4 | 8 | no | — |
| 138 | C8×C79 | — | 4 | 4 | no | — |
| 139 | C22×C79 | — | 2 | 4 | no | — |
| 140 | C22×C79 | — | 2 | 8 | no | — |
| 141 | C22×C79 | — | 4 | 4 | no | — |
| 142 | C22×C79 | — | 4 | 4 | no | — |
| 143 | C22×C79 | — | 4 | 1 | no | — |
| 144 | C22×C79 | — | 4 | 2 | no | — |
| 145 | C22×C79 | — | 4 | 4 | no | — |
| 146 | C22×C79 | C2×#120 | 1 | 2 | yes | — |
| 147 | C22×C79 | #534 | 1 | 1 | yes | characteristic |
| 148 | C22×C79 | #576 | 1 | 4 | yes | — |
| 149 | C2×D79 | — | 4 | 8 | no | — |
| 150 | C2×D79 | — | 4 | 8 | no | — |
| 151 | C2×D79 | — | 4 | 8 | no | — |
| 152 | C2×D79 | — | 8 | 32 | no | — |
| 153 | C2×D79 | — | 8 | 8 | no | — |
| 154 | C2×D79 | — | 8 | 8 | no | — |
| 155 | C4×C79 | — | 2 | 4 | no | — |
| 156 | C4×C79 | — | 4 | 1 | no | — |
| 157 | C4×C79 | D42 | 1 | 4 | yes | — |
| 158 | Dic79 | — | 4 | 8 | no | — |
| 159 | #23925 | — | 79 | 1 | no | maximal, Sylow |
| 160 | C2×C79 | — | 4 | 4 | no | — |
| 161 | C2×C79 | — | 8 | 1 | no | — |
| 162 | C2×C79 | #2403 | 1 | 1 | yes | characteristic |
| 163 | C2×C79 | #3008 | 1 | 4 | yes | — |
| 164 | C2×C79 | #3199 | 1 | 2 | yes | — |
| 165 | D79 | — | 8 | 8 | no | — |
| 166 | C2×#535 | — | 79 | 2 | no | — |
| 167 | C2×#447 | — | 79 | 4 | no | — |
| 168 | #1999 | — | 79 | 4 | no | — |
| 169 | #1875 | — | 79 | 4 | no | — |
| 170 | #1780 | D79 | 1 | 1 | yes | characteristic, complemented |
| 171 | C79 | #23925 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | C22×#41 | C2×D79 | 1 | 1 | yes | characteristic |
| 173 | C23×D4 | — | 79 | 2 | no | — |
| 174 | C22×D8 | — | 79 | 4 | no | — |
| 175 | #535 | — | 158 | 16 | no | — |
| 176 | C2×#127 | — | 158 | 2 | no | — |
| 177 | C2×#124 | — | 158 | 2 | no | — |
| 178 | C2×#121 | — | 79 | 4 | no | — |
| 179 | C2×#121 | — | 79 | 4 | no | — |
| 180 | C2×#120 | — | 158 | 2 | no | — |
| 181 | C2×#120 | — | 158 | 2 | no | — |
| 182 | C2×#117 | — | 79 | 2 | no | — |
| 183 | #447 | — | 158 | 16 | no | — |
| 184 | C2×#102 | — | 79 | 8 | no | — |
| 185 | C2×C22⋊C8 | C2×D79 | 1 | 4 | yes | complemented |
| 186 | C2×C22⋊C8 | C2×D79 | 1 | 2 | yes | complemented |
| 187 | #340 | — | 79 | 4 | no | — |
| 188 | C25 | — | 158 | 2 | no | — |
| 189 | C2×#41 | — | 2 | 8 | no | — |
| 190 | C2×#41 | — | 4 | 2 | no | — |
| 191 | C22×Q8 | D316 | 1 | 1 | yes | characteristic |
| 192 | C22×D4 | — | 316 | 4 | no | — |
| 193 | C22×D4 | — | 79 | 4 | no | — |
| 194 | C22×D4 | — | 158 | 8 | no | — |
| 195 | C22×D4 | — | 158 | 4 | no | — |
| 196 | C22×D4 | — | 158 | 4 | no | — |
| 197 | C22×D4 | — | 158 | 8 | no | — |
| 198 | C22×D4 | — | 158 | 2 | no | — |
| 199 | C22×D4 | D316 | 1 | 1 | yes | characteristic |
| 200 | C22×D4 | D316 | 1 | 2 | yes | — |
| 201 | C23×C4 | C22×D79 | 1 | 1 | yes | characteristic |
| 202 | C23×C4 | C22×D79 | 1 | 2 | yes | — |
| 203 | C2×D8 | — | 316 | 4 | no | — |
| 204 | C2×D8 | — | 158 | 16 | no | — |
| 205 | C22×C8 | C22×D79 | 1 | 4 | yes | — |
| 206 | #127 | — | 158 | 16 | no | — |
| 207 | #124 | — | 158 | 16 | no | — |
| 208 | #121 | — | 79 | 16 | no | — |
| 209 | #121 | — | 158 | 8 | no | — |
| 210 | #121 | — | 158 | 8 | no | — |
| 211 | #121 | — | 158 | 8 | no | — |
| 212 | #120 | — | 158 | 16 | no | — |
| 213 | #120 | — | 158 | 16 | no | — |
| 214 | #117 | — | 158 | 8 | no | — |
| 215 | C2×C4⋊C4 | — | 79 | 4 | no | — |
| 216 | C2×C22⋊C4 | — | 158 | 2 | no | — |
| 217 | C2×C22⋊C4 | — | 158 | 4 | no | — |
| 218 | C2×C22⋊C4 | — | 158 | 4 | no | — |
| 219 | C2×C42 | — | 158 | 2 | no | — |
| 220 | #102 | — | 158 | 16 | no | — |
| 221 | C22⋊C8 | — | 2 | 8 | no | — |
| 222 | C22⋊C8 | — | 4 | 2 | no | — |
| 223 | C24 | — | 316 | 4 | no | — |
| 224 | C24 | — | 316 | 4 | no | — |
| 225 | C24 | — | 632 | 8 | no | — |
| 226 | C24 | — | 158 | 8 | no | — |
| 227 | C24 | — | 158 | 4 | no | — |
| 228 | C24 | C2×D316 | 1 | 1 | yes | characteristic |
| 229 | C24 | C2×D316 | 1 | 2 | yes | — |
| 230 | #41 | — | 4 | 16 | no | — |
| 231 | C2×Q8 | — | 2 | 4 | no | — |
| 232 | C2×Q8 | — | 4 | 1 | no | — |
| 233 | C2×D4 | — | 2 | 4 | no | — |
| 234 | C2×D4 | — | 2 | 4 | no | — |
| 235 | C2×D4 | — | 2 | 4 | no | — |
| 236 | C2×D4 | — | 4 | 2 | no | — |
| 237 | C2×D4 | — | 4 | 1 | no | — |
| 238 | C2×D4 | — | 316 | 4 | no | — |
| 239 | C2×D4 | — | 316 | 32 | no | — |
| 240 | C2×D4 | — | 316 | 4 | no | — |
| 241 | C2×D4 | — | 79 | 8 | no | — |
| 242 | C2×D4 | — | 632 | 8 | no | — |
| 243 | C2×D4 | — | 632 | 8 | no | — |
| 244 | C2×D4 | — | 158 | 8 | no | — |
| 245 | C2×D4 | — | 158 | 16 | no | — |
| 246 | C2×D4 | — | 158 | 16 | no | — |
| 247 | C2×D4 | — | 158 | 8 | no | — |
| 248 | C2×D4 | — | 158 | 4 | no | — |
| 249 | C2×D4 | — | 158 | 8 | no | — |
| 250 | C22×C4 | — | 2 | 4 | no | — |
| 251 | C22×C4 | — | 2 | 4 | no | — |
| 252 | C22×C4 | — | 2 | 2 | no | — |
| 253 | C22×C4 | — | 2 | 4 | no | — |
| 254 | C22×C4 | — | 158 | 4 | no | — |
| 255 | C22×C4 | D4×D79 | 1 | 8 | yes | — |
| 256 | C22×C4 | C2×D316 | 1 | 1 | yes | characteristic |
| 257 | C22×C4 | C2×D316 | 1 | 2 | yes | — |
| 258 | C22×C4 | C23×D79 | 1 | 1 | yes | characteristic |
| 259 | D8 | — | 316 | 16 | no | — |
| 260 | C2×C8 | — | 2 | 4 | no | — |
| 261 | C2×C8 | — | 2 | 8 | no | — |
| 262 | C4⋊C4 | — | 79 | 8 | no | — |
| 263 | C4⋊C4 | — | 158 | 4 | no | — |
| 264 | C22⋊C4 | — | 158 | 16 | no | — |
| 265 | C22⋊C4 | — | 158 | 8 | no | — |
| 266 | C22⋊C4 | — | 158 | 16 | no | — |
| 267 | C42 | — | 158 | 8 | no | — |
| 268 | C23 | — | 2 | 2 | no | — |
| 269 | C23 | — | 2 | 8 | no | — |
| 270 | C23 | — | 2 | 2 | no | — |
| 271 | C23 | — | 2 | 2 | no | — |
| 272 | C23 | — | 2 | 4 | no | — |
| 273 | C23 | — | 2 | 1 | no | — |
| 274 | C23 | — | 316 | 16 | no | — |
| 275 | C23 | — | 316 | 4 | no | — |
| 276 | C23 | — | 316 | 8 | no | — |
| 277 | C23 | — | 316 | 8 | no | — |
| 278 | C23 | — | 316 | 4 | no | — |
| 279 | C23 | — | 632 | 16 | no | — |
| 280 | C23 | — | 632 | 16 | no | — |
| 281 | C23 | — | 632 | 16 | no | — |
| 282 | C23 | — | 158 | 8 | no | — |
| 283 | C23 | #14579131 | 1 | 4 | yes | — |
| 284 | C23 | C22×D316 | 1 | 1 | yes | characteristic |
| 285 | Q8 | — | 4 | 4 | no | — |
| 286 | D4 | — | 4 | 8 | no | — |
| 287 | D4 | — | 4 | 4 | no | — |
| 288 | D4 | — | 316 | 16 | no | — |
| 289 | D4 | — | 316 | 8 | no | — |
| 290 | D4 | — | 632 | 16 | no | — |
| 291 | D4 | — | 632 | 16 | no | — |
| 292 | D4 | — | 158 | 16 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 2 | 2 | no | — |
| 295 | C2×C4 | — | 2 | 16 | no | — |
| 296 | C2×C4 | — | 2 | 2 | no | — |
| 297 | C2×C4 | — | 2 | 1 | no | — |
| 298 | C2×C4 | — | 4 | 4 | no | — |
| 299 | C2×C4 | — | 316 | 8 | no | — |
| 300 | C2×C4 | — | 158 | 8 | no | — |
| 301 | C2×C4 | #14579131 | 1 | 4 | yes | — |
| 302 | C2×C4 | C2×D4×D79 | 1 | 4 | yes | — |
| 303 | C2×C4 | #14579206 | 1 | 2 | yes | — |
| 304 | C8 | — | 4 | 4 | no | — |
| 305 | C22 | — | 2 | 4 | no | — |
| 306 | C22 | — | 2 | 8 | no | — |
| 307 | C22 | — | 4 | 4 | no | — |
| 308 | C22 | — | 4 | 4 | no | — |
| 309 | C22 | — | 4 | 2 | no | — |
| 310 | C22 | — | 4 | 1 | no | — |
| 311 | C22 | — | 4 | 4 | no | — |
| 312 | C22 | — | 316 | 8 | no | — |
| 313 | C22 | — | 316 | 8 | no | — |
| 314 | C22 | — | 316 | 8 | no | — |
| 315 | C22 | — | 632 | 32 | no | — |
| 316 | C22 | — | 632 | 8 | no | — |
| 317 | C22 | — | 632 | 8 | no | — |
| 318 | C22 | C2×#14579131 | 1 | 2 | yes | — |
| 319 | C22 | #19099125 | 1 | 1 | yes | characteristic |
| 320 | C22 | #19099368 | 1 | 4 | yes | — |
| 321 | C4 | — | 2 | 4 | no | — |
| 322 | C4 | — | 4 | 1 | no | — |
| 323 | C4 | — | 316 | 8 | no | — |
| 324 | C4 | #19099195 | 1 | 4 | yes | — |
| 325 | C2 | — | 4 | 4 | no | — |
| 326 | C2 | — | 8 | 1 | no | — |
| 327 | C2 | — | 632 | 8 | no | — |
| 328 | C2 | #23766713 | 1 | 1 | yes | characteristic |
| 329 | C2 | #23772670 | 1 | 4 | yes | — |
| 330 | C2 | #23774265 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19099127 | C2 | semidirect |
| 2. C2×#19098699 | C2 | semidirect |
| 3. #23764398 | C2 | semidirect |
| 4. #23763720 | C2 | semidirect |
| 5. C79×#1780 | C2 | semidirect |
| 6. C23×D316 | C22 | semidirect |
| 12. C2×#14579135 | C22 | semidirect |
| 15. C2×#14579120 | C22 | semidirect |
| 17. C2×#14579083 | C22 | semidirect |
| 18. #19098326 | C22 | semidirect |
| 20. C2×C22⋊C8×C79 | C22 | semidirect |
| 21. C2×C22⋊C8×C79 | C22 | semidirect |
| 37. #14579135 | D4 | semidirect |
| 170. #1780 | D79 | semidirect |
| 171. C79 | #23925 | semidirect |
| 185. C2×C22⋊C8 | C2×D79 | semidirect |
| 186. C2×C22⋊C8 | C2×D79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19099127 | 2 |
| C2×#19098699 | 2 |
| #23764398 | 2 |
| #23763720 | 2 |
| C79×#1780 | 2 |
| #23925 | 79 |