Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1138 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 171 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×#90391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C7×#490 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#89846 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#89338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11729002 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 6 | C2×#89259 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×C7×#127 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×C7×#125 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#1142 | — | 2 | 2 | no | — |
| 10 | C22×#1141 | — | 2 | 2 | no | — |
| 11 | #90391 | — | 2 | 2 | no | — |
| 12 | #90391 | — | 2 | 4 | no | — |
| 13 | #90391 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×#6459 | — | 2 | 2 | no | — |
| 15 | C2×#6458 | — | 2 | 2 | no | — |
| 16 | C2×#6457 | — | 2 | 2 | no | — |
| 17 | C2×#6438 | C22 | 1 | 2 | yes | — |
| 18 | C7×#490 | — | 2 | 2 | no | — |
| 19 | C7×#490 | C22 | 1 | 4 | yes | complemented |
| 20 | C2×C7×#102 | — | 2 | 2 | no | — |
| 21 | C2×C7×#97 | C22 | 1 | 1 | yes | characteristic |
| 22 | #89846 | — | 2 | 4 | no | — |
| 23 | #89846 | C22 | 1 | 8 | yes | complemented |
| 24 | C2×#6377 | C22 | 1 | 2 | yes | — |
| 25 | #89338 | C22 | 1 | 4 | yes | complemented |
| 26 | #89259 | C22 | 1 | 8 | yes | complemented |
| 27 | C2×#1656 | — | 7 | 1 | no | maximal, Sylow |
| 28 | C22×C7×D4 | — | 2 | 2 | no | — |
| 29 | C22×C7×D4 | — | 2 | 1 | no | — |
| 30 | C22×D28 | — | 2 | 2 | no | — |
| 31 | C22×D28 | D4 | 1 | 2 | yes | — |
| 32 | C22×C4×D7 | — | 2 | 2 | no | — |
| 33 | C22×Dic14 | D4 | 1 | 2 | yes | — |
| 34 | C22×C7×C8 | — | 2 | 1 | no | — |
| 35 | C7×#127 | — | 2 | 1 | no | — |
| 36 | C7×#127 | — | 2 | 2 | no | — |
| 37 | C7×#127 | C23 | 1 | 2 | yes | — |
| 38 | C7×#125 | — | 2 | 2 | no | — |
| 39 | C7×#125 | D4 | 1 | 2 | yes | — |
| 40 | C7×#125 | C23 | 1 | 2 | yes | — |
| 41 | C2×C7×C4⋊C4 | — | 2 | 2 | no | — |
| 42 | C2×C7×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 43 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×#1142 | — | 2 | 8 | no | — |
| 45 | C2×#1142 | — | 4 | 8 | no | — |
| 46 | C2×#1141 | — | 2 | 8 | no | — |
| 47 | C2×#1141 | — | 4 | 8 | no | — |
| 48 | C2×(C2×D7)⋊C4 | — | 2 | 2 | no | — |
| 49 | C2×(C2×D7)⋊C4 | — | 2 | 2 | no | — |
| 50 | C2×(C2×D7)⋊C4 | — | 2 | 2 | no | — |
| 51 | C2×Dic7⋊C4 | — | 2 | 2 | no | — |
| 52 | C22×C7⋊C8 | — | 2 | 2 | no | — |
| 53 | #6459 | — | 2 | 8 | no | — |
| 54 | #6459 | — | 2 | 8 | no | — |
| 55 | #6458 | — | 2 | 8 | no | — |
| 56 | #6458 | — | 2 | 8 | no | — |
| 57 | #6457 | — | 2 | 8 | no | — |
| 58 | #6457 | — | 2 | 8 | no | — |
| 59 | #6438 | — | 2 | 2 | no | — |
| 60 | #6438 | — | 2 | 2 | no | — |
| 61 | #6438 | D4 | 1 | 4 | yes | — |
| 62 | #6438 | C23 | 1 | 4 | yes | — |
| 63 | C7×#102 | — | 2 | 8 | no | — |
| 64 | C7×#97 | — | 2 | 1 | no | — |
| 65 | C7×#97 | C23 | 1 | 2 | yes | — |
| 66 | #6377 | — | 2 | 2 | no | — |
| 67 | #6377 | C23 | 1 | 4 | yes | — |
| 68 | C2×#562 | — | 7 | 1 | no | — |
| 69 | C2×#496 | — | 7 | 2 | no | — |
| 70 | C2×#490 | D7 | 1 | 1 | yes | characteristic, complemented |
| 71 | #1656 | — | 7 | 8 | no | — |
| 72 | C2×#330 | — | 7 | 1 | no | — |
| 73 | C2×#329 | — | 7 | 2 | no | — |
| 74 | C24×C7 | — | 4 | 1 | no | — |
| 75 | C23×D7 | — | 2 | 2 | no | — |
| 76 | C2×C7×D4 | — | 2 | 2 | no | — |
| 77 | C2×C7×D4 | — | 2 | 4 | no | — |
| 78 | C2×C7×D4 | — | 4 | 4 | no | — |
| 79 | C2×C7×D4 | — | 4 | 4 | no | — |
| 80 | C2×C7×D4 | — | 4 | 4 | no | — |
| 81 | C2×C7×D4 | — | 4 | 1 | no | — |
| 82 | C2×C7×D4 | — | 4 | 2 | no | — |
| 83 | C22×C4×C7 | — | 2 | 2 | no | — |
| 84 | C22×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 85 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C22×Dic7 | — | 2 | 2 | no | — |
| 87 | C2×D28 | — | 2 | 8 | no | — |
| 88 | C2×D28 | — | 2 | 2 | no | — |
| 89 | C2×D28 | — | 4 | 4 | no | — |
| 90 | C2×D28 | — | 8 | 4 | no | — |
| 91 | C2×D28 | C2×D4 | 1 | 4 | yes | — |
| 92 | C2×C4×D7 | — | 2 | 8 | no | — |
| 93 | C2×C4×D7 | — | 8 | 4 | no | — |
| 94 | C2×Dic14 | — | 2 | 2 | no | — |
| 95 | C2×Dic14 | — | 4 | 4 | no | — |
| 96 | C2×Dic14 | C2×D4 | 1 | 4 | yes | — |
| 97 | C2×C7×C8 | — | 2 | 2 | no | — |
| 98 | C2×C7×C8 | — | 4 | 2 | no | — |
| 99 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 100 | C7×C4⋊C4 | — | 2 | 4 | no | — |
| 101 | C7×C4⋊C4 | — | 2 | 4 | no | — |
| 102 | C7×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 103 | C42×C7 | C2×D4 | 1 | 2 | yes | — |
| 104 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | C42×C7 | C24 | 1 | 1 | yes | characteristic |
| 106 | #1142 | — | 4 | 16 | no | — |
| 107 | #1141 | — | 4 | 16 | no | — |
| 108 | (C2×D7)⋊C4 | — | 2 | 8 | no | — |
| 109 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 110 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 111 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 112 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 113 | Dic7⋊C4 | — | 2 | 8 | no | — |
| 114 | C2×C7⋊C8 | — | 2 | 8 | no | — |
| 115 | C2×C7⋊C8 | — | 2 | 4 | no | — |
| 116 | C22×SD16 | — | 14 | 2 | no | — |
| 117 | C22×D8 | — | 14 | 2 | no | — |
| 118 | #562 | — | 7 | 4 | no | — |
| 119 | #562 | — | 14 | 4 | no | — |
| 120 | #562 | — | 14 | 2 | no | — |
| 121 | C2×#127 | C2×D7 | 1 | 1 | yes | characteristic |
| 122 | C2×#125 | C2×D7 | 1 | 1 | yes | characteristic |
| 123 | C2×#124 | — | 7 | 2 | no | — |
| 124 | C2×#123 | — | 14 | 2 | no | — |
| 125 | C2×C22⋊Q8 | — | 14 | 2 | no | — |
| 126 | C2×#121 | — | 14 | 2 | no | — |
| 127 | #496 | — | 7 | 8 | no | — |
| 128 | #496 | — | 14 | 4 | no | — |
| 129 | #490 | — | 2 | 2 | no | — |
| 130 | #490 | C2×D7 | 1 | 4 | yes | complemented |
| 131 | C2×#102 | — | 2 | 2 | no | — |
| 132 | C2×#97 | — | 7 | 2 | no | — |
| 133 | C2×#97 | C2×D7 | 1 | 1 | yes | characteristic |
| 134 | #330 | — | 7 | 4 | no | — |
| 135 | #329 | — | 7 | 8 | no | — |
| 136 | C23×C7 | — | 4 | 2 | no | — |
| 137 | C23×C7 | — | 8 | 4 | no | — |
| 138 | C23×C7 | — | 8 | 2 | no | — |
| 139 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 140 | C22×D7 | — | 2 | 4 | no | — |
| 141 | C22×D7 | — | 4 | 4 | no | — |
| 142 | C22×D7 | — | 4 | 4 | no | — |
| 143 | C22×D7 | — | 4 | 4 | no | — |
| 144 | C7×D4 | — | 4 | 4 | no | — |
| 145 | C7×D4 | — | 4 | 4 | no | — |
| 146 | C7×D4 | — | 4 | 4 | no | — |
| 147 | C7×D4 | — | 8 | 4 | no | — |
| 148 | C7×D4 | — | 8 | 2 | no | — |
| 149 | C2×C4×C7 | — | 2 | 4 | no | — |
| 150 | C2×C4×C7 | — | 2 | 2 | no | — |
| 151 | C2×C4×C7 | — | 2 | 4 | no | — |
| 152 | C2×C4×C7 | — | 4 | 4 | no | — |
| 153 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C7 | #120 | 1 | 2 | yes | — |
| 155 | C2×C4×C7 | C22×D4 | 1 | 2 | yes | — |
| 156 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 157 | C2×Dic7 | — | 2 | 4 | no | — |
| 158 | C2×Dic7 | — | 4 | 4 | no | — |
| 159 | D28 | — | 4 | 4 | no | — |
| 160 | D28 | — | 8 | 2 | no | — |
| 161 | D28 | — | 8 | 8 | no | — |
| 162 | C4×D7 | — | 8 | 8 | no | — |
| 163 | Dic14 | — | 4 | 4 | no | — |
| 164 | Dic14 | — | 8 | 2 | no | — |
| 165 | C7×C8 | — | 4 | 2 | no | — |
| 166 | C7⋊C8 | — | 2 | 8 | no | — |
| 167 | C22×Q8 | — | 7 | 2 | no | — |
| 168 | C22×D4 | — | 2 | 2 | no | — |
| 169 | C22×D4 | — | 2 | 1 | no | — |
| 170 | C22×D4 | — | 7 | 2 | no | — |
| 171 | C22×D4 | — | 14 | 2 | no | — |
| 172 | C23×C4 | — | 14 | 2 | no | — |
| 173 | C2×SD16 | — | 14 | 8 | no | — |
| 174 | C2×SD16 | — | 28 | 8 | no | — |
| 175 | C2×D8 | — | 14 | 8 | no | — |
| 176 | C2×D8 | — | 28 | 8 | no | — |
| 177 | C22×C8 | — | 2 | 1 | no | — |
| 178 | C22×C8 | — | 14 | 2 | no | — |
| 179 | #127 | — | 2 | 2 | no | — |
| 180 | #127 | — | 2 | 1 | no | — |
| 181 | #127 | C22×D7 | 1 | 2 | yes | — |
| 182 | #125 | — | 2 | 2 | no | — |
| 183 | #125 | C22⋊D7 | 1 | 2 | yes | — |
| 184 | #125 | C22×D7 | 1 | 2 | yes | — |
| 185 | #124 | — | 7 | 4 | no | — |
| 186 | #124 | — | 7 | 4 | no | — |
| 187 | #124 | — | 14 | 2 | no | — |
| 188 | #124 | — | 14 | 2 | no | — |
| 189 | #123 | — | 14 | 8 | no | — |
| 190 | #123 | — | 14 | 8 | no | — |
| 191 | C22⋊Q8 | — | 14 | 8 | no | — |
| 192 | C22⋊Q8 | — | 14 | 8 | no | — |
| 193 | #121 | — | 14 | 8 | no | — |
| 194 | #121 | — | 14 | 8 | no | — |
| 195 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 196 | C2×C4⋊C4 | — | 14 | 2 | no | — |
| 197 | C2×C4⋊C4 | C22⋊D7 | 1 | 2 | yes | — |
| 198 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 199 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 200 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 201 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 202 | #102 | — | 2 | 8 | no | — |
| 203 | #97 | — | 2 | 1 | no | — |
| 204 | #97 | — | 7 | 4 | no | — |
| 205 | #97 | — | 14 | 2 | no | — |
| 206 | #97 | C22×D7 | 1 | 2 | yes | — |
| 207 | C22×C7 | — | 8 | 4 | no | — |
| 208 | C22×C7 | — | 8 | 2 | no | — |
| 209 | C22×C7 | — | 16 | 4 | no | — |
| 210 | C22×C7 | #453 | 1 | 4 | yes | — |
| 211 | C22×C7 | #454 | 1 | 2 | yes | — |
| 212 | C22×C7 | C2×#120 | 1 | 1 | yes | characteristic |
| 213 | C2×D7 | — | 4 | 4 | no | — |
| 214 | C2×D7 | — | 4 | 4 | no | — |
| 215 | C2×D7 | — | 4 | 4 | no | — |
| 216 | C2×D7 | — | 8 | 8 | no | — |
| 217 | C4×C7 | — | 2 | 2 | no | — |
| 218 | C4×C7 | — | 2 | 2 | no | — |
| 219 | C4×C7 | — | 2 | 1 | no | — |
| 220 | C4×C7 | — | 2 | 1 | no | — |
| 221 | C4×C7 | — | 4 | 4 | no | — |
| 222 | Dic7 | — | 4 | 4 | no | — |
| 223 | C24 | — | 4 | 1 | no | — |
| 224 | C24 | — | 14 | 2 | no | — |
| 225 | C2×Q8 | — | 7 | 4 | no | — |
| 226 | C2×Q8 | — | 14 | 2 | no | — |
| 227 | C2×Q8 | — | 28 | 4 | no | — |
| 228 | C2×D4 | — | 2 | 4 | no | — |
| 229 | C2×D4 | — | 2 | 2 | no | — |
| 230 | C2×D4 | — | 4 | 4 | no | — |
| 231 | C2×D4 | — | 4 | 4 | no | — |
| 232 | C2×D4 | — | 4 | 4 | no | — |
| 233 | C2×D4 | — | 4 | 1 | no | — |
| 234 | C2×D4 | — | 4 | 2 | no | — |
| 235 | C2×D4 | — | 7 | 4 | no | — |
| 236 | C2×D4 | — | 14 | 8 | no | — |
| 237 | C2×D4 | — | 14 | 2 | no | — |
| 238 | C2×D4 | — | 28 | 4 | no | — |
| 239 | C2×D4 | — | 56 | 4 | no | — |
| 240 | C22×C4 | — | 2 | 2 | no | — |
| 241 | C22×C4 | — | 14 | 8 | no | — |
| 242 | C22×C4 | — | 14 | 2 | no | — |
| 243 | C22×C4 | — | 56 | 4 | no | — |
| 244 | C22×C4 | D4×D7 | 1 | 2 | yes | — |
| 245 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 246 | SD16 | — | 28 | 16 | no | — |
| 247 | D8 | — | 28 | 16 | no | — |
| 248 | C2×C8 | — | 2 | 2 | no | — |
| 249 | C2×C8 | — | 4 | 2 | no | — |
| 250 | C2×C8 | — | 14 | 8 | no | — |
| 251 | C2×C8 | — | 14 | 4 | no | — |
| 252 | C4⋊C4 | — | 2 | 4 | no | — |
| 253 | C4⋊C4 | — | 2 | 2 | no | — |
| 254 | C4⋊C4 | — | 2 | 4 | no | — |
| 255 | C4⋊C4 | — | 14 | 8 | no | — |
| 256 | C4⋊C4 | C2×C22⋊D7 | 1 | 4 | yes | — |
| 257 | C22⋊C4 | — | 14 | 8 | no | — |
| 258 | C22⋊C4 | — | 14 | 4 | no | — |
| 259 | C22⋊C4 | — | 14 | 4 | no | — |
| 260 | C22⋊C4 | — | 14 | 4 | no | — |
| 261 | C22⋊C4 | — | 14 | 4 | no | — |
| 262 | C42 | D4×D7 | 1 | 2 | yes | — |
| 263 | C42 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 264 | C42 | C23×D7 | 1 | 1 | yes | characteristic |
| 265 | C2×C7 | — | 16 | 2 | no | — |
| 266 | C2×C7 | #1656 | 1 | 4 | yes | complemented |
| 267 | C2×C7 | C2×#453 | 1 | 2 | yes | — |
| 268 | C2×C7 | C2×#454 | 1 | 1 | yes | characteristic |
| 269 | D7 | — | 8 | 4 | no | — |
| 270 | C23 | — | 4 | 2 | no | — |
| 271 | C23 | — | 8 | 4 | no | — |
| 272 | C23 | — | 8 | 2 | no | — |
| 273 | C23 | — | 14 | 4 | no | — |
| 274 | C23 | — | 28 | 4 | no | — |
| 275 | C23 | — | 28 | 4 | no | — |
| 276 | C23 | — | 28 | 4 | no | — |
| 277 | C23 | #6500 | 1 | 1 | yes | characteristic |
| 278 | Q8 | — | 28 | 4 | no | — |
| 279 | Q8 | — | 56 | 2 | no | — |
| 280 | D4 | — | 4 | 4 | no | — |
| 281 | D4 | — | 4 | 4 | no | — |
| 282 | D4 | — | 4 | 4 | no | — |
| 283 | D4 | — | 8 | 4 | no | — |
| 284 | D4 | — | 8 | 2 | no | — |
| 285 | D4 | — | 28 | 4 | no | — |
| 286 | D4 | — | 56 | 2 | no | — |
| 287 | D4 | — | 56 | 8 | no | — |
| 288 | C2×C4 | — | 2 | 4 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 4 | no | — |
| 291 | C2×C4 | — | 4 | 4 | no | — |
| 292 | C2×C4 | — | 14 | 4 | no | — |
| 293 | C2×C4 | — | 28 | 4 | no | — |
| 294 | C2×C4 | — | 56 | 8 | no | — |
| 295 | C2×C4 | #6500 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | #6500 | 1 | 2 | yes | — |
| 297 | C2×C4 | C2×D4×D7 | 1 | 2 | yes | — |
| 298 | C2×C4 | C22×C22⋊D7 | 1 | 1 | yes | characteristic |
| 299 | C8 | — | 4 | 2 | no | — |
| 300 | C8 | — | 14 | 8 | no | — |
| 301 | C7 | C2×#1656 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 302 | C22 | — | 8 | 4 | no | — |
| 303 | C22 | — | 8 | 2 | no | — |
| 304 | C22 | — | 16 | 4 | no | — |
| 305 | C22 | — | 28 | 4 | no | — |
| 306 | C22 | — | 28 | 4 | no | — |
| 307 | C22 | — | 28 | 4 | no | — |
| 308 | C22 | — | 56 | 8 | no | — |
| 309 | C22 | #89834 | 1 | 2 | yes | — |
| 310 | C22 | #89970 | 1 | 4 | yes | — |
| 311 | C22 | C2×#6500 | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 2 | 2 | no | — |
| 313 | C4 | — | 2 | 2 | no | — |
| 314 | C4 | — | 2 | 1 | no | — |
| 315 | C4 | — | 2 | 1 | no | — |
| 316 | C4 | — | 4 | 4 | no | — |
| 317 | C4 | — | 28 | 4 | no | — |
| 318 | C2 | — | 16 | 2 | no | — |
| 319 | C2 | — | 56 | 4 | no | — |
| 320 | C2 | #11729002 | 1 | 4 | yes | direct product factor |
| 321 | C2 | C2×#89834 | 1 | 1 | yes | characteristic |
| 322 | C2 | C2×#89970 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#90391 | C2 | semidirect |
| 2. C2×C7×#490 | C2 | semidirect |
| 3. C2×#89846 | C2 | semidirect |
| 4. C2×#89338 | C2 | semidirect |
| 5. #11729002 | C2 | direct (N × H) |
| 6. C2×#89259 | C2 | semidirect |
| 13. #90391 | C22 | semidirect |
| 19. C7×#490 | C22 | semidirect |
| 23. #89846 | C22 | semidirect |
| 25. #89338 | C22 | semidirect |
| 26. #89259 | C22 | semidirect |
| 70. C2×#490 | D7 | semidirect |
| 130. #490 | C2×D7 | semidirect |
| 266. C2×C7 | #1656 | semidirect |
| 301. C7 | C2×#1656 | semidirect |
| 320. C2 | #11729002 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#90391 | 2 |
| C2×C7×#490 | 2 |
| C2×#89846 | 2 |
| C2×#89338 | 2 |
| #11729002 | 2 |
| C2×#89259 | 2 |
| C2×#1656 | 7 |