Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
434 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 71 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 | C7×#3245 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11728829 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11728235 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11725580 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11725515 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C7×#532 | — | 2 | 1 | no | — |
| 7 | C22×D56 | C22 | 1 | 1 | yes | characteristic |
| 8 | C7×#474 | — | 2 | 2 | no | — |
| 9 | C7×#466 | — | 2 | 1 | no | — |
| 10 | C7×#466 | C22 | 1 | 2 | yes | — |
| 11 | C7×#459 | — | 2 | 2 | no | — |
| 12 | C2×C7×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#6400 | — | 2 | 2 | no | — |
| 14 | C2×#6395 | — | 2 | 1 | no | — |
| 15 | #89711 | — | 2 | 1 | no | — |
| 16 | #89346 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22⋊(C7⋊C16) | C22 | 1 | 2 | yes | — |
| 18 | #89286 | — | 2 | 2 | no | — |
| 19 | #12158 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C2×C7×#41 | — | 4 | 1 | no | — |
| 21 | C22×D28 | — | 2 | 1 | no | — |
| 22 | C2×C7×D8 | — | 2 | 1 | no | — |
| 23 | C2×C7×D8 | D4 | 1 | 2 | yes | — |
| 24 | C22×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 25 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 26 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 27 | C7×#127 | — | 2 | 1 | no | — |
| 28 | C7×#124 | — | 4 | 1 | no | — |
| 29 | C7×#121 | — | 2 | 2 | no | — |
| 30 | C7×#121 | — | 4 | 1 | no | — |
| 31 | C4×C7×D4 | — | 2 | 2 | no | — |
| 32 | C4×C7×D4 | — | 4 | 1 | no | — |
| 33 | C2×C42×C7 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×(C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 35 | C2×(C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 36 | C2×C4⋊Dic7 | — | 2 | 1 | no | — |
| 37 | C2×D56 | — | 2 | 2 | no | — |
| 38 | C2×D56 | — | 4 | 2 | no | — |
| 39 | C7×#107 | — | 2 | 1 | no | — |
| 40 | C7×#107 | D4 | 1 | 2 | yes | — |
| 41 | C7×C4⋊C8 | — | 2 | 2 | no | — |
| 42 | C7×#102 | — | 2 | 2 | no | — |
| 43 | C7×#102 | — | 2 | 2 | no | — |
| 44 | #6400 | — | 4 | 4 | no | — |
| 45 | #6395 | — | 4 | 2 | no | — |
| 46 | C2×C7⋊C16 | — | 2 | 2 | no | — |
| 47 | #3245 | D7 | 1 | 1 | yes | characteristic, complemented |
| 48 | #2184 | — | 7 | 2 | no | — |
| 49 | #2054 | — | 7 | 1 | no | — |
| 50 | #1347 | — | 7 | 2 | no | — |
| 51 | #1322 | — | 7 | 1 | no | — |
| 52 | C23×D7 | — | 4 | 1 | no | — |
| 53 | C7×#41 | — | 8 | 2 | no | — |
| 54 | C7×#41 | — | 8 | 2 | no | — |
| 55 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 56 | C2×C7×D4 | — | 2 | 2 | no | — |
| 57 | C2×C7×D4 | — | 4 | 1 | no | — |
| 58 | C2×C7×D4 | — | 4 | 1 | no | — |
| 59 | C2×C7×D4 | — | 4 | 2 | no | — |
| 60 | C22×C4×C7 | — | 4 | 2 | no | — |
| 61 | C22×C4×C7 | D8 | 1 | 1 | yes | characteristic |
| 62 | C22×C4×C7 | D8 | 1 | 1 | yes | characteristic |
| 63 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C22×Dic7 | — | 4 | 1 | no | — |
| 65 | C2×D28 | — | 2 | 2 | no | — |
| 66 | C2×D28 | — | 4 | 1 | no | — |
| 67 | C2×D28 | — | 4 | 1 | no | — |
| 68 | C2×D28 | — | 4 | 1 | no | — |
| 69 | C2×D28 | — | 4 | 1 | no | — |
| 70 | C2×D28 | — | 4 | 1 | no | — |
| 71 | C7×D8 | — | 4 | 2 | no | — |
| 72 | C7×D8 | — | 8 | 1 | no | — |
| 73 | C2×C7×C8 | — | 2 | 1 | no | — |
| 74 | C2×C7×C8 | — | 4 | 1 | no | — |
| 75 | C2×C7×C8 | — | 4 | 1 | no | — |
| 76 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 77 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 78 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 79 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 80 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 81 | C42×C7 | — | 2 | 1 | no | — |
| 82 | C42×C7 | — | 2 | 1 | no | — |
| 83 | (C2×D7)⋊C4 | — | 8 | 2 | no | — |
| 84 | (C2×D7)⋊C4 | — | 8 | 2 | no | — |
| 85 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 86 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 87 | D56 | — | 2 | 4 | no | — |
| 88 | D56 | — | 4 | 2 | no | — |
| 89 | C7⋊C16 | — | 4 | 2 | no | — |
| 90 | C22×D8 | — | 7 | 1 | no | — |
| 91 | #532 | — | 2 | 1 | no | — |
| 92 | C2×D16 | — | 14 | 2 | no | — |
| 93 | #474 | — | 2 | 2 | no | — |
| 94 | #466 | — | 2 | 1 | no | — |
| 95 | #466 | C2×D7 | 1 | 2 | yes | — |
| 96 | #459 | — | 2 | 2 | no | — |
| 97 | C2×C4⋊C8 | C2×D7 | 1 | 1 | yes | characteristic |
| 98 | C2×#102 | — | 14 | 1 | no | — |
| 99 | #390 | — | 14 | 1 | no | — |
| 100 | #357 | — | 14 | 2 | no | — |
| 101 | C22⋊C16 | — | 7 | 2 | no | — |
| 102 | #340 | — | 7 | 1 | no | — |
| 103 | C23×C7 | — | 4 | 2 | no | — |
| 104 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 105 | C22×D7 | — | 4 | 2 | no | — |
| 106 | C22×D7 | — | 8 | 1 | no | — |
| 107 | C22×D7 | — | 8 | 1 | no | — |
| 108 | C22×D7 | — | 8 | 1 | no | — |
| 109 | C22×D7 | — | 8 | 1 | no | — |
| 110 | C22×D7 | — | 8 | 1 | no | — |
| 111 | C22×D7 | — | 8 | 1 | no | — |
| 112 | C7×Q8 | — | 8 | 2 | no | — |
| 113 | C7×D4 | — | 4 | 2 | no | — |
| 114 | C7×D4 | — | 4 | 2 | no | — |
| 115 | C7×D4 | — | 8 | 2 | no | — |
| 116 | C7×D4 | — | 8 | 2 | no | — |
| 117 | C7×D4 | — | 8 | 2 | no | — |
| 118 | C7×D4 | — | 8 | 2 | no | — |
| 119 | C2×C4×C7 | — | 2 | 1 | no | — |
| 120 | C2×C4×C7 | — | 2 | 1 | no | — |
| 121 | C2×C4×C7 | — | 2 | 1 | no | — |
| 122 | C2×C4×C7 | — | 2 | 1 | no | — |
| 123 | C2×C4×C7 | — | 2 | 1 | no | — |
| 124 | C2×C4×C7 | — | 4 | 1 | no | — |
| 125 | C2×C4×C7 | — | 4 | 2 | no | — |
| 126 | C2×C4×C7 | — | 8 | 4 | no | — |
| 127 | C2×C4×C7 | D16 | 1 | 2 | yes | complemented |
| 128 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 130 | C2×Dic7 | — | 8 | 1 | no | — |
| 131 | C2×Dic7 | — | 8 | 1 | no | — |
| 132 | C2×Dic7 | — | 8 | 1 | no | — |
| 133 | D28 | — | 4 | 2 | no | — |
| 134 | D28 | — | 4 | 2 | no | — |
| 135 | D28 | — | 8 | 1 | no | — |
| 136 | D28 | — | 8 | 1 | no | — |
| 137 | C7×C8 | — | 2 | 2 | no | — |
| 138 | C7×C8 | — | 4 | 1 | no | — |
| 139 | C2×#41 | — | 4 | 1 | no | — |
| 140 | C22×D4 | — | 14 | 1 | no | — |
| 141 | C2×D8 | — | 2 | 1 | no | — |
| 142 | C2×D8 | — | 14 | 2 | no | — |
| 143 | C2×D8 | — | 28 | 2 | no | — |
| 144 | C2×D8 | C22⋊D7 | 1 | 2 | yes | — |
| 145 | C22×C8 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 146 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 147 | C4⋊Q8 | — | 2 | 1 | no | — |
| 148 | #127 | — | 2 | 1 | no | — |
| 149 | #124 | — | 4 | 1 | no | — |
| 150 | #121 | — | 2 | 2 | no | — |
| 151 | #121 | — | 4 | 1 | no | — |
| 152 | C4×D4 | — | 2 | 2 | no | — |
| 153 | C4×D4 | — | 4 | 1 | no | — |
| 154 | C2×C4⋊C4 | — | 14 | 1 | no | — |
| 155 | C2×C22⋊C4 | — | 28 | 1 | no | — |
| 156 | C2×C22⋊C4 | — | 28 | 1 | no | — |
| 157 | C2×C42 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 158 | D16 | — | 28 | 4 | no | — |
| 159 | C2×C16 | — | 14 | 2 | no | — |
| 160 | #107 | — | 2 | 1 | no | — |
| 161 | #107 | C22⋊D7 | 1 | 2 | yes | — |
| 162 | C4⋊C8 | — | 2 | 2 | no | — |
| 163 | #102 | — | 2 | 2 | no | — |
| 164 | #102 | — | 2 | 2 | no | — |
| 165 | #102 | — | 28 | 2 | no | — |
| 166 | C22×C7 | — | 2 | 1 | no | — |
| 167 | C22×C7 | — | 2 | 1 | no | — |
| 168 | C22×C7 | — | 8 | 2 | no | — |
| 169 | C22×C7 | — | 8 | 2 | no | — |
| 170 | C22×C7 | — | 8 | 2 | no | — |
| 171 | C22×C7 | #447 | 1 | 1 | yes | characteristic |
| 172 | C22×C7 | C2×D16 | 1 | 1 | yes | characteristic |
| 173 | C22×C7 | #509 | 1 | 1 | yes | characteristic |
| 174 | C2×D7 | — | 8 | 2 | no | — |
| 175 | C2×D7 | — | 8 | 2 | no | — |
| 176 | C2×D7 | — | 8 | 2 | no | — |
| 177 | C2×D7 | — | 16 | 1 | no | — |
| 178 | C2×D7 | — | 16 | 1 | no | — |
| 179 | C2×D7 | — | 16 | 1 | no | — |
| 180 | C2×D7 | — | 16 | 1 | no | — |
| 181 | C4×C7 | — | 2 | 1 | no | — |
| 182 | C4×C7 | — | 2 | 2 | no | — |
| 183 | C4×C7 | — | 4 | 1 | no | — |
| 184 | C4×C7 | — | 8 | 2 | no | — |
| 185 | C4×C7 | #454 | 1 | 1 | yes | characteristic |
| 186 | C4×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 187 | Dic7 | — | 16 | 1 | no | — |
| 188 | C24 | — | 28 | 1 | no | — |
| 189 | #41 | — | 8 | 2 | no | — |
| 190 | #41 | — | 8 | 2 | no | — |
| 191 | C2×Q8 | — | 4 | 1 | no | — |
| 192 | C2×D4 | — | 2 | 2 | no | — |
| 193 | C2×D4 | — | 4 | 1 | no | — |
| 194 | C2×D4 | — | 4 | 1 | no | — |
| 195 | C2×D4 | — | 4 | 2 | no | — |
| 196 | C2×D4 | — | 14 | 2 | no | — |
| 197 | C2×D4 | — | 28 | 1 | no | — |
| 198 | C2×D4 | — | 28 | 1 | no | — |
| 199 | C2×D4 | — | 28 | 1 | no | — |
| 200 | C2×D4 | — | 28 | 1 | no | — |
| 201 | C2×D4 | — | 28 | 1 | no | — |
| 202 | C22×C4 | — | 4 | 2 | no | — |
| 203 | C22×C4 | — | 28 | 1 | no | — |
| 204 | C22×C4 | #1141 | 1 | 1 | yes | characteristic |
| 205 | C22×C4 | #1141 | 1 | 1 | yes | characteristic |
| 206 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 207 | D8 | — | 4 | 2 | no | — |
| 208 | D8 | — | 8 | 1 | no | — |
| 209 | D8 | — | 14 | 4 | no | — |
| 210 | D8 | — | 28 | 2 | no | — |
| 211 | C2×C8 | — | 2 | 1 | no | — |
| 212 | C2×C8 | — | 4 | 1 | no | — |
| 213 | C2×C8 | — | 4 | 1 | no | — |
| 214 | C2×C8 | C2×C22⋊D7 | 1 | 2 | yes | — |
| 215 | C4⋊C4 | — | 2 | 2 | no | — |
| 216 | C4⋊C4 | — | 4 | 1 | no | — |
| 217 | C4⋊C4 | — | 28 | 1 | no | — |
| 218 | C4⋊C4 | — | 28 | 1 | no | — |
| 219 | C22⋊C4 | — | 4 | 2 | no | — |
| 220 | C22⋊C4 | — | 4 | 2 | no | — |
| 221 | C22⋊C4 | — | 56 | 2 | no | — |
| 222 | C22⋊C4 | — | 56 | 2 | no | — |
| 223 | C42 | — | 2 | 1 | no | — |
| 224 | C42 | — | 2 | 1 | no | — |
| 225 | C16 | — | 28 | 2 | no | — |
| 226 | C2×C7 | — | 2 | 1 | no | — |
| 227 | C2×C7 | — | 2 | 1 | no | — |
| 228 | C2×C7 | — | 16 | 2 | no | — |
| 229 | C2×C7 | #1598 | 1 | 1 | yes | characteristic |
| 230 | C2×C7 | #2184 | 1 | 1 | yes | characteristic |
| 231 | C2×C7 | #2190 | 1 | 1 | yes | characteristic |
| 232 | D7 | — | 16 | 2 | no | — |
| 233 | C23 | — | 4 | 2 | no | — |
| 234 | C23 | — | 28 | 2 | no | — |
| 235 | C23 | — | 56 | 1 | no | — |
| 236 | C23 | — | 56 | 1 | no | — |
| 237 | C23 | — | 56 | 1 | no | — |
| 238 | C23 | — | 56 | 1 | no | — |
| 239 | C23 | — | 56 | 1 | no | — |
| 240 | C23 | — | 56 | 1 | no | — |
| 241 | C23 | C2×#1141 | 1 | 1 | yes | characteristic |
| 242 | Q8 | — | 8 | 2 | no | — |
| 243 | D4 | — | 4 | 2 | no | — |
| 244 | D4 | — | 4 | 2 | no | — |
| 245 | D4 | — | 8 | 2 | no | — |
| 246 | D4 | — | 8 | 2 | no | — |
| 247 | D4 | — | 8 | 2 | no | — |
| 248 | D4 | — | 8 | 2 | no | — |
| 249 | D4 | — | 28 | 2 | no | — |
| 250 | D4 | — | 28 | 2 | no | — |
| 251 | D4 | — | 56 | 1 | no | — |
| 252 | D4 | — | 56 | 1 | no | — |
| 253 | C2×C4 | — | 2 | 1 | no | — |
| 254 | C2×C4 | — | 2 | 1 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 1 | no | — |
| 259 | C2×C4 | — | 4 | 2 | no | — |
| 260 | C2×C4 | — | 8 | 4 | no | — |
| 261 | C2×C4 | — | 56 | 1 | no | — |
| 262 | C2×C4 | — | 56 | 1 | no | — |
| 263 | C2×C4 | — | 56 | 1 | no | — |
| 264 | C2×C4 | #6400 | 1 | 2 | yes | complemented |
| 265 | C2×C4 | #6495 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | #6516 | 1 | 1 | yes | characteristic |
| 267 | C8 | — | 2 | 2 | no | — |
| 268 | C8 | — | 4 | 1 | no | — |
| 269 | C7 | #12158 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 270 | C22 | — | 2 | 1 | no | — |
| 271 | C22 | — | 2 | 1 | no | — |
| 272 | C22 | — | 8 | 2 | no | — |
| 273 | C22 | — | 8 | 2 | no | — |
| 274 | C22 | — | 8 | 2 | no | — |
| 275 | C22 | — | 56 | 2 | no | — |
| 276 | C22 | — | 56 | 2 | no | — |
| 277 | C22 | — | 56 | 2 | no | — |
| 278 | C22 | — | 112 | 1 | no | — |
| 279 | C22 | — | 112 | 1 | no | — |
| 280 | C22 | — | 112 | 1 | no | — |
| 281 | C22 | — | 112 | 1 | no | — |
| 282 | C22 | C2×#6400 | 1 | 1 | yes | characteristic |
| 283 | C22 | #89919 | 1 | 1 | yes | characteristic |
| 284 | C22 | #89989 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 2 | no | — |
| 286 | C4 | — | 2 | 1 | no | — |
| 287 | C4 | — | 4 | 1 | no | — |
| 288 | C4 | — | 8 | 2 | no | — |
| 289 | C4 | — | 112 | 1 | no | — |
| 290 | C4 | #90016 | 1 | 1 | yes | characteristic |
| 291 | C4 | #90017 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 2 | 1 | no | — |
| 293 | C2 | — | 2 | 1 | no | — |
| 294 | C2 | — | 16 | 2 | no | — |
| 295 | C2 | — | 112 | 2 | no | — |
| 296 | C2 | #11729632 | 1 | 1 | yes | characteristic |
| 297 | C2 | #11729789 | 1 | 1 | yes | characteristic |
| 298 | C2 | #11729866 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3245 | C2 | semidirect |
| 2. #11728829 | C2 | semidirect |
| 3. #11728235 | C2 | semidirect |
| 4. #11725580 | C2 | semidirect |
| 5. #11725515 | C2 | semidirect |
| 47. #3245 | D7 | semidirect |
| 127. C2×C4×C7 | D16 | semidirect |
| 264. C2×C4 | #6400 | semidirect |
| 269. C7 | #12158 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3245 | 2 |
| #11728829 | 2 |
| #11728235 | 2 |
| #11725580 | 2 |
| #11725515 | 2 |
| #12158 | 7 |