Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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 | #11739454 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #11735944 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11729202 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11727599 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C7×#1375 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#6467 | — | 2 | 1 | no | — |
| 7 | C2×#6467 | C22 | 1 | 2 | yes | — |
| 8 | C2×#6467 | C22 | 1 | 1 | yes | characteristic |
| 9 | #90168 | — | 2 | 1 | no | — |
| 10 | C2×C7×M32 | C22 | 1 | 2 | yes | — |
| 11 | C2×C7×#97 | C22 | 1 | 1 | yes | characteristic |
| 12 | #89882 | — | 2 | 1 | no | — |
| 13 | C2×#6393 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#6379 | — | 2 | 1 | no | — |
| 15 | #89681 | — | 2 | 2 | no | — |
| 16 | #89681 | — | 2 | 2 | no | — |
| 17 | #89680 | — | 2 | 2 | no | — |
| 18 | #89680 | — | 2 | 2 | no | — |
| 19 | #89676 | — | 2 | 4 | no | — |
| 20 | #89487 | — | 2 | 2 | no | — |
| 21 | #89484 | — | 2 | 2 | no | — |
| 22 | #89305 | — | 2 | 4 | no | — |
| 23 | #13340 | — | 7 | 1 | no | maximal, Sylow |
| 24 | C2×#1157 | — | 2 | 1 | no | — |
| 25 | C2×#1157 | — | 2 | 2 | no | — |
| 26 | C22×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 28 | C2×C42×C7 | D4 | 1 | 1 | yes | characteristic |
| 29 | #6493 | — | 4 | 2 | no | — |
| 30 | C2×#1136 | — | 2 | 1 | no | — |
| 31 | C2×Dic28 | — | 2 | 1 | no | — |
| 32 | C2×Dic28 | D4 | 1 | 2 | yes | — |
| 33 | C2×Dic28 | D4 | 1 | 1 | yes | characteristic, complemented |
| 34 | #6467 | — | 2 | 2 | no | — |
| 35 | #6467 | — | 2 | 2 | no | — |
| 36 | #6467 | — | 2 | 2 | no | — |
| 37 | #6467 | — | 2 | 2 | no | — |
| 38 | #6467 | — | 2 | 2 | no | — |
| 39 | #6467 | — | 2 | 2 | no | — |
| 40 | #6467 | — | 4 | 2 | no | — |
| 41 | #6467 | — | 4 | 2 | no | — |
| 42 | C2×D56 | — | 2 | 1 | no | — |
| 43 | C2×D56 | D4 | 1 | 2 | yes | — |
| 44 | C2×D56 | D4 | 1 | 1 | yes | characteristic, complemented |
| 45 | C2×#1132 | — | 2 | 2 | no | — |
| 46 | C2×#1132 | — | 2 | 2 | no | — |
| 47 | C2×#1132 | — | 2 | 1 | no | — |
| 48 | #6438 | — | 2 | 2 | no | — |
| 49 | #6437 | — | 2 | 1 | no | — |
| 50 | C4×D28 | — | 4 | 2 | no | — |
| 51 | #6432 | — | 2 | 1 | no | — |
| 52 | C7×M32 | — | 2 | 2 | no | — |
| 53 | C7×M32 | — | 2 | 2 | no | — |
| 54 | C2×C7×C16 | D4 | 1 | 4 | yes | complemented |
| 55 | C7×#97 | — | 2 | 2 | no | — |
| 56 | (C2×D7)⋊C8 | — | 4 | 2 | no | — |
| 57 | #6393 | — | 2 | 2 | no | — |
| 58 | #6379 | — | 4 | 2 | no | — |
| 59 | Dic56 | — | 2 | 4 | no | — |
| 60 | #6374 | — | 2 | 4 | no | — |
| 61 | #6374 | — | 2 | 4 | no | — |
| 62 | D112 | — | 2 | 4 | no | — |
| 63 | #3414 | — | 7 | 2 | no | — |
| 64 | #3150 | — | 7 | 1 | no | — |
| 65 | #2143 | — | 7 | 2 | no | — |
| 66 | #1975 | — | 7 | 1 | no | — |
| 67 | #1375 | D7 | 1 | 1 | yes | characteristic, complemented |
| 68 | C22×C4×C7 | — | 2 | 1 | no | — |
| 69 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 71 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 72 | #1157 | — | 4 | 1 | no | — |
| 73 | #1157 | — | 4 | 2 | no | — |
| 74 | #1157 | — | 4 | 1 | no | — |
| 75 | #1157 | — | 4 | 2 | no | — |
| 76 | #1157 | — | 4 | 2 | no | — |
| 77 | #1157 | — | 4 | 1 | no | — |
| 78 | #1157 | — | 4 | 2 | no | — |
| 79 | #1157 | — | 4 | 1 | no | — |
| 80 | C2×D28 | — | 2 | 2 | no | — |
| 81 | C2×D28 | — | 2 | 1 | no | — |
| 82 | C2×D28 | — | 4 | 2 | no | — |
| 83 | C2×C4×D7 | — | 4 | 2 | no | — |
| 84 | C2×C4×D7 | — | 4 | 1 | no | — |
| 85 | C2×Dic14 | — | 2 | 2 | no | — |
| 86 | C2×Dic14 | — | 2 | 1 | no | — |
| 87 | C2×C7×C8 | — | 2 | 1 | no | — |
| 88 | C2×C7×C8 | — | 2 | 1 | no | — |
| 89 | C2×C7×C8 | D8 | 1 | 2 | yes | — |
| 90 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 92 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 93 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 94 | C42×C7 | — | 2 | 1 | no | — |
| 95 | C42×C7 | D8 | 1 | 2 | yes | — |
| 96 | (C2×D7)⋊C4 | — | 4 | 4 | no | — |
| 97 | C4⋊Dic7 | — | 4 | 2 | no | — |
| 98 | #1136 | — | 4 | 1 | no | — |
| 99 | #1136 | — | 4 | 1 | no | — |
| 100 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 101 | Dic28 | — | 2 | 2 | no | — |
| 102 | Dic28 | — | 2 | 2 | no | — |
| 103 | Dic28 | — | 2 | 2 | no | — |
| 104 | Dic28 | — | 4 | 2 | no | — |
| 105 | D56 | — | 2 | 2 | no | — |
| 106 | D56 | — | 2 | 2 | no | — |
| 107 | D56 | — | 2 | 2 | no | — |
| 108 | D56 | — | 4 | 2 | no | — |
| 109 | #1132 | — | 4 | 4 | no | — |
| 110 | #1132 | — | 4 | 2 | no | — |
| 111 | #1132 | — | 4 | 2 | no | — |
| 112 | #1132 | — | 4 | 2 | no | — |
| 113 | C7×C16 | — | 2 | 4 | no | — |
| 114 | C2×#135 | — | 7 | 2 | no | — |
| 115 | C2×#135 | — | 7 | 1 | no | — |
| 116 | C2×#135 | — | 14 | 1 | no | — |
| 117 | #532 | — | 14 | 1 | no | — |
| 118 | #510 | — | 14 | 2 | no | — |
| 119 | #510 | — | 14 | 2 | no | — |
| 120 | #509 | — | 14 | 2 | no | — |
| 121 | #509 | — | 14 | 2 | no | — |
| 122 | #508 | — | 14 | 4 | no | — |
| 123 | C2×M32 | C2×D7 | 1 | 2 | yes | — |
| 124 | #497 | — | 14 | 2 | no | — |
| 125 | #496 | — | 14 | 2 | no | — |
| 126 | C2×#108 | — | 7 | 1 | no | — |
| 127 | C2×#104 | — | 14 | 1 | no | — |
| 128 | #408 | — | 14 | 1 | no | — |
| 129 | C2×#97 | C2×D7 | 1 | 1 | yes | characteristic |
| 130 | #359 | — | 14 | 4 | no | — |
| 131 | C23×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 132 | C22×D7 | — | 4 | 2 | no | — |
| 133 | C22×D7 | — | 4 | 1 | no | — |
| 134 | C2×C4×C7 | — | 2 | 2 | no | — |
| 135 | C2×C4×C7 | — | 8 | 1 | no | — |
| 136 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C7 | #127 | 1 | 1 | yes | characteristic |
| 139 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 140 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 141 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 142 | C22⋊D7 | — | 8 | 1 | no | — |
| 143 | C22⋊D7 | — | 8 | 2 | no | — |
| 144 | C22⋊D7 | — | 8 | 2 | no | — |
| 145 | C22⋊D7 | — | 8 | 1 | no | — |
| 146 | C2×Dic7 | — | 4 | 2 | no | — |
| 147 | C2×Dic7 | — | 4 | 1 | no | — |
| 148 | D28 | — | 4 | 2 | no | — |
| 149 | D28 | — | 4 | 1 | no | — |
| 150 | D28 | — | 4 | 2 | no | — |
| 151 | D28 | — | 4 | 1 | no | — |
| 152 | D28 | — | 8 | 4 | no | — |
| 153 | C4×D7 | — | 8 | 1 | no | — |
| 154 | C4×D7 | — | 8 | 2 | no | — |
| 155 | C4×D7 | — | 8 | 2 | no | — |
| 156 | C4×D7 | — | 8 | 1 | no | — |
| 157 | Dic14 | — | 4 | 2 | no | — |
| 158 | Dic14 | — | 4 | 1 | no | — |
| 159 | Dic14 | — | 4 | 2 | no | — |
| 160 | Dic14 | — | 4 | 1 | no | — |
| 161 | C7×C8 | — | 2 | 2 | no | — |
| 162 | C7×C8 | #120 | 1 | 2 | yes | — |
| 163 | C7×C8 | #121 | 1 | 2 | yes | — |
| 164 | C7⋊C8 | — | 8 | 1 | no | — |
| 165 | C2×#41 | — | 14 | 2 | no | — |
| 166 | C2×#41 | — | 14 | 1 | no | — |
| 167 | #135 | — | 14 | 2 | no | — |
| 168 | #135 | — | 14 | 2 | no | — |
| 169 | #135 | — | 14 | 2 | no | — |
| 170 | #135 | — | 14 | 2 | no | — |
| 171 | #135 | — | 14 | 2 | no | — |
| 172 | #135 | — | 14 | 2 | no | — |
| 173 | #135 | — | 28 | 2 | no | — |
| 174 | #135 | — | 28 | 2 | no | — |
| 175 | C2×Q16 | — | 7 | 2 | no | — |
| 176 | C2×Q16 | — | 7 | 1 | no | — |
| 177 | C2×Q16 | — | 14 | 1 | no | — |
| 178 | C2×SD16 | — | 14 | 2 | no | — |
| 179 | C2×SD16 | — | 14 | 2 | no | — |
| 180 | C2×SD16 | — | 14 | 1 | no | — |
| 181 | C2×D8 | — | 7 | 2 | no | — |
| 182 | C2×D8 | — | 7 | 1 | no | — |
| 183 | C2×D8 | — | 14 | 1 | no | — |
| 184 | C2×M16 | — | 14 | 1 | no | — |
| 185 | C22×C8 | D28 | 1 | 1 | yes | characteristic |
| 186 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 187 | C4⋊Q8 | — | 14 | 1 | no | — |
| 188 | #127 | — | 14 | 1 | no | — |
| 189 | #124 | — | 14 | 2 | no | — |
| 190 | #121 | — | 28 | 2 | no | — |
| 191 | C4×D4 | — | 28 | 2 | no | — |
| 192 | C2×C42 | D28 | 1 | 1 | yes | characteristic |
| 193 | Q32 | — | 14 | 4 | no | — |
| 194 | SD32 | — | 14 | 4 | no | — |
| 195 | SD32 | — | 14 | 4 | no | — |
| 196 | D16 | — | 14 | 4 | no | — |
| 197 | M32 | — | 2 | 2 | no | — |
| 198 | M32 | — | 2 | 2 | no | — |
| 199 | C2×C16 | D28 | 1 | 4 | yes | complemented |
| 200 | #108 | — | 14 | 2 | no | — |
| 201 | #104 | — | 28 | 2 | no | — |
| 202 | C22⋊C8 | — | 28 | 2 | no | — |
| 203 | #97 | — | 2 | 2 | no | — |
| 204 | C22×C7 | — | 4 | 1 | no | — |
| 205 | C22×C7 | #447 | 1 | 1 | yes | characteristic |
| 206 | C22×C7 | #459 | 1 | 1 | yes | characteristic |
| 207 | C22×C7 | #496 | 1 | 1 | yes | characteristic |
| 208 | C2×D7 | — | 8 | 4 | no | — |
| 209 | C2×D7 | — | 8 | 2 | no | — |
| 210 | C2×D7 | — | 8 | 1 | no | — |
| 211 | C2×D7 | — | 8 | 2 | no | — |
| 212 | C4×C7 | — | 4 | 2 | no | — |
| 213 | C4×C7 | #392 | 1 | 1 | yes | characteristic |
| 214 | C4×C7 | #447 | 1 | 1 | yes | characteristic |
| 215 | C4×C7 | #459 | 1 | 1 | yes | characteristic |
| 216 | C4×C7 | #493 | 1 | 1 | yes | characteristic |
| 217 | Dic7 | — | 8 | 2 | no | — |
| 218 | Dic7 | — | 8 | 1 | no | — |
| 219 | #41 | — | 28 | 2 | no | — |
| 220 | #41 | — | 28 | 1 | no | — |
| 221 | #41 | — | 28 | 1 | no | — |
| 222 | #41 | — | 28 | 2 | no | — |
| 223 | #41 | — | 28 | 1 | no | — |
| 224 | #41 | — | 28 | 2 | no | — |
| 225 | #41 | — | 28 | 2 | no | — |
| 226 | #41 | — | 28 | 1 | no | — |
| 227 | C2×Q8 | — | 14 | 2 | no | — |
| 228 | C2×Q8 | — | 14 | 1 | no | — |
| 229 | C2×D4 | — | 14 | 2 | no | — |
| 230 | C2×D4 | — | 14 | 1 | no | — |
| 231 | C2×D4 | — | 28 | 2 | no | — |
| 232 | C2×D4 | — | 28 | 2 | no | — |
| 233 | C2×D4 | — | 28 | 1 | no | — |
| 234 | C22×C4 | — | 2 | 1 | no | — |
| 235 | C22×C4 | — | 28 | 1 | no | — |
| 236 | C22×C4 | — | 28 | 2 | no | — |
| 237 | C22×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 238 | Q16 | — | 14 | 2 | no | — |
| 239 | Q16 | — | 14 | 2 | no | — |
| 240 | Q16 | — | 14 | 2 | no | — |
| 241 | Q16 | — | 28 | 2 | no | — |
| 242 | SD16 | — | 28 | 4 | no | — |
| 243 | SD16 | — | 28 | 2 | no | — |
| 244 | SD16 | — | 28 | 2 | no | — |
| 245 | SD16 | — | 28 | 2 | no | — |
| 246 | D8 | — | 14 | 2 | no | — |
| 247 | D8 | — | 14 | 2 | no | — |
| 248 | D8 | — | 14 | 2 | no | — |
| 249 | D8 | — | 28 | 2 | no | — |
| 250 | M16 | — | 28 | 1 | no | — |
| 251 | M16 | — | 28 | 1 | no | — |
| 252 | C2×C8 | — | 2 | 1 | no | — |
| 253 | C2×C8 | — | 2 | 1 | no | — |
| 254 | C2×C8 | — | 28 | 1 | no | — |
| 255 | C2×C8 | D56 | 1 | 2 | yes | — |
| 256 | C2×C8 | C2×D28 | 1 | 2 | yes | — |
| 257 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 259 | C2×C8 | #1161 | 1 | 1 | yes | characteristic |
| 260 | C4⋊C4 | — | 28 | 2 | no | — |
| 261 | C22⋊C4 | — | 28 | 4 | no | — |
| 262 | C42 | — | 2 | 1 | no | — |
| 263 | C42 | D56 | 1 | 2 | yes | — |
| 264 | C16 | — | 2 | 4 | no | — |
| 265 | C2×C7 | — | 2 | 1 | no | — |
| 266 | C2×C7 | — | 2 | 1 | no | — |
| 267 | C2×C7 | — | 2 | 1 | no | — |
| 268 | C2×C7 | #2011 | 1 | 1 | yes | characteristic |
| 269 | D7 | — | 16 | 2 | no | — |
| 270 | D7 | — | 16 | 1 | no | — |
| 271 | C23 | — | 28 | 1 | no | — |
| 272 | C23 | — | 28 | 2 | no | — |
| 273 | C23 | #6471 | 1 | 1 | yes | characteristic |
| 274 | Q8 | — | 28 | 1 | no | — |
| 275 | Q8 | — | 28 | 2 | no | — |
| 276 | Q8 | — | 28 | 1 | no | — |
| 277 | Q8 | — | 28 | 2 | no | — |
| 278 | D4 | — | 28 | 1 | no | — |
| 279 | D4 | — | 28 | 1 | no | — |
| 280 | D4 | — | 28 | 2 | no | — |
| 281 | D4 | — | 28 | 2 | no | — |
| 282 | D4 | — | 56 | 4 | no | — |
| 283 | D4 | — | 56 | 2 | no | — |
| 284 | D4 | — | 56 | 2 | no | — |
| 285 | D4 | — | 56 | 1 | no | — |
| 286 | D4 | — | 56 | 1 | no | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 8 | 1 | no | — |
| 289 | C2×C4 | — | 28 | 1 | no | — |
| 290 | C2×C4 | — | 28 | 2 | no | — |
| 291 | C2×C4 | — | 56 | 2 | no | — |
| 292 | C2×C4 | — | 56 | 1 | no | — |
| 293 | C2×C4 | — | 56 | 2 | no | — |
| 294 | C2×C4 | — | 56 | 1 | no | — |
| 295 | C2×C4 | #6437 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | #6445 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #6458 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #6471 | 1 | 1 | yes | characteristic |
| 301 | C8 | — | 2 | 2 | no | — |
| 302 | C8 | — | 56 | 1 | no | — |
| 303 | C8 | #6445 | 1 | 2 | yes | — |
| 304 | C8 | #6458 | 1 | 2 | yes | — |
| 305 | C7 | #13340 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 306 | C22 | — | 4 | 1 | no | — |
| 307 | C22 | — | 56 | 4 | no | — |
| 308 | C22 | — | 56 | 2 | no | — |
| 309 | C22 | — | 56 | 2 | no | — |
| 310 | C22 | — | 56 | 1 | no | — |
| 311 | C22 | #89484 | 1 | 1 | yes | characteristic |
| 312 | C22 | #89504 | 1 | 1 | yes | characteristic |
| 313 | C22 | #89615 | 1 | 1 | yes | characteristic |
| 314 | C4 | — | 4 | 2 | no | — |
| 315 | C4 | — | 56 | 1 | no | — |
| 316 | C4 | — | 56 | 2 | no | — |
| 317 | C4 | #89443 | 1 | 1 | yes | characteristic |
| 318 | C4 | #89467 | 1 | 1 | yes | characteristic |
| 319 | C4 | #89504 | 1 | 1 | yes | characteristic |
| 320 | C4 | #89615 | 1 | 1 | yes | characteristic |
| 321 | C2 | — | 2 | 1 | no | — |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 2 | 1 | no | — |
| 324 | C2 | — | 112 | 2 | no | — |
| 325 | C2 | — | 112 | 1 | no | — |
| 326 | C2 | #11726428 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11739454 | C2 | semidirect |
| 2. #11735944 | C2 | semidirect |
| 3. #11729202 | C2 | semidirect |
| 4. #11727599 | C2 | semidirect |
| 5. C7×#1375 | C2 | semidirect |
| 33. C2×Dic28 | D4 | semidirect |
| 44. C2×D56 | D4 | semidirect |
| 54. C2×C7×C16 | D4 | semidirect |
| 67. #1375 | D7 | semidirect |
| 199. C2×C16 | D28 | semidirect |
| 305. C7 | #13340 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11739454 | 2 |
| #11735944 | 2 |
| #11729202 | 2 |
| #11727599 | 2 |
| C7×#1375 | 2 |
| #13340 | 7 |