Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
384 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 65 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 | #11737175 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11728668 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11726835 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11726294 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C7×#1334 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11725321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11725273 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#6481 | — | 2 | 1 | no | — |
| 9 | #90267 | — | 2 | 1 | no | — |
| 10 | C4×C7×Q16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#6402 | — | 2 | 1 | no | — |
| 12 | #89800 | — | 2 | 1 | no | — |
| 13 | C2×#6374 | — | 2 | 1 | no | — |
| 14 | C2×D112 | — | 2 | 1 | no | — |
| 15 | #89658 | — | 2 | 1 | no | — |
| 16 | #89615 | — | 2 | 1 | no | — |
| 17 | #89583 | — | 2 | 1 | no | — |
| 18 | #89580 | — | 2 | 1 | no | — |
| 19 | #89468 | C22 | 1 | 1 | yes | characteristic |
| 20 | #89467 | C22 | 1 | 1 | yes | characteristic |
| 21 | #89460 | C22 | 1 | 1 | yes | characteristic |
| 22 | C7×#346 | C22 | 1 | 1 | yes | characteristic |
| 23 | C7×#334 | C22 | 1 | 1 | yes | characteristic |
| 24 | #89286 | — | 2 | 1 | no | — |
| 25 | #89280 | — | 2 | 1 | no | — |
| 26 | C4⋊(C7⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 27 | #12325 | — | 7 | 1 | no | maximal, Sylow |
| 28 | C2×#1161 | — | 2 | 1 | no | — |
| 29 | C2×Q8×D7 | — | 2 | 1 | no | — |
| 30 | C2×C7×Q16 | D4 | 1 | 1 | yes | characteristic |
| 31 | C4×C7×Q8 | — | 2 | 1 | no | — |
| 32 | C2×#1144 | — | 2 | 1 | no | — |
| 33 | C2×#1143 | — | 2 | 1 | no | — |
| 34 | #6481 | — | 4 | 2 | no | — |
| 35 | #6481 | — | 4 | 2 | no | — |
| 36 | C2×Dic28 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×D56 | — | 2 | 1 | no | — |
| 38 | C2×D56 | — | 2 | 1 | no | — |
| 39 | C2×D56 | — | 4 | 1 | no | — |
| 40 | C2×D56 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×#1132 | — | 2 | 1 | no | — |
| 42 | C2×#1131 | — | 2 | 1 | no | — |
| 43 | #6460 | — | 2 | 1 | no | — |
| 44 | #6460 | — | 4 | 1 | no | — |
| 45 | #6458 | — | 2 | 1 | no | — |
| 46 | #6458 | — | 4 | 1 | no | — |
| 47 | #6438 | — | 2 | 1 | no | — |
| 48 | #6438 | — | 4 | 1 | no | — |
| 49 | #6437 | — | 2 | 1 | no | — |
| 50 | C4×D28 | — | 4 | 1 | no | — |
| 51 | C4×D28 | D4 | 1 | 1 | yes | characteristic |
| 52 | C2×C7×C16 | — | 2 | 1 | no | — |
| 53 | C7×#107 | D4 | 1 | 1 | yes | characteristic |
| 54 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 55 | C7×#103 | — | 2 | 1 | no | — |
| 56 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 57 | #6402 | — | 4 | 2 | no | — |
| 58 | #6395 | — | 4 | 1 | no | — |
| 59 | (C2×D7)⋊C8 | — | 2 | 1 | no | — |
| 60 | C2×C7⋊C16 | — | 2 | 1 | no | — |
| 61 | #6384 | — | 2 | 1 | no | — |
| 62 | #6383 | — | 2 | 1 | no | — |
| 63 | C4⋊(C7⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 64 | #6374 | — | 4 | 2 | no | — |
| 65 | D112 | — | 4 | 2 | no | — |
| 66 | #3277 | — | 7 | 1 | no | — |
| 67 | #2250 | — | 7 | 1 | no | — |
| 68 | #2207 | — | 7 | 1 | no | — |
| 69 | #1687 | — | 7 | 1 | no | — |
| 70 | #1360 | — | 7 | 1 | no | — |
| 71 | #1334 | D7 | 1 | 1 | yes | characteristic, complemented |
| 72 | #1330 | — | 7 | 1 | no | — |
| 73 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 74 | #1161 | — | 4 | 2 | no | — |
| 75 | #1161 | — | 8 | 1 | no | — |
| 76 | Q8×D7 | — | 4 | 2 | no | — |
| 77 | Q8×D7 | — | 8 | 1 | no | — |
| 78 | C2×D28 | — | 2 | 1 | no | — |
| 79 | C2×D28 | — | 4 | 1 | no | — |
| 80 | C2×D28 | — | 4 | 1 | no | — |
| 81 | C2×D28 | — | 4 | 1 | no | — |
| 82 | C2×D28 | — | 4 | 1 | no | — |
| 83 | C2×D28 | — | 4 | 1 | no | — |
| 84 | C2×D28 | D8 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×D7 | — | 2 | 1 | no | — |
| 86 | C2×C4×D7 | — | 4 | 1 | no | — |
| 87 | C2×C4×D7 | — | 4 | 1 | no | — |
| 88 | C2×Dic14 | — | 2 | 1 | no | — |
| 89 | C2×Dic14 | — | 4 | 1 | no | — |
| 90 | C7×Q16 | — | 2 | 1 | no | — |
| 91 | C7×Q16 | D8 | 1 | 2 | yes | complemented |
| 92 | C2×C7×C8 | — | 4 | 1 | no | — |
| 93 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 96 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 97 | C42×C7 | — | 4 | 1 | no | — |
| 98 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | #1144 | — | 4 | 2 | no | — |
| 100 | #1143 | — | 4 | 2 | no | — |
| 101 | (C2×D7)⋊C4 | — | 2 | 1 | no | — |
| 102 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 103 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 104 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 105 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 106 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 107 | C4⋊Dic7 | D8 | 1 | 1 | yes | characteristic |
| 108 | C2×C7⋊C8 | — | 2 | 1 | no | — |
| 109 | Dic28 | — | 4 | 1 | no | — |
| 110 | D56 | — | 4 | 1 | no | — |
| 111 | D56 | — | 4 | 1 | no | — |
| 112 | D56 | — | 4 | 1 | no | — |
| 113 | D56 | — | 4 | 1 | no | — |
| 114 | D56 | — | 4 | 1 | no | — |
| 115 | D56 | — | 8 | 2 | no | — |
| 116 | #1132 | — | 4 | 1 | no | — |
| 117 | #1132 | — | 4 | 1 | no | — |
| 118 | #1131 | — | 4 | 1 | no | — |
| 119 | #1131 | — | 4 | 1 | no | — |
| 120 | C7×C16 | — | 2 | 2 | no | — |
| 121 | C7⋊C16 | — | 4 | 1 | no | — |
| 122 | C2×#137 | — | 14 | 1 | no | — |
| 123 | #548 | — | 14 | 1 | no | — |
| 124 | C2×SD32 | — | 14 | 1 | no | — |
| 125 | C2×SD32 | — | 14 | 1 | no | — |
| 126 | C2×D16 | — | 14 | 1 | no | — |
| 127 | #495 | — | 7 | 1 | no | — |
| 128 | #493 | — | 7 | 1 | no | — |
| 129 | #468 | — | 14 | 1 | no | — |
| 130 | #464 | — | 14 | 1 | no | — |
| 131 | #459 | — | 14 | 1 | no | — |
| 132 | #451 | — | 14 | 1 | no | — |
| 133 | #449 | — | 14 | 1 | no | — |
| 134 | C4×Q16 | C2×D7 | 1 | 1 | yes | characteristic |
| 135 | #436 | — | 7 | 1 | no | — |
| 136 | C4⋊C16 | — | 7 | 1 | no | — |
| 137 | #357 | — | 14 | 1 | no | — |
| 138 | #346 | C2×D7 | 1 | 1 | yes | characteristic |
| 139 | #334 | C2×D7 | 1 | 1 | yes | characteristic |
| 140 | #331 | — | 14 | 1 | no | — |
| 141 | C22×D7 | — | 2 | 1 | no | — |
| 142 | C22×D7 | — | 4 | 1 | no | — |
| 143 | C22×D7 | — | 8 | 1 | no | — |
| 144 | C7×Q8 | — | 2 | 2 | no | — |
| 145 | C7×Q8 | — | 4 | 1 | no | — |
| 146 | C2×C4×C7 | — | 4 | 1 | no | — |
| 147 | C2×C4×C7 | — | 4 | 1 | no | — |
| 148 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 149 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 151 | C2×Dic7 | — | 2 | 1 | no | — |
| 152 | C2×Dic7 | — | 4 | 1 | no | — |
| 153 | D28 | — | 4 | 1 | no | — |
| 154 | D28 | — | 4 | 1 | no | — |
| 155 | D28 | — | 4 | 1 | no | — |
| 156 | D28 | — | 8 | 1 | no | — |
| 157 | D28 | — | 8 | 1 | no | — |
| 158 | D28 | — | 8 | 2 | no | — |
| 159 | D28 | — | 8 | 2 | no | — |
| 160 | D28 | — | 8 | 1 | no | — |
| 161 | D28 | — | 8 | 1 | no | — |
| 162 | D28 | — | 16 | 1 | no | — |
| 163 | C4×D7 | — | 4 | 1 | no | — |
| 164 | C4×D7 | — | 4 | 1 | no | — |
| 165 | C4×D7 | — | 8 | 2 | no | — |
| 166 | C4×D7 | — | 8 | 2 | no | — |
| 167 | Dic14 | — | 4 | 1 | no | — |
| 168 | Dic14 | — | 4 | 1 | no | — |
| 169 | Dic14 | — | 8 | 2 | no | — |
| 170 | C7×C8 | — | 2 | 1 | no | — |
| 171 | C7×C8 | — | 8 | 1 | no | — |
| 172 | C7×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 173 | C7×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 174 | C7⋊C8 | — | 4 | 1 | no | — |
| 175 | C2×#41 | — | 14 | 1 | no | — |
| 176 | C22×Q8 | — | 14 | 1 | no | — |
| 177 | #137 | — | 28 | 2 | no | — |
| 178 | #137 | — | 28 | 2 | no | — |
| 179 | C2×Q16 | — | 7 | 1 | no | — |
| 180 | C2×Q16 | — | 14 | 1 | no | — |
| 181 | C2×Q16 | D28 | 1 | 1 | yes | characteristic |
| 182 | C2×SD16 | — | 14 | 1 | no | — |
| 183 | C2×SD16 | — | 14 | 1 | no | — |
| 184 | C2×D8 | — | 7 | 1 | no | — |
| 185 | C2×D8 | — | 14 | 1 | no | — |
| 186 | C2×D8 | — | 14 | 1 | no | — |
| 187 | C2×D8 | — | 28 | 1 | no | — |
| 188 | C2×M16 | — | 14 | 1 | no | — |
| 189 | #127 | — | 14 | 1 | no | — |
| 190 | #124 | — | 14 | 1 | no | — |
| 191 | #124 | — | 28 | 1 | no | — |
| 192 | C22⋊Q8 | — | 14 | 1 | no | — |
| 193 | C22⋊Q8 | — | 28 | 1 | no | — |
| 194 | #121 | — | 14 | 1 | no | — |
| 195 | #121 | — | 28 | 1 | no | — |
| 196 | C4×Q8 | — | 2 | 1 | no | — |
| 197 | C4×D4 | — | 7 | 1 | no | — |
| 198 | C4×D4 | — | 28 | 1 | no | — |
| 199 | SD32 | — | 28 | 2 | no | — |
| 200 | SD32 | — | 28 | 2 | no | — |
| 201 | D16 | — | 28 | 2 | no | — |
| 202 | C2×C16 | — | 2 | 1 | no | — |
| 203 | C2×C16 | — | 14 | 1 | no | — |
| 204 | #107 | D28 | 1 | 1 | yes | characteristic |
| 205 | C4⋊C8 | — | 2 | 1 | no | — |
| 206 | C4⋊C8 | — | 7 | 1 | no | — |
| 207 | #103 | — | 2 | 1 | no | — |
| 208 | #103 | — | 14 | 1 | no | — |
| 209 | #102 | — | 14 | 1 | no | — |
| 210 | #102 | — | 28 | 1 | no | — |
| 211 | C22⋊C8 | — | 14 | 1 | no | — |
| 212 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 213 | C22×C7 | #447 | 1 | 1 | yes | characteristic |
| 214 | C2×D7 | — | 4 | 1 | no | — |
| 215 | C2×D7 | — | 4 | 1 | no | — |
| 216 | C2×D7 | — | 4 | 1 | no | — |
| 217 | C2×D7 | — | 8 | 1 | no | — |
| 218 | C2×D7 | — | 8 | 1 | no | — |
| 219 | C2×D7 | — | 8 | 1 | no | — |
| 220 | C2×D7 | — | 16 | 1 | no | — |
| 221 | C2×D7 | — | 16 | 1 | no | — |
| 222 | C2×D7 | — | 16 | 1 | no | — |
| 223 | C4×C7 | — | 2 | 1 | no | — |
| 224 | C4×C7 | — | 2 | 1 | no | — |
| 225 | C4×C7 | — | 4 | 2 | no | — |
| 226 | C4×C7 | — | 8 | 1 | no | — |
| 227 | C4×C7 | #447 | 1 | 1 | yes | characteristic |
| 228 | C4×C7 | #453 | 1 | 1 | yes | characteristic |
| 229 | Dic7 | — | 4 | 1 | no | — |
| 230 | Dic7 | — | 8 | 1 | no | — |
| 231 | #41 | — | 28 | 2 | no | — |
| 232 | #41 | — | 56 | 1 | no | — |
| 233 | C2×Q8 | — | 2 | 1 | no | — |
| 234 | C2×Q8 | — | 14 | 1 | no | — |
| 235 | C2×Q8 | — | 28 | 1 | no | — |
| 236 | C2×Q8 | — | 28 | 2 | no | — |
| 237 | C2×Q8 | — | 56 | 1 | no | — |
| 238 | C2×D4 | — | 7 | 1 | no | — |
| 239 | C2×D4 | — | 14 | 1 | no | — |
| 240 | C2×D4 | — | 28 | 1 | no | — |
| 241 | C2×D4 | — | 28 | 1 | no | — |
| 242 | C2×D4 | — | 28 | 1 | no | — |
| 243 | C2×D4 | — | 28 | 1 | no | — |
| 244 | C2×D4 | — | 28 | 1 | no | — |
| 245 | C22×C4 | — | 14 | 1 | no | — |
| 246 | C22×C4 | — | 28 | 1 | no | — |
| 247 | C22×C4 | — | 28 | 1 | no | — |
| 248 | Q16 | — | 2 | 1 | no | — |
| 249 | Q16 | — | 28 | 1 | no | — |
| 250 | Q16 | — | 28 | 2 | no | — |
| 251 | Q16 | D56 | 1 | 2 | yes | complemented |
| 252 | SD16 | — | 28 | 2 | no | — |
| 253 | SD16 | — | 28 | 1 | no | — |
| 254 | SD16 | — | 28 | 1 | no | — |
| 255 | D8 | — | 28 | 1 | no | — |
| 256 | D8 | — | 28 | 1 | no | — |
| 257 | D8 | — | 28 | 1 | no | — |
| 258 | D8 | — | 28 | 1 | no | — |
| 259 | D8 | — | 28 | 1 | no | — |
| 260 | D8 | — | 56 | 2 | no | — |
| 261 | M16 | — | 28 | 1 | no | — |
| 262 | M16 | — | 28 | 1 | no | — |
| 263 | C2×C8 | — | 4 | 1 | no | — |
| 264 | C2×C8 | — | 14 | 1 | no | — |
| 265 | C2×C8 | C2×D28 | 1 | 1 | yes | characteristic |
| 266 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 267 | C4⋊C4 | — | 2 | 1 | no | — |
| 268 | C4⋊C4 | — | 4 | 1 | no | — |
| 269 | C4⋊C4 | — | 7 | 1 | no | — |
| 270 | C4⋊C4 | — | 28 | 1 | no | — |
| 271 | C22⋊C4 | — | 14 | 1 | no | — |
| 272 | C22⋊C4 | — | 28 | 1 | no | — |
| 273 | C22⋊C4 | — | 28 | 1 | no | — |
| 274 | C22⋊C4 | — | 28 | 1 | no | — |
| 275 | C22⋊C4 | — | 28 | 1 | no | — |
| 276 | C42 | — | 4 | 1 | no | — |
| 277 | C42 | D4×D7 | 1 | 1 | yes | characteristic |
| 278 | C16 | — | 2 | 2 | no | — |
| 279 | C16 | — | 28 | 1 | no | — |
| 280 | C2×C7 | #1619 | 1 | 1 | yes | characteristic |
| 281 | C2×C7 | #2190 | 1 | 1 | yes | characteristic |
| 282 | C2×C7 | #2193 | 1 | 1 | yes | characteristic |
| 283 | D7 | — | 8 | 1 | no | — |
| 284 | D7 | — | 16 | 1 | no | — |
| 285 | D7 | — | 32 | 1 | no | — |
| 286 | C23 | — | 14 | 1 | no | — |
| 287 | C23 | — | 28 | 1 | no | — |
| 288 | C23 | — | 56 | 1 | no | — |
| 289 | Q8 | — | 2 | 2 | no | — |
| 290 | Q8 | — | 4 | 1 | no | — |
| 291 | Q8 | — | 28 | 1 | no | — |
| 292 | Q8 | — | 28 | 1 | no | — |
| 293 | Q8 | — | 56 | 2 | no | — |
| 294 | D4 | — | 28 | 1 | no | — |
| 295 | D4 | — | 28 | 1 | no | — |
| 296 | D4 | — | 28 | 1 | no | — |
| 297 | D4 | — | 56 | 1 | no | — |
| 298 | D4 | — | 56 | 1 | no | — |
| 299 | D4 | — | 56 | 2 | no | — |
| 300 | D4 | — | 56 | 2 | no | — |
| 301 | D4 | — | 56 | 1 | no | — |
| 302 | D4 | — | 56 | 1 | no | — |
| 303 | D4 | — | 112 | 1 | no | — |
| 304 | C2×C4 | — | 4 | 1 | no | — |
| 305 | C2×C4 | — | 4 | 1 | no | — |
| 306 | C2×C4 | — | 14 | 1 | no | — |
| 307 | C2×C4 | — | 28 | 1 | no | — |
| 308 | C2×C4 | — | 28 | 1 | no | — |
| 309 | C2×C4 | — | 28 | 1 | no | — |
| 310 | C2×C4 | — | 56 | 2 | no | — |
| 311 | C2×C4 | — | 56 | 2 | no | — |
| 312 | C2×C4 | #6445 | 1 | 1 | yes | characteristic |
| 313 | C2×C4 | D7×D8 | 1 | 1 | yes | characteristic |
| 314 | C2×C4 | #6477 | 1 | 1 | yes | characteristic |
| 315 | C8 | — | 2 | 1 | no | — |
| 316 | C8 | — | 8 | 1 | no | — |
| 317 | C8 | — | 28 | 1 | no | — |
| 318 | C8 | C2×D56 | 1 | 1 | yes | characteristic |
| 319 | C8 | #6471 | 1 | 1 | yes | characteristic |
| 320 | C7 | #12325 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 321 | C22 | — | 28 | 1 | no | — |
| 322 | C22 | — | 28 | 1 | no | — |
| 323 | C22 | — | 28 | 1 | no | — |
| 324 | C22 | — | 56 | 1 | no | — |
| 325 | C22 | — | 56 | 1 | no | — |
| 326 | C22 | — | 56 | 1 | no | — |
| 327 | C22 | — | 112 | 1 | no | — |
| 328 | C22 | — | 112 | 1 | no | — |
| 329 | C22 | — | 112 | 1 | no | — |
| 330 | C22 | #89545 | 1 | 1 | yes | characteristic |
| 331 | C4 | — | 2 | 1 | no | — |
| 332 | C4 | — | 2 | 1 | no | — |
| 333 | C4 | — | 4 | 2 | no | — |
| 334 | C4 | — | 8 | 1 | no | — |
| 335 | C4 | — | 28 | 1 | no | — |
| 336 | C4 | — | 56 | 1 | no | — |
| 337 | C4 | #89504 | 1 | 1 | yes | characteristic |
| 338 | C4 | #89594 | 1 | 1 | yes | characteristic |
| 339 | C2 | — | 56 | 1 | no | — |
| 340 | C2 | — | 112 | 1 | no | — |
| 341 | C2 | — | 224 | 1 | no | — |
| 342 | C2 | #11725948 | 1 | 1 | yes | characteristic |
| 343 | C2 | #11727006 | 1 | 1 | yes | characteristic |
| 344 | C2 | #11727022 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11737175 | C2 | semidirect |
| 2. #11728668 | C2 | semidirect |
| 3. #11726835 | C2 | semidirect |
| 4. #11726294 | C2 | semidirect |
| 5. C7×#1334 | C2 | semidirect |
| 6. #11725321 | C2 | semidirect |
| 7. #11725273 | C2 | semidirect |
| 71. #1334 | D7 | semidirect |
| 91. C7×Q16 | D8 | semidirect |
| 251. Q16 | D56 | semidirect |
| 320. C7 | #12325 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11737175 | 2 |
| #11728668 | 2 |
| #11726835 | 2 |
| #11726294 | 2 |
| C7×#1334 | 2 |
| #11725321 | 2 |
| #11725273 | 2 |
| #12325 | 7 |