Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
342 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #11736542 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11728297 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11726867 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11726866 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C7×#1323 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11725335 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11725333 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C22×Dic28 | C22 | 1 | 1 | yes | characteristic |
| 9 | #90225 | — | 2 | 1 | no | — |
| 10 | C7×#424 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#6398 | — | 2 | 1 | no | — |
| 12 | #89906 | — | 2 | 1 | no | — |
| 13 | #89888 | C22 | 1 | 1 | yes | characteristic |
| 14 | #89887 | C22 | 1 | 1 | yes | characteristic |
| 15 | #89776 | — | 2 | 1 | no | — |
| 16 | C2×Dic56 | — | 2 | 1 | no | — |
| 17 | C2×#6374 | — | 2 | 1 | no | — |
| 18 | #89622 | — | 2 | 1 | no | — |
| 19 | #89618 | — | 2 | 1 | no | — |
| 20 | #89617 | — | 2 | 1 | no | — |
| 21 | #89616 | — | 2 | 1 | no | — |
| 22 | C7×C22⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 23 | C7×C22⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 24 | #89347 | C22 | 1 | 1 | yes | characteristic |
| 25 | #89298 | — | 2 | 1 | no | — |
| 26 | #89298 | — | 2 | 1 | no | — |
| 27 | #12169 | — | 7 | 1 | no | maximal, Sylow |
| 28 | C22×Dic14 | — | 2 | 1 | no | — |
| 29 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 30 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 31 | C7×#117 | D4 | 1 | 1 | yes | characteristic |
| 32 | #6493 | — | 2 | 1 | no | — |
| 33 | #6487 | — | 2 | 1 | no | — |
| 34 | C2×#1136 | — | 2 | 1 | no | — |
| 35 | C2×Dic28 | — | 2 | 1 | no | — |
| 36 | C2×Dic28 | — | 2 | 1 | no | — |
| 37 | C2×Dic28 | — | 4 | 2 | no | — |
| 38 | C2×Dic28 | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×D56 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×#1132 | — | 2 | 1 | no | — |
| 41 | #6461 | — | 4 | 1 | no | — |
| 42 | #6459 | — | 4 | 1 | no | — |
| 43 | #6448 | — | 4 | 1 | no | — |
| 44 | #6446 | — | 4 | 1 | no | — |
| 45 | #6438 | — | 2 | 1 | no | — |
| 46 | C4×D28 | — | 2 | 1 | no | — |
| 47 | #6432 | — | 2 | 1 | no | — |
| 48 | C4×Dic14 | — | 2 | 1 | no | — |
| 49 | C2×C7×C16 | — | 2 | 1 | no | — |
| 50 | C2×C7×C16 | — | 2 | 1 | no | — |
| 51 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 52 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 53 | #6398 | — | 4 | 2 | no | — |
| 54 | #6395 | — | 2 | 1 | no | — |
| 55 | #6395 | — | 2 | 1 | no | — |
| 56 | #6392 | — | 2 | 1 | no | — |
| 57 | #6392 | D4 | 1 | 1 | yes | characteristic |
| 58 | #6392 | D4 | 1 | 1 | yes | characteristic |
| 59 | #6390 | — | 2 | 1 | no | — |
| 60 | #6390 | — | 2 | 1 | no | — |
| 61 | #6384 | — | 4 | 1 | no | — |
| 62 | #6384 | — | 4 | 1 | no | — |
| 63 | Dic56 | — | 4 | 2 | no | — |
| 64 | #6374 | — | 4 | 2 | no | — |
| 65 | #3262 | — | 7 | 1 | no | — |
| 66 | #2188 | — | 7 | 1 | no | — |
| 67 | #2185 | — | 7 | 1 | no | — |
| 68 | #2047 | — | 7 | 1 | no | — |
| 69 | #1350 | — | 7 | 1 | no | — |
| 70 | #1349 | — | 7 | 1 | no | — |
| 71 | #1323 | D7 | 1 | 1 | yes | characteristic, complemented |
| 72 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 74 | C22×Dic7 | — | 4 | 1 | no | — |
| 75 | C2×D28 | — | 2 | 1 | no | — |
| 76 | C2×C4×D7 | — | 4 | 1 | no | — |
| 77 | C2×Dic14 | — | 2 | 2 | no | — |
| 78 | C2×Dic14 | — | 2 | 1 | no | — |
| 79 | C2×Dic14 | — | 4 | 1 | no | — |
| 80 | C2×Dic14 | — | 4 | 1 | no | — |
| 81 | C2×Dic14 | — | 4 | 1 | no | — |
| 82 | C2×Dic14 | — | 4 | 1 | no | — |
| 83 | C2×Dic14 | — | 4 | 1 | no | — |
| 84 | C2×Dic14 | — | 4 | 1 | no | — |
| 85 | C7×M16 | — | 4 | 1 | no | — |
| 86 | C2×C7×C8 | — | 2 | 1 | no | — |
| 87 | C2×C7×C8 | — | 4 | 1 | no | — |
| 88 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C7×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 91 | C7×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 92 | C7×C22⋊C4 | — | 2 | 1 | no | — |
| 93 | C42×C7 | — | 2 | 1 | no | — |
| 94 | C22⋊Dic7 | — | 4 | 1 | no | — |
| 95 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 96 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 97 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 98 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 99 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 100 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 101 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 102 | Dic7⋊C4 | — | 4 | 1 | no | — |
| 103 | Dic7⋊C4 | — | 4 | 1 | no | — |
| 104 | C4×Dic7 | — | 4 | 1 | no | — |
| 105 | #1136 | — | 4 | 1 | no | — |
| 106 | #1136 | — | 4 | 1 | no | — |
| 107 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 108 | Dic28 | — | 2 | 2 | no | — |
| 109 | Dic28 | — | 2 | 2 | no | — |
| 110 | Dic28 | — | 4 | 1 | no | — |
| 111 | Dic28 | — | 4 | 1 | no | — |
| 112 | Dic28 | — | 4 | 1 | no | — |
| 113 | D56 | — | 4 | 1 | no | — |
| 114 | #1132 | — | 8 | 1 | no | — |
| 115 | C7×C16 | — | 4 | 1 | no | — |
| 116 | C7×C16 | — | 4 | 1 | no | — |
| 117 | C22×Q16 | — | 7 | 1 | no | — |
| 118 | #542 | — | 14 | 1 | no | — |
| 119 | C2×Q32 | — | 14 | 1 | no | — |
| 120 | C2×SD32 | — | 14 | 1 | no | — |
| 121 | #477 | — | 14 | 1 | no | — |
| 122 | #474 | — | 14 | 1 | no | — |
| 123 | #468 | — | 14 | 1 | no | — |
| 124 | #467 | — | 7 | 1 | no | — |
| 125 | #466 | — | 7 | 1 | no | — |
| 126 | #464 | — | 14 | 1 | no | — |
| 127 | #463 | — | 14 | 1 | no | — |
| 128 | #424 | C2×D7 | 1 | 1 | yes | characteristic |
| 129 | #419 | — | 14 | 1 | no | — |
| 130 | C2×#101 | — | 14 | 1 | no | — |
| 131 | #358 | — | 14 | 1 | no | — |
| 132 | #358 | — | 14 | 1 | no | — |
| 133 | C22⋊C16 | C2×D7 | 1 | 1 | yes | characteristic |
| 134 | C22⋊C16 | C2×D7 | 1 | 1 | yes | characteristic |
| 135 | #341 | — | 7 | 1 | no | — |
| 136 | C23×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 137 | C22×D7 | — | 4 | 1 | no | — |
| 138 | C2×C4×C7 | — | 2 | 1 | no | — |
| 139 | C2×C4×C7 | — | 2 | 1 | no | — |
| 140 | C2×C4×C7 | — | 2 | 1 | no | — |
| 141 | C2×C4×C7 | — | 2 | 1 | no | — |
| 142 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 143 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 144 | C22⋊D7 | — | 16 | 1 | no | — |
| 145 | C2×Dic7 | — | 4 | 2 | no | — |
| 146 | C2×Dic7 | — | 4 | 1 | no | — |
| 147 | C2×Dic7 | — | 4 | 1 | no | — |
| 148 | C2×Dic7 | — | 4 | 1 | no | — |
| 149 | C2×Dic7 | — | 8 | 1 | no | — |
| 150 | C2×Dic7 | — | 8 | 1 | no | — |
| 151 | D28 | — | 4 | 1 | no | — |
| 152 | D28 | — | 4 | 1 | no | — |
| 153 | C4×D7 | — | 16 | 1 | no | — |
| 154 | Dic14 | — | 4 | 2 | no | — |
| 155 | Dic14 | — | 4 | 1 | no | — |
| 156 | Dic14 | — | 4 | 1 | no | — |
| 157 | Dic14 | — | 4 | 2 | no | — |
| 158 | Dic14 | — | 8 | 1 | no | — |
| 159 | Dic14 | — | 8 | 1 | no | — |
| 160 | Dic14 | — | 16 | 1 | no | — |
| 161 | C7×C8 | — | 2 | 1 | no | — |
| 162 | C7×C8 | — | 2 | 1 | no | — |
| 163 | C7×C8 | — | 4 | 1 | no | — |
| 164 | C7⋊C8 | — | 8 | 1 | no | — |
| 165 | C22×Q8 | — | 14 | 1 | no | — |
| 166 | C2×Q16 | — | 7 | 1 | no | — |
| 167 | C2×Q16 | — | 14 | 1 | no | — |
| 168 | C2×Q16 | — | 14 | 1 | no | — |
| 169 | C2×Q16 | — | 28 | 2 | no | — |
| 170 | C2×SD16 | — | 14 | 1 | no | — |
| 171 | C2×D8 | — | 7 | 1 | no | — |
| 172 | C2×M16 | — | 14 | 1 | no | — |
| 173 | C2×M16 | D28 | 1 | 1 | yes | characteristic |
| 174 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 175 | C4⋊Q8 | — | 14 | 1 | no | — |
| 176 | #126 | — | 28 | 1 | no | — |
| 177 | #124 | — | 14 | 1 | no | — |
| 178 | #124 | — | 28 | 1 | no | — |
| 179 | #123 | — | 28 | 1 | no | — |
| 180 | C22⋊Q8 | — | 14 | 1 | no | — |
| 181 | C22⋊Q8 | — | 28 | 1 | no | — |
| 182 | #121 | — | 14 | 1 | no | — |
| 183 | C4×Q8 | — | 14 | 1 | no | — |
| 184 | C4×D4 | — | 14 | 1 | no | — |
| 185 | #117 | D28 | 1 | 1 | yes | characteristic |
| 186 | Q32 | — | 28 | 2 | no | — |
| 187 | SD32 | — | 28 | 2 | no | — |
| 188 | C2×C16 | — | 2 | 1 | no | — |
| 189 | C2×C16 | — | 2 | 1 | no | — |
| 190 | #107 | — | 7 | 1 | no | — |
| 191 | #107 | — | 7 | 1 | no | — |
| 192 | #107 | — | 14 | 1 | no | — |
| 193 | C4⋊C8 | — | 2 | 1 | no | — |
| 194 | C4⋊C8 | — | 2 | 1 | no | — |
| 195 | #103 | — | 14 | 1 | no | — |
| 196 | #103 | — | 14 | 1 | no | — |
| 197 | #103 | — | 28 | 1 | no | — |
| 198 | #103 | — | 28 | 1 | no | — |
| 199 | #102 | — | 14 | 1 | no | — |
| 200 | #102 | — | 14 | 1 | no | — |
| 201 | #101 | — | 28 | 2 | no | — |
| 202 | C22×C7 | — | 2 | 1 | no | — |
| 203 | C22×C7 | — | 2 | 1 | no | — |
| 204 | C22×C7 | — | 2 | 1 | no | — |
| 205 | C22×C7 | #447 | 1 | 1 | yes | characteristic |
| 206 | C2×D7 | — | 8 | 1 | no | — |
| 207 | C2×D7 | — | 8 | 1 | no | — |
| 208 | C2×D7 | — | 8 | 1 | no | — |
| 209 | C4×C7 | — | 2 | 1 | no | — |
| 210 | C4×C7 | — | 4 | 1 | no | — |
| 211 | C4×C7 | — | 4 | 1 | no | — |
| 212 | C4×C7 | #454 | 1 | 1 | yes | characteristic |
| 213 | C4×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 214 | Dic7 | — | 8 | 2 | no | — |
| 215 | Dic7 | — | 8 | 1 | no | — |
| 216 | Dic7 | — | 8 | 1 | no | — |
| 217 | Dic7 | — | 8 | 1 | no | — |
| 218 | C2×Q8 | — | 14 | 2 | no | — |
| 219 | C2×Q8 | — | 14 | 1 | no | — |
| 220 | C2×Q8 | — | 28 | 1 | no | — |
| 221 | C2×Q8 | — | 28 | 1 | no | — |
| 222 | C2×Q8 | — | 28 | 1 | no | — |
| 223 | C2×Q8 | — | 28 | 1 | no | — |
| 224 | C2×Q8 | — | 28 | 1 | no | — |
| 225 | C2×Q8 | — | 28 | 1 | no | — |
| 226 | C2×D4 | — | 14 | 1 | no | — |
| 227 | C2×D4 | — | 28 | 1 | no | — |
| 228 | C22×C4 | — | 28 | 1 | no | — |
| 229 | C22×C4 | — | 28 | 1 | no | — |
| 230 | C22×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 231 | Q16 | — | 14 | 2 | no | — |
| 232 | Q16 | — | 14 | 2 | no | — |
| 233 | Q16 | — | 28 | 1 | no | — |
| 234 | Q16 | — | 28 | 1 | no | — |
| 235 | Q16 | — | 28 | 1 | no | — |
| 236 | SD16 | — | 56 | 1 | no | — |
| 237 | D8 | — | 28 | 1 | no | — |
| 238 | M16 | — | 4 | 1 | no | — |
| 239 | M16 | — | 28 | 1 | no | — |
| 240 | M16 | — | 28 | 1 | no | — |
| 241 | C2×C8 | — | 2 | 1 | no | — |
| 242 | C2×C8 | — | 4 | 1 | no | — |
| 243 | C2×C8 | — | 28 | 1 | no | — |
| 244 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 246 | C4⋊C4 | — | 14 | 1 | no | — |
| 247 | C4⋊C4 | — | 14 | 1 | no | — |
| 248 | C4⋊C4 | — | 28 | 1 | no | — |
| 249 | C4⋊C4 | — | 28 | 1 | no | — |
| 250 | C4⋊C4 | — | 28 | 1 | no | — |
| 251 | C4⋊C4 | D56 | 1 | 1 | yes | characteristic |
| 252 | C4⋊C4 | D56 | 1 | 1 | yes | characteristic |
| 253 | C22⋊C4 | — | 2 | 1 | no | — |
| 254 | C22⋊C4 | — | 28 | 1 | no | — |
| 255 | C22⋊C4 | — | 28 | 1 | no | — |
| 256 | C22⋊C4 | — | 28 | 1 | no | — |
| 257 | C22⋊C4 | — | 28 | 1 | no | — |
| 258 | C22⋊C4 | — | 28 | 1 | no | — |
| 259 | C42 | — | 2 | 1 | no | — |
| 260 | C42 | — | 28 | 1 | no | — |
| 261 | C16 | — | 4 | 1 | no | — |
| 262 | C16 | — | 4 | 1 | no | — |
| 263 | C2×C7 | — | 4 | 1 | no | — |
| 264 | C2×C7 | #1598 | 1 | 1 | yes | characteristic |
| 265 | C2×C7 | #2194 | 1 | 1 | yes | characteristic |
| 266 | C2×C7 | #2195 | 1 | 1 | yes | characteristic |
| 267 | D7 | — | 16 | 1 | no | — |
| 268 | C23 | — | 28 | 1 | no | — |
| 269 | C23 | #6471 | 1 | 1 | yes | characteristic |
| 270 | Q8 | — | 28 | 2 | no | — |
| 271 | Q8 | — | 28 | 1 | no | — |
| 272 | Q8 | — | 28 | 2 | no | — |
| 273 | Q8 | — | 28 | 1 | no | — |
| 274 | Q8 | — | 56 | 1 | no | — |
| 275 | Q8 | — | 56 | 1 | no | — |
| 276 | Q8 | — | 112 | 1 | no | — |
| 277 | D4 | — | 28 | 1 | no | — |
| 278 | D4 | — | 28 | 1 | no | — |
| 279 | D4 | — | 112 | 1 | no | — |
| 280 | C2×C4 | — | 2 | 1 | no | — |
| 281 | C2×C4 | — | 2 | 1 | no | — |
| 282 | C2×C4 | — | 2 | 1 | no | — |
| 283 | C2×C4 | — | 2 | 1 | no | — |
| 284 | C2×C4 | — | 28 | 2 | no | — |
| 285 | C2×C4 | — | 28 | 1 | no | — |
| 286 | C2×C4 | — | 28 | 1 | no | — |
| 287 | C2×C4 | — | 28 | 1 | no | — |
| 288 | C2×C4 | — | 56 | 1 | no | — |
| 289 | C2×C4 | — | 56 | 1 | no | — |
| 290 | C2×C4 | — | 112 | 1 | no | — |
| 291 | C2×C4 | #6445 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 293 | C8 | — | 2 | 1 | no | — |
| 294 | C8 | — | 2 | 1 | no | — |
| 295 | C8 | — | 4 | 1 | no | — |
| 296 | C8 | — | 56 | 1 | no | — |
| 297 | C7 | #12169 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 298 | C22 | — | 2 | 1 | no | — |
| 299 | C22 | — | 2 | 1 | no | — |
| 300 | C22 | — | 2 | 1 | no | — |
| 301 | C22 | — | 56 | 1 | no | — |
| 302 | C22 | — | 56 | 1 | no | — |
| 303 | C22 | — | 56 | 1 | no | — |
| 304 | C22 | #89504 | 1 | 1 | yes | characteristic |
| 305 | C4 | — | 2 | 1 | no | — |
| 306 | C4 | — | 4 | 1 | no | — |
| 307 | C4 | — | 4 | 1 | no | — |
| 308 | C4 | — | 56 | 1 | no | — |
| 309 | C4 | — | 56 | 1 | no | — |
| 310 | C4 | — | 56 | 2 | no | — |
| 311 | C4 | — | 56 | 1 | no | — |
| 312 | C4 | #89513 | 1 | 1 | yes | characteristic |
| 313 | C4 | #89525 | 1 | 1 | yes | characteristic |
| 314 | C2 | — | 4 | 1 | no | — |
| 315 | C2 | — | 112 | 1 | no | — |
| 316 | C2 | #11725885 | 1 | 1 | yes | characteristic |
| 317 | C2 | #11726882 | 1 | 1 | yes | characteristic |
| 318 | C2 | #11726883 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11728297 | C2 | semidirect |
| 3. #11726867 | C2 | semidirect |
| 5. C7×#1323 | C2 | semidirect |
| 6. #11725335 | C2 | semidirect |
| 71. #1323 | D7 | semidirect |
| 297. C7 | #12169 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11736542 | 2 |
| #11728297 | 2 |
| #11726867 | 2 |
| #11726866 | 2 |
| C7×#1323 | 2 |
| #11725335 | 2 |
| #11725333 | 2 |
| #12169 | 7 |