Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
476 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 67 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 | #11736056 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11727718 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11726866 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C7×#1314 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11725332 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#6493 | — | 2 | 1 | no | — |
| 7 | C2×#6487 | — | 2 | 1 | no | — |
| 8 | C22×Dic28 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×C7×C22⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 10 | #89887 | — | 2 | 1 | no | — |
| 11 | #89887 | C22 | 1 | 2 | yes | — |
| 12 | C2×#6390 | — | 2 | 1 | no | — |
| 13 | C2×#6374 | — | 2 | 2 | no | — |
| 14 | #89508 | — | 2 | 2 | no | — |
| 15 | #89504 | — | 2 | 2 | no | — |
| 16 | C7×C22⋊C16 | C22 | 1 | 2 | yes | complemented |
| 17 | #89346 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #89298 | — | 2 | 2 | no | — |
| 19 | #12105 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C23×C4×C7 | D4 | 1 | 1 | yes | characteristic |
| 21 | C22×C22⋊D7 | — | 4 | 1 | no | — |
| 22 | C22×D28 | — | 2 | 1 | no | — |
| 23 | C22×Dic14 | — | 2 | 1 | no | — |
| 24 | C22×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 25 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 26 | C2×C22⋊Dic7 | — | 4 | 1 | no | — |
| 27 | #6493 | — | 2 | 2 | no | — |
| 28 | #6493 | — | 2 | 2 | no | — |
| 29 | #6493 | — | 4 | 1 | no | — |
| 30 | #6493 | — | 4 | 1 | no | — |
| 31 | C2×(C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 32 | C2×C4⋊Dic7 | — | 2 | 1 | no | — |
| 33 | C2×C4⋊Dic7 | — | 2 | 1 | no | — |
| 34 | #6487 | — | 4 | 2 | no | — |
| 35 | #6487 | — | 4 | 2 | no | — |
| 36 | C2×Dic7⋊C4 | — | 4 | 1 | no | — |
| 37 | C2×Dic28 | — | 2 | 2 | no | — |
| 38 | C2×Dic28 | — | 4 | 2 | no | — |
| 39 | C2×D56 | — | 2 | 1 | no | — |
| 40 | C2×D56 | D4 | 1 | 2 | yes | — |
| 41 | C2×C7×C16 | — | 2 | 2 | no | — |
| 42 | C7×C22⋊C8 | — | 2 | 2 | no | — |
| 43 | #6395 | — | 2 | 2 | no | — |
| 44 | #6395 | — | 2 | 2 | no | — |
| 45 | #6392 | — | 2 | 1 | no | — |
| 46 | #6392 | D4 | 1 | 2 | yes | — |
| 47 | #6390 | — | 4 | 2 | no | — |
| 48 | #6374 | — | 4 | 4 | no | — |
| 49 | #3186 | — | 7 | 1 | no | — |
| 50 | #2185 | — | 7 | 2 | no | — |
| 51 | #2001 | — | 7 | 1 | no | — |
| 52 | #1347 | — | 7 | 2 | no | — |
| 53 | #1314 | D7 | 1 | 1 | yes | characteristic, complemented |
| 54 | C24×C7 | D8 | 1 | 1 | yes | characteristic |
| 55 | C23×D7 | — | 4 | 1 | no | — |
| 56 | C22×C4×C7 | — | 2 | 1 | no | — |
| 57 | C22×C4×C7 | — | 2 | 1 | no | — |
| 58 | C22×C4×C7 | — | 4 | 1 | no | — |
| 59 | C22×C4×C7 | — | 4 | 1 | no | — |
| 60 | C22×C4×C7 | D8 | 1 | 1 | yes | characteristic |
| 61 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 63 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 64 | C2×C22⋊D7 | — | 8 | 2 | no | — |
| 65 | C2×C22⋊D7 | — | 8 | 2 | no | — |
| 66 | C22×Dic7 | — | 4 | 1 | no | — |
| 67 | C22×Dic7 | — | 4 | 1 | no | — |
| 68 | C22×Dic7 | — | 4 | 1 | no | — |
| 69 | C2×D28 | — | 2 | 2 | no | — |
| 70 | C2×D28 | — | 4 | 1 | no | — |
| 71 | C2×D28 | — | 8 | 1 | no | — |
| 72 | C2×D28 | — | 8 | 1 | no | — |
| 73 | C2×Dic14 | — | 2 | 2 | no | — |
| 74 | C2×Dic14 | — | 4 | 1 | no | — |
| 75 | C2×Dic14 | — | 4 | 1 | no | — |
| 76 | C2×Dic14 | — | 4 | 1 | no | — |
| 77 | C2×Dic14 | — | 4 | 1 | no | — |
| 78 | C2×Dic14 | — | 4 | 1 | no | — |
| 79 | C2×C7×C8 | — | 2 | 1 | no | — |
| 80 | C2×C7×C8 | — | 4 | 1 | no | — |
| 81 | C2×C7×C8 | — | 4 | 1 | no | — |
| 82 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 83 | C22⋊Dic7 | — | 4 | 4 | no | — |
| 84 | (C2×D7)⋊C4 | — | 4 | 2 | no | — |
| 85 | (C2×D7)⋊C4 | — | 4 | 2 | no | — |
| 86 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 87 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 88 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 89 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 90 | Dic7⋊C4 | — | 4 | 4 | no | — |
| 91 | Dic28 | — | 2 | 4 | no | — |
| 92 | Dic28 | — | 4 | 2 | no | — |
| 93 | D56 | — | 4 | 2 | no | — |
| 94 | D56 | — | 8 | 1 | no | — |
| 95 | C7×C16 | — | 4 | 2 | no | — |
| 96 | C22×Q16 | — | 7 | 1 | no | — |
| 97 | C2×C22⋊Q8 | — | 14 | 1 | no | — |
| 98 | C2×#121 | — | 14 | 1 | no | — |
| 99 | C2×SD32 | — | 14 | 2 | no | — |
| 100 | #480 | — | 14 | 2 | no | — |
| 101 | #466 | — | 7 | 2 | no | — |
| 102 | #466 | — | 14 | 1 | no | — |
| 103 | #447 | — | 14 | 2 | no | — |
| 104 | C2×#103 | — | 14 | 1 | no | — |
| 105 | C2×C22⋊C8 | C2×D7 | 1 | 1 | yes | characteristic |
| 106 | #358 | — | 14 | 2 | no | — |
| 107 | C22⋊C16 | C2×D7 | 1 | 2 | yes | complemented |
| 108 | #340 | — | 7 | 1 | no | — |
| 109 | C23×C7 | — | 2 | 1 | no | — |
| 110 | C23×C7 | — | 2 | 1 | no | — |
| 111 | C23×C7 | — | 4 | 1 | no | — |
| 112 | C23×C7 | — | 4 | 1 | no | — |
| 113 | C23×C7 | SD32 | 1 | 2 | yes | complemented |
| 114 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 115 | C22×D7 | — | 4 | 2 | no | — |
| 116 | C22×D7 | — | 8 | 2 | no | — |
| 117 | C22×D7 | — | 8 | 2 | no | — |
| 118 | C22×D7 | — | 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 | — | 4 | 1 | no | — |
| 123 | C2×C4×C7 | — | 4 | 1 | no | — |
| 124 | C2×C4×C7 | — | 4 | 1 | no | — |
| 125 | C2×C4×C7 | — | 8 | 1 | no | — |
| 126 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 128 | C22⋊D7 | — | 8 | 4 | no | — |
| 129 | C22⋊D7 | — | 8 | 2 | no | — |
| 130 | C22⋊D7 | — | 8 | 2 | no | — |
| 131 | C2×Dic7 | — | 4 | 2 | no | — |
| 132 | C2×Dic7 | — | 4 | 2 | no | — |
| 133 | C2×Dic7 | — | 4 | 2 | no | — |
| 134 | C2×Dic7 | — | 8 | 1 | no | — |
| 135 | C2×Dic7 | — | 8 | 1 | no | — |
| 136 | C2×Dic7 | — | 8 | 2 | no | — |
| 137 | C2×Dic7 | — | 8 | 1 | no | — |
| 138 | C2×Dic7 | — | 8 | 1 | no | — |
| 139 | D28 | — | 4 | 2 | no | — |
| 140 | D28 | — | 4 | 2 | no | — |
| 141 | D28 | — | 8 | 2 | no | — |
| 142 | Dic14 | — | 4 | 2 | no | — |
| 143 | Dic14 | — | 4 | 2 | no | — |
| 144 | Dic14 | — | 8 | 1 | no | — |
| 145 | Dic14 | — | 8 | 1 | no | — |
| 146 | C7×C8 | — | 2 | 2 | no | — |
| 147 | C7×C8 | — | 4 | 1 | no | — |
| 148 | C22×Q8 | — | 14 | 1 | no | — |
| 149 | C22×D4 | — | 14 | 1 | no | — |
| 150 | C22×D4 | — | 28 | 1 | no | — |
| 151 | C23×C4 | D28 | 1 | 1 | yes | characteristic |
| 152 | C2×Q16 | — | 14 | 2 | no | — |
| 153 | C2×Q16 | — | 28 | 2 | no | — |
| 154 | C2×D8 | — | 7 | 2 | no | — |
| 155 | C2×D8 | — | 14 | 1 | no | — |
| 156 | C22×C8 | D28 | 1 | 1 | yes | characteristic |
| 157 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 158 | C22⋊Q8 | — | 28 | 2 | no | — |
| 159 | C22⋊Q8 | — | 28 | 2 | no | — |
| 160 | #121 | — | 14 | 2 | no | — |
| 161 | #121 | — | 14 | 2 | no | — |
| 162 | #121 | — | 28 | 1 | no | — |
| 163 | #121 | — | 28 | 1 | no | — |
| 164 | C2×C4⋊C4 | — | 14 | 1 | no | — |
| 165 | C2×C4⋊C4 | — | 14 | 1 | no | — |
| 166 | C2×C4⋊C4 | — | 28 | 1 | no | — |
| 167 | C2×C22⋊C4 | — | 28 | 1 | no | — |
| 168 | C2×C22⋊C4 | — | 28 | 1 | no | — |
| 169 | SD32 | — | 28 | 4 | no | — |
| 170 | C2×C16 | — | 2 | 2 | no | — |
| 171 | #107 | — | 7 | 2 | no | — |
| 172 | #107 | — | 14 | 1 | no | — |
| 173 | #103 | — | 28 | 2 | no | — |
| 174 | #102 | — | 14 | 2 | no | — |
| 175 | #102 | — | 14 | 2 | no | — |
| 176 | C22⋊C8 | — | 2 | 2 | no | — |
| 177 | C22×C7 | — | 2 | 1 | no | — |
| 178 | C22×C7 | — | 2 | 1 | no | — |
| 179 | C22×C7 | — | 2 | 2 | no | — |
| 180 | C22×C7 | — | 4 | 2 | no | — |
| 181 | C22×C7 | — | 4 | 1 | no | — |
| 182 | C22×C7 | — | 4 | 1 | no | — |
| 183 | C22×C7 | — | 8 | 1 | no | — |
| 184 | C22×C7 | #447 | 1 | 1 | yes | characteristic |
| 185 | C22×C7 | C2×SD32 | 1 | 1 | yes | characteristic |
| 186 | C22×C7 | #509 | 1 | 1 | yes | characteristic |
| 187 | C2×D7 | — | 8 | 2 | no | — |
| 188 | C2×D7 | — | 8 | 2 | no | — |
| 189 | C2×D7 | — | 8 | 2 | no | — |
| 190 | C2×D7 | — | 16 | 4 | no | — |
| 191 | C4×C7 | — | 2 | 1 | no | — |
| 192 | C4×C7 | — | 4 | 1 | no | — |
| 193 | C4×C7 | #453 | 1 | 1 | yes | characteristic |
| 194 | C4×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 195 | Dic7 | — | 8 | 2 | no | — |
| 196 | Dic7 | — | 8 | 2 | no | — |
| 197 | Dic7 | — | 8 | 2 | no | — |
| 198 | C24 | — | 28 | 1 | no | — |
| 199 | C24 | D56 | 1 | 1 | yes | characteristic |
| 200 | C2×Q8 | — | 14 | 2 | no | — |
| 201 | C2×Q8 | — | 28 | 1 | no | — |
| 202 | C2×Q8 | — | 28 | 1 | no | — |
| 203 | C2×Q8 | — | 28 | 1 | no | — |
| 204 | C2×Q8 | — | 28 | 1 | no | — |
| 205 | C2×Q8 | — | 28 | 1 | no | — |
| 206 | C2×D4 | — | 14 | 2 | no | — |
| 207 | C2×D4 | — | 28 | 1 | no | — |
| 208 | C2×D4 | — | 28 | 2 | no | — |
| 209 | C2×D4 | — | 28 | 2 | no | — |
| 210 | C2×D4 | — | 56 | 1 | no | — |
| 211 | C2×D4 | — | 56 | 1 | no | — |
| 212 | C2×D4 | — | 56 | 2 | no | — |
| 213 | C2×D4 | — | 56 | 2 | no | — |
| 214 | C22×C4 | — | 2 | 1 | no | — |
| 215 | C22×C4 | — | 2 | 1 | no | — |
| 216 | C22×C4 | — | 4 | 1 | no | — |
| 217 | C22×C4 | — | 4 | 1 | no | — |
| 218 | C22×C4 | — | 28 | 1 | no | — |
| 219 | C22×C4 | — | 28 | 1 | no | — |
| 220 | C22×C4 | — | 28 | 1 | no | — |
| 221 | C22×C4 | D56 | 1 | 1 | yes | characteristic |
| 222 | C22×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 223 | Q16 | — | 14 | 4 | no | — |
| 224 | Q16 | — | 28 | 2 | no | — |
| 225 | D8 | — | 28 | 2 | no | — |
| 226 | D8 | — | 56 | 1 | no | — |
| 227 | C2×C8 | — | 2 | 1 | no | — |
| 228 | C2×C8 | — | 4 | 1 | no | — |
| 229 | C2×C8 | — | 4 | 1 | no | — |
| 230 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 231 | C4⋊C4 | — | 14 | 2 | no | — |
| 232 | C4⋊C4 | — | 14 | 2 | no | — |
| 233 | C4⋊C4 | — | 28 | 1 | no | — |
| 234 | C4⋊C4 | — | 28 | 1 | no | — |
| 235 | C4⋊C4 | — | 28 | 4 | no | — |
| 236 | C22⋊C4 | — | 28 | 2 | no | — |
| 237 | C22⋊C4 | — | 28 | 4 | no | — |
| 238 | C22⋊C4 | — | 28 | 2 | no | — |
| 239 | C16 | — | 4 | 2 | no | — |
| 240 | C2×C7 | — | 2 | 1 | no | — |
| 241 | C2×C7 | — | 2 | 1 | no | — |
| 242 | C2×C7 | — | 4 | 2 | no | — |
| 243 | C2×C7 | #1595 | 1 | 1 | yes | characteristic |
| 244 | C2×C7 | #2187 | 1 | 1 | yes | characteristic |
| 245 | C2×C7 | #2191 | 1 | 1 | yes | characteristic |
| 246 | D7 | — | 16 | 2 | no | — |
| 247 | C23 | — | 2 | 1 | no | — |
| 248 | C23 | — | 2 | 1 | no | — |
| 249 | C23 | — | 4 | 1 | no | — |
| 250 | C23 | — | 4 | 1 | no | — |
| 251 | C23 | — | 28 | 2 | no | — |
| 252 | C23 | — | 56 | 2 | no | — |
| 253 | C23 | — | 56 | 2 | no | — |
| 254 | C23 | — | 56 | 2 | no | — |
| 255 | C23 | #6374 | 1 | 2 | yes | complemented |
| 256 | C23 | C2×D56 | 1 | 1 | yes | characteristic |
| 257 | Q8 | — | 28 | 2 | no | — |
| 258 | Q8 | — | 28 | 2 | no | — |
| 259 | Q8 | — | 56 | 1 | no | — |
| 260 | Q8 | — | 56 | 1 | no | — |
| 261 | D4 | — | 28 | 2 | no | — |
| 262 | D4 | — | 28 | 2 | no | — |
| 263 | D4 | — | 56 | 2 | no | — |
| 264 | D4 | — | 56 | 4 | no | — |
| 265 | D4 | — | 56 | 2 | no | — |
| 266 | D4 | — | 56 | 2 | no | — |
| 267 | C2×C4 | — | 2 | 1 | no | — |
| 268 | C2×C4 | — | 2 | 1 | no | — |
| 269 | C2×C4 | — | 2 | 1 | no | — |
| 270 | C2×C4 | — | 4 | 1 | no | — |
| 271 | C2×C4 | — | 4 | 1 | no | — |
| 272 | C2×C4 | — | 4 | 1 | no | — |
| 273 | C2×C4 | — | 8 | 1 | no | — |
| 274 | C2×C4 | — | 28 | 2 | no | — |
| 275 | C2×C4 | — | 28 | 2 | no | — |
| 276 | C2×C4 | — | 28 | 2 | no | — |
| 277 | C2×C4 | — | 56 | 1 | no | — |
| 278 | C2×C4 | — | 56 | 2 | no | — |
| 279 | C2×C4 | — | 56 | 1 | no | — |
| 280 | C2×C4 | — | 56 | 1 | no | — |
| 281 | C2×C4 | — | 56 | 1 | no | — |
| 282 | C2×C4 | #6445 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #6471 | 1 | 1 | yes | characteristic |
| 284 | C8 | — | 2 | 2 | no | — |
| 285 | C8 | — | 4 | 1 | no | — |
| 286 | C7 | #12105 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 287 | C22 | — | 2 | 1 | no | — |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 2 | 2 | no | — |
| 290 | C22 | — | 4 | 2 | no | — |
| 291 | C22 | — | 4 | 1 | no | — |
| 292 | C22 | — | 4 | 1 | no | — |
| 293 | C22 | — | 8 | 1 | no | — |
| 294 | C22 | — | 56 | 2 | no | — |
| 295 | C22 | — | 56 | 2 | no | — |
| 296 | C22 | — | 56 | 2 | no | — |
| 297 | C22 | — | 112 | 4 | no | — |
| 298 | C22 | #89504 | 1 | 1 | yes | characteristic |
| 299 | C22 | C2×#6374 | 1 | 1 | yes | characteristic |
| 300 | C22 | #89680 | 1 | 1 | yes | characteristic |
| 301 | C4 | — | 2 | 1 | no | — |
| 302 | C4 | — | 4 | 1 | no | — |
| 303 | C4 | — | 56 | 2 | no | — |
| 304 | C4 | — | 56 | 2 | no | — |
| 305 | C4 | — | 56 | 2 | no | — |
| 306 | C4 | #89513 | 1 | 1 | yes | characteristic |
| 307 | C4 | #89519 | 1 | 1 | yes | characteristic |
| 308 | C2 | — | 2 | 1 | no | — |
| 309 | C2 | — | 2 | 1 | no | — |
| 310 | C2 | — | 4 | 2 | no | — |
| 311 | C2 | — | 112 | 2 | no | — |
| 312 | C2 | #11725860 | 1 | 1 | yes | characteristic |
| 313 | C2 | #11726864 | 1 | 1 | yes | characteristic |
| 314 | C2 | #11726875 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11727718 | C2 | semidirect |
| 3. #11726866 | C2 | semidirect |
| 4. C7×#1314 | C2 | semidirect |
| 5. #11725332 | C2 | semidirect |
| 8. C22×Dic28 | C22 | semidirect |
| 16. C7×C22⋊C16 | C22 | semidirect |
| 17. #89346 | C22 | semidirect |
| 53. #1314 | D7 | semidirect |
| 107. C22⋊C16 | C2×D7 | semidirect |
| 113. C23×C7 | SD32 | semidirect |
| 255. C23 | #6374 | semidirect |
| 286. C7 | #12105 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11736056 | 2 |
| #11727718 | 2 |
| #11726866 | 2 |
| C7×#1314 | 2 |
| #11725332 | 2 |
| #12105 | 7 |