Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
820 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 107 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 | #11743320 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#1963 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11729553 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11728258 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11727186 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C7×#532 | C22 | 1 | 1 | yes | characteristic |
| 7 | #90521 | — | 2 | 2 | no | — |
| 8 | #90425 | C4 | 1 | 2 | yes | complemented |
| 9 | #90408 | C4 | 1 | 2 | yes | complemented |
| 10 | #90374 | — | 2 | 2 | no | — |
| 11 | #90298 | — | 2 | 4 | no | — |
| 12 | #90254 | — | 2 | 2 | no | — |
| 13 | D4×D28 | — | 2 | 2 | no | — |
| 14 | #90168 | C22 | 1 | 2 | yes | — |
| 15 | C7×#423 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×C7×#104 | — | 2 | 2 | no | — |
| 17 | C7×#376 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #89698 | C22 | 1 | 2 | yes | complemented |
| 19 | #89335 | — | 2 | 2 | no | — |
| 20 | #89281 | — | 2 | 2 | no | — |
| 21 | #89280 | — | 2 | 2 | no | — |
| 22 | #89246 | — | 2 | 4 | no | — |
| 23 | #8491 | — | 7 | 1 | no | maximal, Sylow |
| 24 | C2×C7×#41 | D4 | 1 | 2 | yes | — |
| 25 | C2×D4×D7 | — | 2 | 4 | no | — |
| 26 | C2×#1157 | — | 2 | 2 | no | — |
| 27 | C22×D28 | — | 2 | 2 | no | — |
| 28 | C2×C7×M16 | — | 2 | 1 | no | — |
| 29 | C7×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C7×#127 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C7×#124 | D4 | 1 | 2 | yes | — |
| 32 | C7×#121 | — | 2 | 2 | no | — |
| 33 | C4×C7×D4 | — | 2 | 2 | no | — |
| 34 | C2×C42×C7 | — | 2 | 1 | no | — |
| 35 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 36 | #6510 | — | 2 | 4 | no | — |
| 37 | #6503 | — | 2 | 4 | no | — |
| 38 | #6501 | — | 2 | 4 | no | — |
| 39 | #6500 | — | 2 | 4 | no | — |
| 40 | #6500 | — | 2 | 4 | no | — |
| 41 | #6493 | — | 2 | 4 | no | — |
| 42 | #6493 | — | 2 | 2 | no | — |
| 43 | #6493 | D4 | 1 | 4 | yes | — |
| 44 | #6489 | — | 2 | 2 | no | — |
| 45 | #6458 | — | 2 | 4 | no | — |
| 46 | #6455 | — | 2 | 4 | no | — |
| 47 | #6448 | — | 2 | 8 | no | — |
| 48 | #6445 | — | 2 | 8 | no | — |
| 49 | #6438 | — | 2 | 2 | no | — |
| 50 | #6437 | D4 | 1 | 2 | yes | — |
| 51 | C4×D28 | — | 2 | 2 | no | — |
| 52 | C4×D28 | C2×C4 | 1 | 4 | yes | — |
| 53 | #6432 | D4 | 1 | 2 | yes | — |
| 54 | C7×C4⋊C8 | — | 2 | 2 | no | — |
| 55 | C7×#104 | — | 4 | 4 | no | — |
| 56 | C7×#95 | — | 2 | 2 | no | — |
| 57 | C22⋊(C7⋊C8) | — | 2 | 2 | no | — |
| 58 | C4⋊(C7⋊C8) | — | 2 | 2 | no | — |
| 59 | #6377 | — | 2 | 2 | no | — |
| 60 | #3549 | — | 7 | 1 | no | — |
| 61 | #1963 | D7 | 1 | 1 | yes | characteristic, complemented |
| 62 | #1557 | — | 7 | 2 | no | — |
| 63 | #1536 | — | 7 | 1 | no | — |
| 64 | #1505 | — | 7 | 2 | no | — |
| 65 | C23×D7 | — | 2 | 4 | no | — |
| 66 | C7×#41 | — | 4 | 2 | no | — |
| 67 | C7×#41 | — | 4 | 2 | no | — |
| 68 | C2×C7×Q8 | C22⋊C4 | 1 | 2 | yes | — |
| 69 | C2×C7×D4 | — | 2 | 2 | no | — |
| 70 | C2×C7×D4 | — | 2 | 2 | no | — |
| 71 | C2×C7×D4 | C22⋊C4 | 1 | 2 | yes | — |
| 72 | C22×C4×C7 | — | 2 | 2 | no | — |
| 73 | C22×C4×C7 | — | 4 | 1 | no | — |
| 74 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 76 | C2×C22⋊D7 | — | 2 | 8 | no | — |
| 77 | C2×C22⋊D7 | — | 2 | 4 | no | — |
| 78 | D4×D7 | — | 8 | 8 | no | — |
| 79 | #1157 | — | 8 | 4 | no | — |
| 80 | C2×D28 | — | 2 | 2 | no | — |
| 81 | C2×D28 | — | 2 | 4 | no | — |
| 82 | C2×D28 | — | 2 | 2 | no | — |
| 83 | C2×D28 | — | 8 | 2 | no | — |
| 84 | C2×D28 | — | 8 | 2 | no | — |
| 85 | C2×D28 | C22⋊C4 | 1 | 4 | yes | — |
| 86 | C2×C4×D7 | — | 2 | 4 | no | — |
| 87 | C2×Dic14 | — | 2 | 2 | no | — |
| 88 | C7×M16 | — | 4 | 2 | no | — |
| 89 | C2×C7×C8 | — | 4 | 1 | no | — |
| 90 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 91 | C7×C22⋊C4 | — | 2 | 4 | no | — |
| 92 | C42×C7 | — | 2 | 2 | no | — |
| 93 | C42×C7 | — | 4 | 1 | no | — |
| 94 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 95 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C42×C7 | C2×D4 | 1 | 2 | yes | — |
| 97 | C22⋊Dic7 | — | 2 | 4 | no | — |
| 98 | (C2×D7)⋊C4 | — | 2 | 8 | no | — |
| 99 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 100 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 101 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 102 | C4⋊Dic7 | C22⋊C4 | 1 | 4 | yes | — |
| 103 | C4×Dic7 | — | 2 | 4 | no | — |
| 104 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 105 | H(3,𝔽4) | — | 14 | 2 | no | — |
| 106 | #555 | — | 7 | 2 | no | — |
| 107 | #555 | — | 14 | 2 | no | — |
| 108 | D42 | — | 7 | 2 | no | — |
| 109 | D42 | — | 14 | 2 | no | — |
| 110 | #535 | — | 14 | 4 | no | — |
| 111 | #535 | — | 14 | 2 | no | — |
| 112 | #532 | — | 7 | 2 | no | — |
| 113 | #532 | C2×D7 | 1 | 1 | yes | characteristic |
| 114 | #432 | — | 7 | 2 | no | — |
| 115 | #423 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 116 | C2×#104 | — | 2 | 2 | no | — |
| 117 | #376 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 118 | #332 | — | 14 | 2 | no | — |
| 119 | #331 | — | 14 | 2 | no | — |
| 120 | #329 | — | 14 | 2 | no | — |
| 121 | #327 | — | 14 | 4 | no | — |
| 122 | C23×C7 | — | 2 | 2 | no | — |
| 123 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 124 | C22×D7 | — | 2 | 4 | no | — |
| 125 | C22×D7 | — | 8 | 4 | no | — |
| 126 | C22×D7 | — | 8 | 8 | no | — |
| 127 | C7×Q8 | — | 4 | 2 | no | — |
| 128 | C7×D4 | — | 4 | 2 | no | — |
| 129 | C7×D4 | — | 4 | 4 | no | — |
| 130 | C7×D4 | — | 8 | 2 | no | — |
| 131 | C2×C4×C7 | — | 2 | 1 | no | — |
| 132 | C2×C4×C7 | — | 2 | 1 | no | — |
| 133 | C2×C4×C7 | — | 2 | 2 | no | — |
| 134 | C2×C4×C7 | — | 4 | 2 | no | — |
| 135 | C2×C4×C7 | — | 4 | 2 | no | — |
| 136 | C2×C4×C7 | — | 4 | 2 | no | — |
| 137 | C2×C4×C7 | — | 4 | 2 | no | — |
| 138 | C2×C4×C7 | — | 8 | 2 | no | — |
| 139 | C2×C4×C7 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 140 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 141 | C2×C4×C7 | #120 | 1 | 2 | yes | — |
| 142 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 143 | C22⋊D7 | — | 8 | 4 | no | — |
| 144 | C22⋊D7 | — | 8 | 8 | no | — |
| 145 | C2×Dic7 | — | 2 | 4 | no | — |
| 146 | D28 | — | 4 | 4 | no | — |
| 147 | D28 | — | 4 | 2 | no | — |
| 148 | D28 | — | 4 | 2 | no | — |
| 149 | D28 | — | 8 | 2 | no | — |
| 150 | D28 | — | 8 | 4 | no | — |
| 151 | C4×D7 | — | 4 | 8 | no | — |
| 152 | Dic14 | — | 8 | 2 | no | — |
| 153 | C7×C8 | — | 8 | 1 | no | — |
| 154 | C7⋊C8 | — | 8 | 2 | no | — |
| 155 | C2×#41 | — | 14 | 2 | no | — |
| 156 | C2×#41 | C22⋊D7 | 1 | 2 | yes | — |
| 157 | C22×D4 | — | 14 | 2 | no | — |
| 158 | C22×D4 | — | 14 | 4 | no | — |
| 159 | C2×M16 | — | 2 | 1 | no | — |
| 160 | C4⋊Q8 | — | 7 | 2 | no | — |
| 161 | C4⋊Q8 | C4×D7 | 1 | 1 | yes | characteristic |
| 162 | #127 | — | 7 | 2 | no | — |
| 163 | #127 | — | 14 | 4 | no | — |
| 164 | #127 | C4×D7 | 1 | 1 | yes | characteristic |
| 165 | #124 | — | 14 | 8 | no | — |
| 166 | #124 | — | 14 | 4 | no | — |
| 167 | #124 | — | 14 | 2 | no | — |
| 168 | #124 | D28 | 1 | 2 | yes | — |
| 169 | #121 | — | 2 | 2 | no | — |
| 170 | #121 | — | 7 | 4 | no | — |
| 171 | #121 | — | 14 | 4 | no | — |
| 172 | #121 | — | 14 | 4 | no | — |
| 173 | #121 | — | 14 | 4 | no | — |
| 174 | #121 | — | 14 | 2 | no | — |
| 175 | #120 | — | 14 | 8 | no | — |
| 176 | #120 | — | 14 | 4 | no | — |
| 177 | #120 | — | 14 | 4 | no | — |
| 178 | C4×D4 | — | 2 | 2 | no | — |
| 179 | C4×D4 | — | 7 | 4 | no | — |
| 180 | C4×D4 | — | 14 | 2 | no | — |
| 181 | #117 | — | 14 | 4 | no | — |
| 182 | #117 | — | 14 | 2 | no | — |
| 183 | C2×C42 | — | 2 | 1 | no | — |
| 184 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 185 | C4⋊C8 | — | 2 | 2 | no | — |
| 186 | C4⋊C8 | — | 14 | 2 | no | — |
| 187 | #104 | — | 4 | 4 | no | — |
| 188 | C22⋊C8 | — | 14 | 2 | no | — |
| 189 | #97 | — | 14 | 2 | no | — |
| 190 | #95 | — | 2 | 2 | no | — |
| 191 | C22×C7 | — | 2 | 1 | no | — |
| 192 | C22×C7 | — | 2 | 2 | no | — |
| 193 | C22×C7 | — | 4 | 2 | no | — |
| 194 | C22×C7 | — | 4 | 4 | no | — |
| 195 | C22×C7 | #379 | 1 | 1 | yes | characteristic |
| 196 | C2×D7 | — | 4 | 4 | no | — |
| 197 | C2×D7 | — | 4 | 4 | no | — |
| 198 | C2×D7 | — | 4 | 4 | no | — |
| 199 | C2×D7 | — | 16 | 8 | no | — |
| 200 | C4×C7 | — | 2 | 1 | no | — |
| 201 | C4×C7 | — | 2 | 2 | no | — |
| 202 | C4×C7 | — | 2 | 2 | no | — |
| 203 | C4×C7 | — | 4 | 2 | no | — |
| 204 | C4×C7 | — | 8 | 2 | no | — |
| 205 | C4×C7 | #453 | 1 | 1 | yes | characteristic |
| 206 | C4×C7 | #455 | 1 | 1 | yes | characteristic |
| 207 | Dic7 | — | 4 | 4 | no | — |
| 208 | C24 | — | 14 | 4 | no | — |
| 209 | #41 | — | 4 | 2 | no | — |
| 210 | #41 | — | 4 | 2 | no | — |
| 211 | #41 | — | 56 | 4 | no | — |
| 212 | C2×Q8 | — | 14 | 2 | no | — |
| 213 | C2×Q8 | (C2×D7)⋊C4 | 1 | 2 | yes | — |
| 214 | C2×D4 | — | 2 | 2 | no | — |
| 215 | C2×D4 | — | 2 | 2 | no | — |
| 216 | C2×D4 | — | 7 | 4 | no | — |
| 217 | C2×D4 | — | 14 | 4 | no | — |
| 218 | C2×D4 | — | 14 | 8 | no | — |
| 219 | C2×D4 | — | 14 | 4 | no | — |
| 220 | C2×D4 | — | 14 | 2 | no | — |
| 221 | C2×D4 | — | 14 | 2 | no | — |
| 222 | C2×D4 | — | 56 | 2 | no | — |
| 223 | C2×D4 | — | 56 | 2 | no | — |
| 224 | C2×D4 | — | 56 | 8 | no | — |
| 225 | C2×D4 | (C2×D7)⋊C4 | 1 | 2 | yes | — |
| 226 | C22×C4 | — | 2 | 2 | no | — |
| 227 | C22×C4 | — | 4 | 1 | no | — |
| 228 | C22×C4 | — | 14 | 4 | no | — |
| 229 | C22×C4 | D4×D7 | 1 | 2 | yes | — |
| 230 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 231 | M16 | — | 4 | 2 | no | — |
| 232 | C2×C8 | — | 4 | 1 | no | — |
| 233 | C2×C8 | — | 28 | 2 | no | — |
| 234 | C4⋊C4 | — | 2 | 2 | no | — |
| 235 | C4⋊C4 | — | 7 | 4 | no | — |
| 236 | C4⋊C4 | — | 14 | 2 | no | — |
| 237 | C22⋊C4 | — | 2 | 4 | no | — |
| 238 | C22⋊C4 | — | 14 | 8 | no | — |
| 239 | C22⋊C4 | — | 14 | 4 | no | — |
| 240 | C22⋊C4 | — | 14 | 4 | no | — |
| 241 | C22⋊C4 | — | 14 | 4 | no | — |
| 242 | C42 | — | 2 | 2 | no | — |
| 243 | C42 | — | 4 | 1 | no | — |
| 244 | C42 | — | 14 | 4 | no | — |
| 245 | C42 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 246 | C42 | C2×D28 | 1 | 1 | yes | characteristic |
| 247 | C42 | D4×D7 | 1 | 2 | yes | — |
| 248 | C2×C7 | — | 4 | 1 | no | — |
| 249 | C2×C7 | — | 8 | 2 | no | — |
| 250 | C2×C7 | #1792 | 1 | 1 | yes | characteristic |
| 251 | C2×C7 | #1795 | 1 | 2 | yes | — |
| 252 | D7 | — | 8 | 4 | no | — |
| 253 | C23 | — | 2 | 2 | no | — |
| 254 | C23 | — | 14 | 4 | no | — |
| 255 | C23 | — | 56 | 8 | no | — |
| 256 | C23 | — | 56 | 4 | no | — |
| 257 | C23 | #6500 | 1 | 1 | yes | characteristic |
| 258 | Q8 | — | 4 | 2 | no | — |
| 259 | Q8 | — | 56 | 2 | no | — |
| 260 | D4 | — | 4 | 2 | no | — |
| 261 | D4 | — | 4 | 4 | no | — |
| 262 | D4 | — | 8 | 2 | no | — |
| 263 | D4 | — | 28 | 4 | no | — |
| 264 | D4 | — | 28 | 2 | no | — |
| 265 | D4 | — | 28 | 2 | no | — |
| 266 | D4 | — | 56 | 2 | no | — |
| 267 | D4 | — | 56 | 4 | no | — |
| 268 | D4 | — | 56 | 4 | no | — |
| 269 | D4 | — | 56 | 8 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 2 | no | — |
| 273 | C2×C4 | — | 4 | 2 | no | — |
| 274 | C2×C4 | — | 4 | 2 | no | — |
| 275 | C2×C4 | — | 4 | 2 | no | — |
| 276 | C2×C4 | — | 4 | 2 | no | — |
| 277 | C2×C4 | — | 8 | 2 | no | — |
| 278 | C2×C4 | — | 14 | 4 | no | — |
| 279 | C2×C4 | — | 28 | 8 | no | — |
| 280 | C2×C4 | D7×C22⋊C4 | 1 | 2 | yes | — |
| 281 | C2×C4 | #6445 | 1 | 2 | yes | — |
| 282 | C2×C4 | C2×(C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #6500 | 1 | 1 | yes | characteristic |
| 284 | C8 | — | 8 | 1 | no | — |
| 285 | C8 | — | 56 | 2 | no | — |
| 286 | C7 | #8491 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 287 | C22 | — | 2 | 2 | no | — |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 4 | 4 | no | — |
| 290 | C22 | — | 4 | 2 | no | — |
| 291 | C22 | — | 28 | 4 | no | — |
| 292 | C22 | — | 28 | 4 | no | — |
| 293 | C22 | — | 28 | 4 | no | — |
| 294 | C22 | — | 112 | 8 | no | — |
| 295 | C22 | #89727 | 1 | 1 | yes | characteristic |
| 296 | C4 | — | 2 | 2 | no | — |
| 297 | C4 | — | 2 | 2 | no | — |
| 298 | C4 | — | 2 | 1 | no | — |
| 299 | C4 | — | 4 | 2 | no | — |
| 300 | C4 | — | 8 | 2 | no | — |
| 301 | C4 | — | 28 | 4 | no | — |
| 302 | C4 | #89833 | 1 | 1 | yes | characteristic |
| 303 | C4 | #89849 | 1 | 1 | yes | characteristic |
| 304 | C2 | — | 4 | 1 | no | — |
| 305 | C2 | — | 8 | 2 | no | — |
| 306 | C2 | — | 56 | 4 | no | — |
| 307 | C2 | #11727416 | 1 | 1 | yes | characteristic |
| 308 | C2 | #11728168 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1963 | C2 | semidirect |
| 3. #11729553 | C2 | semidirect |
| 4. #11728258 | C2 | semidirect |
| 5. #11727186 | C2 | semidirect |
| 8. #90425 | C4 | semidirect |
| 9. #90408 | C4 | semidirect |
| 15. C7×#423 | C22 | semidirect |
| 17. C7×#376 | C22 | semidirect |
| 18. #89698 | C22 | semidirect |
| 61. #1963 | D7 | semidirect |
| 115. #423 | C2×D7 | semidirect |
| 117. #376 | C2×D7 | semidirect |
| 286. C7 | #8491 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11743320 | 2 |
| C7×#1963 | 2 |
| #11729553 | 2 |
| #11728258 | 2 |
| #11727186 | 2 |
| #8491 | 7 |