Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
346 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 61 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 | #11737338 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11728673 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11727061 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11725813 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C7×#1335 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11725376 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11725247 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #90461 | — | 2 | 1 | no | — |
| 9 | #90442 | — | 2 | 1 | no | — |
| 10 | C7×#443 | C22 | 1 | 1 | yes | characteristic |
| 11 | #89913 | — | 2 | 1 | no | — |
| 12 | #89670 | — | 2 | 1 | no | — |
| 13 | #89598 | — | 2 | 1 | no | — |
| 14 | #89594 | — | 2 | 1 | no | — |
| 15 | #89472 | C22 | 1 | 1 | yes | characteristic |
| 16 | #89466 | — | 2 | 1 | no | — |
| 17 | #89465 | C22 | 1 | 1 | yes | characteristic |
| 18 | #89453 | — | 2 | 1 | no | — |
| 19 | #89452 | C22 | 1 | 1 | yes | characteristic, complemented |
| 20 | C7×#364 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | C7×C82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | #89361 | — | 2 | 1 | no | — |
| 23 | #89313 | — | 2 | 1 | no | — |
| 24 | #89308 | — | 2 | 1 | no | — |
| 25 | #89289 | — | 2 | 1 | no | — |
| 26 | #89258 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | #12365 | — | 7 | 1 | no | maximal, Sylow |
| 28 | #6554 | — | 2 | 1 | no | — |
| 29 | #6553 | — | 2 | 1 | no | — |
| 30 | C7×#135 | D4 | 1 | 1 | yes | characteristic |
| 31 | C7×#131 | — | 2 | 1 | no | — |
| 32 | #6514 | — | 2 | 1 | no | — |
| 33 | #6512 | — | 2 | 1 | no | — |
| 34 | #6481 | — | 4 | 1 | no | — |
| 35 | #6479 | — | 4 | 1 | no | — |
| 36 | D7×SD16 | — | 4 | 1 | no | — |
| 37 | #6474 | — | 4 | 1 | no | — |
| 38 | #6472 | — | 2 | 1 | no | — |
| 39 | #6472 | — | 4 | 1 | no | — |
| 40 | #6471 | — | 2 | 1 | no | — |
| 41 | #6471 | — | 4 | 1 | no | — |
| 42 | #6467 | — | 4 | 1 | no | — |
| 43 | #6467 | D4 | 1 | 1 | yes | characteristic |
| 44 | C2×#1132 | — | 2 | 1 | no | — |
| 45 | C2×#1132 | — | 2 | 1 | no | — |
| 46 | C2×#1132 | — | 4 | 1 | no | — |
| 47 | #6464 | — | 2 | 1 | no | — |
| 48 | #6437 | D4 | 1 | 1 | yes | characteristic |
| 49 | #6432 | — | 2 | 1 | no | — |
| 50 | #6432 | D4 | 1 | 1 | yes | characteristic |
| 51 | C7×M32 | — | 2 | 1 | no | — |
| 52 | C7×#108 | D4 | 1 | 1 | yes | characteristic |
| 53 | C7×#104 | — | 2 | 1 | no | — |
| 54 | C4×C7×C8 | — | 2 | 1 | no | — |
| 55 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 56 | #6398 | — | 2 | 1 | no | — |
| 57 | #6397 | — | 2 | 1 | no | — |
| 58 | #6395 | — | 4 | 2 | no | — |
| 59 | #6393 | D4 | 1 | 1 | yes | characteristic |
| 60 | #6391 | — | 2 | 1 | no | — |
| 61 | #6391 | — | 2 | 1 | no | — |
| 62 | #6390 | — | 4 | 2 | no | — |
| 63 | #6386 | — | 2 | 1 | no | — |
| 64 | #6379 | — | 2 | 1 | no | — |
| 65 | #3291 | — | 7 | 1 | no | — |
| 66 | #2230 | — | 7 | 1 | no | — |
| 67 | #2230 | — | 7 | 1 | no | — |
| 68 | #1708 | — | 7 | 1 | no | — |
| 69 | #1359 | — | 7 | 1 | no | — |
| 70 | #1335 | — | 7 | 1 | no | — |
| 71 | #1335 | D7 | 1 | 1 | yes | characteristic, complemented |
| 72 | C7×#41 | — | 2 | 1 | no | — |
| 73 | #1161 | — | 4 | 1 | no | — |
| 74 | Q8×D7 | — | 4 | 1 | no | — |
| 75 | #1159 | — | 4 | 1 | no | — |
| 76 | #1159 | — | 8 | 1 | no | — |
| 77 | D4×D7 | — | 4 | 1 | no | — |
| 78 | D4×D7 | — | 8 | 1 | no | — |
| 79 | #1157 | — | 2 | 1 | no | — |
| 80 | #1157 | — | 4 | 1 | no | — |
| 81 | #1157 | — | 4 | 1 | no | — |
| 82 | C2×D28 | — | 2 | 1 | no | — |
| 83 | C2×D28 | — | 4 | 1 | no | — |
| 84 | C2×D28 | — | 4 | 1 | no | — |
| 85 | C2×Dic14 | — | 2 | 1 | no | — |
| 86 | C2×Dic14 | — | 4 | 1 | no | — |
| 87 | C2×Dic14 | — | 4 | 1 | no | — |
| 88 | C7×Q16 | SD16 | 1 | 1 | yes | characteristic |
| 89 | C7×SD16 | — | 2 | 1 | no | — |
| 90 | C7×D8 | SD16 | 1 | 1 | yes | characteristic |
| 91 | C7×M16 | — | 2 | 1 | no | — |
| 92 | C7×M16 | — | 4 | 1 | no | — |
| 93 | C2×C7×C8 | — | 2 | 2 | no | — |
| 94 | C2×C7×C8 | — | 4 | 1 | no | — |
| 95 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | #1144 | — | 4 | 1 | no | — |
| 99 | #1143 | — | 4 | 1 | no | — |
| 100 | #1142 | — | 4 | 1 | no | — |
| 101 | #1141 | — | 4 | 1 | no | — |
| 102 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 103 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 104 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 105 | #1136 | — | 2 | 1 | no | — |
| 106 | Dic28 | — | 4 | 1 | no | — |
| 107 | Dic28 | SD16 | 1 | 1 | yes | characteristic |
| 108 | D56 | — | 4 | 1 | no | — |
| 109 | D56 | SD16 | 1 | 1 | yes | characteristic |
| 110 | #1132 | — | 2 | 1 | no | — |
| 111 | #1132 | — | 4 | 1 | no | — |
| 112 | #1132 | — | 4 | 1 | no | — |
| 113 | #1132 | — | 4 | 1 | no | — |
| 114 | #1132 | — | 4 | 1 | no | — |
| 115 | #1132 | — | 4 | 1 | no | — |
| 116 | #1132 | — | 4 | 1 | no | — |
| 117 | #1131 | — | 4 | 1 | no | — |
| 118 | C8×D7 | — | 4 | 1 | no | — |
| 119 | C7×C16 | — | 4 | 1 | no | — |
| 120 | C7⋊C16 | — | 4 | 1 | no | — |
| 121 | #577 | — | 14 | 1 | no | — |
| 122 | #577 | — | 14 | 1 | no | — |
| 123 | #498 | — | 7 | 1 | no | — |
| 124 | #492 | — | 7 | 1 | no | — |
| 125 | #487 | — | 14 | 1 | no | — |
| 126 | #486 | — | 14 | 1 | no | — |
| 127 | #471 | — | 14 | 1 | no | — |
| 128 | #471 | — | 14 | 1 | no | — |
| 129 | #456 | — | 14 | 1 | no | — |
| 130 | #453 | — | 14 | 1 | no | — |
| 131 | #443 | — | 7 | 1 | no | — |
| 132 | #443 | C2×D7 | 1 | 1 | yes | characteristic |
| 133 | #365 | — | 14 | 1 | no | — |
| 134 | #365 | — | 14 | 1 | no | — |
| 135 | #364 | — | 7 | 1 | no | — |
| 136 | #364 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 137 | #360 | — | 14 | 1 | no | — |
| 138 | #360 | — | 14 | 1 | no | — |
| 139 | C82 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 140 | C22×D7 | — | 4 | 1 | no | — |
| 141 | C22×D7 | — | 8 | 1 | no | — |
| 142 | C7×Q8 | — | 2 | 1 | no | — |
| 143 | C7×D4 | — | 2 | 1 | no | — |
| 144 | C7×D4 | — | 4 | 1 | no | — |
| 145 | C2×C4×C7 | — | 2 | 1 | no | — |
| 146 | C2×C4×C7 | — | 4 | 1 | no | — |
| 147 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 148 | C22⋊D7 | — | 4 | 1 | no | — |
| 149 | C22⋊D7 | — | 8 | 1 | no | — |
| 150 | C22⋊D7 | — | 8 | 1 | no | — |
| 151 | C2×Dic7 | — | 4 | 1 | no | — |
| 152 | C2×Dic7 | — | 8 | 1 | no | — |
| 153 | C2×Dic7 | — | 8 | 1 | no | — |
| 154 | D28 | — | 2 | 1 | no | — |
| 155 | D28 | — | 4 | 1 | no | — |
| 156 | D28 | — | 4 | 1 | no | — |
| 157 | D28 | — | 8 | 2 | no | — |
| 158 | D28 | — | 8 | 1 | no | — |
| 159 | C4×D7 | — | 4 | 1 | no | — |
| 160 | C4×D7 | — | 8 | 1 | no | — |
| 161 | C4×D7 | — | 8 | 1 | no | — |
| 162 | Dic14 | — | 2 | 1 | no | — |
| 163 | Dic14 | — | 4 | 1 | no | — |
| 164 | Dic14 | — | 4 | 1 | no | — |
| 165 | Dic14 | — | 8 | 1 | no | — |
| 166 | Dic14 | — | 8 | 2 | no | — |
| 167 | C7×C8 | — | 4 | 2 | no | — |
| 168 | C7×C8 | — | 4 | 1 | no | — |
| 169 | C7×C8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 170 | C7×C8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 171 | C7×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 172 | C7×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 173 | C7⋊C8 | — | 4 | 1 | no | — |
| 174 | 2−1+4 | — | 14 | 1 | no | — |
| 175 | H(5,𝔽2) | — | 14 | 1 | no | — |
| 176 | #137 | — | 14 | 1 | no | — |
| 177 | #137 | — | 14 | 1 | no | — |
| 178 | #137 | — | 28 | 1 | no | — |
| 179 | #137 | — | 28 | 1 | no | — |
| 180 | AGL(1,ℤ/8) | — | 14 | 1 | no | — |
| 181 | AGL(1,ℤ/8) | — | 14 | 1 | no | — |
| 182 | AGL(1,ℤ/8) | — | 28 | 1 | no | — |
| 183 | AGL(1,ℤ/8) | — | 28 | 1 | no | — |
| 184 | #135 | — | 7 | 1 | no | — |
| 185 | #135 | — | 28 | 1 | no | — |
| 186 | #135 | — | 28 | 1 | no | — |
| 187 | #135 | D28 | 1 | 1 | yes | characteristic |
| 188 | C2×SD16 | — | 14 | 1 | no | — |
| 189 | C2×SD16 | — | 14 | 1 | no | — |
| 190 | C2×SD16 | — | 28 | 1 | no | — |
| 191 | C2×SD16 | — | 28 | 1 | no | — |
| 192 | #131 | — | 2 | 1 | no | — |
| 193 | #131 | — | 14 | 1 | no | — |
| 194 | C4⋊Q8 | — | 7 | 1 | no | — |
| 195 | C4⋊Q8 | — | 14 | 1 | no | — |
| 196 | #127 | — | 7 | 1 | no | — |
| 197 | M32 | — | 2 | 1 | no | — |
| 198 | M32 | — | 14 | 1 | no | — |
| 199 | #108 | — | 7 | 1 | no | — |
| 200 | #108 | D28 | 1 | 1 | yes | characteristic |
| 201 | #106 | — | 14 | 1 | no | — |
| 202 | #106 | — | 14 | 1 | no | — |
| 203 | #104 | — | 2 | 1 | no | — |
| 204 | #104 | — | 14 | 1 | no | — |
| 205 | #103 | — | 28 | 2 | no | — |
| 206 | #102 | — | 28 | 2 | no | — |
| 207 | #101 | — | 14 | 1 | no | — |
| 208 | #100 | — | 14 | 1 | no | — |
| 209 | C4×C8 | — | 2 | 1 | no | — |
| 210 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 211 | C22×C7 | — | 4 | 1 | no | — |
| 212 | C22×C7 | #453 | 1 | 1 | yes | characteristic |
| 213 | C2×D7 | — | 4 | 1 | no | — |
| 214 | C2×D7 | — | 8 | 1 | no | — |
| 215 | C2×D7 | — | 16 | 1 | no | — |
| 216 | C2×D7 | — | 32 | 1 | no | — |
| 217 | C4×C7 | — | 2 | 2 | no | — |
| 218 | C4×C7 | — | 4 | 1 | no | — |
| 219 | C4×C7 | #450 | 1 | 1 | yes | characteristic |
| 220 | C4×C7 | #450 | 1 | 1 | yes | characteristic |
| 221 | Dic7 | — | 4 | 1 | no | — |
| 222 | Dic7 | — | 8 | 1 | no | — |
| 223 | Dic7 | — | 16 | 1 | no | — |
| 224 | #41 | — | 2 | 1 | no | — |
| 225 | #41 | — | 14 | 1 | no | — |
| 226 | #41 | — | 28 | 1 | no | — |
| 227 | #41 | — | 28 | 1 | no | — |
| 228 | #41 | — | 28 | 1 | no | — |
| 229 | #41 | — | 28 | 1 | no | — |
| 230 | #41 | — | 56 | 1 | no | — |
| 231 | C2×Q8 | — | 14 | 1 | no | — |
| 232 | C2×Q8 | — | 28 | 1 | no | — |
| 233 | C2×Q8 | — | 28 | 1 | no | — |
| 234 | C2×Q8 | — | 28 | 1 | no | — |
| 235 | C2×D4 | — | 14 | 1 | no | — |
| 236 | C2×D4 | — | 28 | 1 | no | — |
| 237 | C2×D4 | — | 28 | 1 | no | — |
| 238 | C2×D4 | — | 28 | 1 | no | — |
| 239 | C2×D4 | — | 56 | 1 | no | — |
| 240 | Q16 | — | 7 | 1 | no | — |
| 241 | Q16 | — | 28 | 1 | no | — |
| 242 | Q16 | — | 28 | 1 | no | — |
| 243 | Q16 | #1132 | 1 | 1 | yes | characteristic |
| 244 | SD16 | — | 2 | 1 | no | — |
| 245 | SD16 | — | 14 | 1 | no | — |
| 246 | SD16 | — | 28 | 1 | no | — |
| 247 | SD16 | — | 28 | 1 | no | — |
| 248 | SD16 | — | 28 | 1 | no | — |
| 249 | SD16 | — | 28 | 1 | no | — |
| 250 | SD16 | — | 28 | 1 | no | — |
| 251 | SD16 | — | 28 | 1 | no | — |
| 252 | SD16 | — | 28 | 1 | no | — |
| 253 | SD16 | — | 28 | 1 | no | — |
| 254 | D8 | — | 7 | 1 | no | — |
| 255 | D8 | — | 28 | 1 | no | — |
| 256 | D8 | — | 28 | 1 | no | — |
| 257 | D8 | #1132 | 1 | 1 | yes | characteristic |
| 258 | M16 | — | 2 | 1 | no | — |
| 259 | M16 | — | 4 | 1 | no | — |
| 260 | M16 | — | 14 | 1 | no | — |
| 261 | M16 | — | 28 | 1 | no | — |
| 262 | C2×C8 | — | 2 | 2 | no | — |
| 263 | C2×C8 | — | 4 | 1 | no | — |
| 264 | C2×C8 | — | 28 | 1 | no | — |
| 265 | C2×C8 | C2×D28 | 1 | 1 | yes | characteristic |
| 266 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 267 | C4⋊C4 | — | 28 | 1 | no | — |
| 268 | C4⋊C4 | — | 28 | 1 | no | — |
| 269 | C4⋊C4 | — | 28 | 1 | no | — |
| 270 | C42 | D4×D7 | 1 | 1 | yes | characteristic |
| 271 | C16 | — | 4 | 1 | no | — |
| 272 | C16 | — | 28 | 1 | no | — |
| 273 | C2×C7 | — | 2 | 1 | no | — |
| 274 | C2×C7 | — | 8 | 1 | no | — |
| 275 | C2×C7 | #1629 | 1 | 1 | yes | characteristic |
| 276 | D7 | — | 8 | 1 | no | — |
| 277 | D7 | — | 16 | 1 | no | — |
| 278 | C23 | — | 28 | 1 | no | — |
| 279 | C23 | — | 56 | 1 | no | — |
| 280 | Q8 | — | 2 | 1 | no | — |
| 281 | Q8 | — | 14 | 1 | no | — |
| 282 | Q8 | — | 28 | 1 | no | — |
| 283 | Q8 | — | 28 | 1 | no | — |
| 284 | Q8 | — | 56 | 2 | no | — |
| 285 | Q8 | — | 56 | 1 | no | — |
| 286 | D4 | — | 2 | 1 | no | — |
| 287 | D4 | — | 4 | 1 | no | — |
| 288 | D4 | — | 14 | 1 | no | — |
| 289 | D4 | — | 28 | 1 | no | — |
| 290 | D4 | — | 28 | 1 | no | — |
| 291 | D4 | — | 28 | 1 | no | — |
| 292 | D4 | — | 56 | 2 | no | — |
| 293 | D4 | — | 56 | 1 | no | — |
| 294 | D4 | — | 56 | 1 | no | — |
| 295 | D4 | — | 56 | 1 | no | — |
| 296 | C2×C4 | — | 2 | 1 | no | — |
| 297 | C2×C4 | — | 4 | 1 | no | — |
| 298 | C2×C4 | — | 28 | 1 | no | — |
| 299 | C2×C4 | — | 28 | 1 | no | — |
| 300 | C2×C4 | — | 56 | 1 | no | — |
| 301 | C2×C4 | — | 56 | 1 | no | — |
| 302 | C2×C4 | — | 56 | 1 | no | — |
| 303 | C2×C4 | — | 56 | 1 | no | — |
| 304 | C2×C4 | #6445 | 1 | 1 | yes | characteristic |
| 305 | C8 | — | 4 | 2 | no | — |
| 306 | C8 | — | 4 | 1 | no | — |
| 307 | C8 | — | 28 | 1 | no | — |
| 308 | C8 | C2×#1132 | 1 | 1 | yes | characteristic |
| 309 | C8 | #6471 | 1 | 1 | yes | characteristic |
| 310 | C8 | #6474 | 1 | 1 | yes | characteristic |
| 311 | C8 | D7×SD16 | 1 | 1 | yes | characteristic |
| 312 | C7 | #12365 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 313 | C22 | — | 4 | 1 | no | — |
| 314 | C22 | — | 28 | 1 | no | — |
| 315 | C22 | — | 56 | 1 | no | — |
| 316 | C22 | — | 112 | 1 | no | — |
| 317 | C22 | — | 224 | 1 | no | — |
| 318 | C22 | #89594 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 2 | 2 | no | — |
| 320 | C4 | — | 4 | 1 | no | — |
| 321 | C4 | — | 28 | 1 | no | — |
| 322 | C4 | — | 56 | 1 | no | — |
| 323 | C4 | — | 112 | 1 | no | — |
| 324 | C4 | #89503 | 1 | 1 | yes | characteristic |
| 325 | C4 | #89548 | 1 | 1 | yes | characteristic |
| 326 | C2 | — | 2 | 1 | no | — |
| 327 | C2 | — | 8 | 1 | no | — |
| 328 | C2 | — | 56 | 1 | no | — |
| 329 | C2 | — | 112 | 1 | no | — |
| 330 | C2 | #11725949 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11728673 | C2 | semidirect |
| 3. #11727061 | C2 | semidirect |
| 4. #11725813 | C2 | semidirect |
| 5. C7×#1335 | C2 | semidirect |
| 6. #11725376 | C2 | semidirect |
| 7. #11725247 | C2 | semidirect |
| 19. #89452 | C22 | semidirect |
| 20. C7×#364 | C22 | semidirect |
| 21. C7×C82 | C22 | semidirect |
| 26. #89258 | C22 | semidirect |
| 71. #1335 | D7 | semidirect |
| 136. #364 | C2×D7 | semidirect |
| 139. C82 | C2×D7 | semidirect |
| 312. C7 | #12365 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11737338 | 2 |
| #11728673 | 2 |
| #11727061 | 2 |
| #11725813 | 2 |
| C7×#1335 | 2 |
| #11725376 | 2 |
| #11725247 | 2 |
| #12365 | 7 |