Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
960 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 205 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 | D7×#430 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#89523 | C2 | 1 | 3 | yes | maximal |
| 3 | C7×#2070 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11728484 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #11727865 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #11726476 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | D7×#338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×#1161 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×D7×M16 | C22 | 1 | 3 | yes | — |
| 10 | C2×D7×M16 | C22 | 1 | 3 | yes | — |
| 11 | C7×#430 | C22 | 1 | 3 | yes | — |
| 12 | C2×C7×#101 | C22 | 1 | 3 | yes | — |
| 13 | C2×#6410 | C22 | 1 | 3 | yes | — |
| 14 | C2×#6397 | C22 | 1 | 3 | yes | — |
| 15 | C2×#6397 | C22 | 1 | 3 | yes | — |
| 16 | #89897 | C22 | 1 | 3 | yes | — |
| 17 | #89896 | C22 | 1 | 3 | yes | — |
| 18 | #89896 | C22 | 1 | 3 | yes | — |
| 19 | #89665 | — | 2 | 6 | no | — |
| 20 | D7×#108 | — | 2 | 6 | no | — |
| 21 | #89523 | — | 2 | 6 | no | — |
| 22 | #89523 | — | 2 | 6 | no | — |
| 23 | C7×#338 | C22 | 1 | 1 | yes | characteristic |
| 24 | C7⋊#338 | C22 | 1 | 3 | yes | — |
| 25 | #23622 | — | 7 | 1 | no | maximal, Sylow |
| 26 | C22×C7×Q8 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×#1161 | — | 2 | 3 | no | — |
| 28 | C2×#1161 | — | 2 | 3 | no | — |
| 29 | C2×#1161 | — | 2 | 3 | no | — |
| 30 | C2×#1161 | — | 2 | 3 | no | — |
| 31 | C22×D28 | Q8 | 1 | 2 | yes | — |
| 32 | C22×D28 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×C4×D7 | Q8 | 1 | 2 | yes | — |
| 34 | C22×C4×D7 | C23 | 1 | 1 | yes | characteristic |
| 35 | C2×C7×M16 | C23 | 1 | 3 | yes | — |
| 36 | C2×C7×M16 | C23 | 1 | 3 | yes | — |
| 37 | C2×#1136 | C23 | 1 | 3 | yes | — |
| 38 | C2×#1136 | C23 | 1 | 3 | yes | — |
| 39 | D7×M16 | — | 2 | 3 | no | — |
| 40 | D7×M16 | — | 2 | 3 | no | — |
| 41 | D7×M16 | — | 2 | 3 | no | — |
| 42 | D7×M16 | — | 2 | 3 | no | — |
| 43 | D7×M16 | — | 2 | 3 | no | — |
| 44 | D7×M16 | — | 2 | 3 | no | — |
| 45 | D7×M16 | D4 | 1 | 6 | yes | — |
| 46 | D7×M16 | D4 | 1 | 6 | yes | — |
| 47 | C2×#1131 | — | 2 | 3 | no | — |
| 48 | C2×#1131 | — | 2 | 3 | no | — |
| 49 | C2×C8×D7 | — | 2 | 3 | no | — |
| 50 | C2×C8×D7 | — | 2 | 3 | no | — |
| 51 | C7×#108 | — | 2 | 6 | no | — |
| 52 | C7×#101 | — | 2 | 6 | no | — |
| 53 | #6410 | — | 2 | 6 | no | — |
| 54 | #6397 | — | 2 | 6 | no | — |
| 55 | #6397 | — | 2 | 6 | no | — |
| 56 | #6396 | — | 2 | 6 | no | — |
| 57 | #6396 | — | 2 | 6 | no | — |
| 58 | #6393 | — | 2 | 6 | no | — |
| 59 | C2×#430 | — | 7 | 3 | no | — |
| 60 | C2×#413 | — | 7 | 3 | no | — |
| 61 | #2070 | — | 7 | 3 | no | — |
| 62 | #2070 | D7 | 1 | 1 | yes | characteristic, complemented |
| 63 | #2069 | — | 7 | 3 | no | — |
| 64 | #2069 | — | 7 | 1 | no | — |
| 65 | C2×#338 | — | 7 | 1 | no | — |
| 66 | C23×D7 | C2×Q8 | 1 | 2 | yes | — |
| 67 | C23×D7 | C2×Q8 | 1 | 1 | yes | characteristic |
| 68 | C2×C7×Q8 | — | 2 | 3 | no | — |
| 69 | C2×C7×Q8 | — | 2 | 3 | no | — |
| 70 | C22×C4×C7 | C2×Q8 | 1 | 2 | yes | — |
| 71 | C22×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 72 | C22×Dic7 | C2×Q8 | 1 | 1 | yes | characteristic |
| 73 | #1161 | — | 4 | 3 | no | — |
| 74 | #1161 | — | 4 | 3 | no | — |
| 75 | #1161 | — | 4 | 3 | no | — |
| 76 | #1161 | — | 4 | 3 | no | — |
| 77 | #1161 | — | 4 | 1 | no | — |
| 78 | #1161 | — | 4 | 3 | no | — |
| 79 | C2×D28 | — | 2 | 6 | no | — |
| 80 | C2×D28 | — | 2 | 6 | no | — |
| 81 | C2×D28 | — | 2 | 3 | no | — |
| 82 | C2×D28 | — | 2 | 3 | no | — |
| 83 | C2×C4×D7 | — | 2 | 6 | no | — |
| 84 | C2×C4×D7 | — | 2 | 6 | no | — |
| 85 | C2×C4×D7 | C2×D4 | 1 | 3 | yes | — |
| 86 | C2×C4×D7 | C2×D4 | 1 | 3 | yes | — |
| 87 | C2×C4×D7 | #41 | 1 | 3 | yes | — |
| 88 | C2×C4×D7 | #41 | 1 | 3 | yes | — |
| 89 | C7×M16 | — | 2 | 3 | no | — |
| 90 | C7×M16 | — | 2 | 3 | no | — |
| 91 | C7×M16 | — | 2 | 3 | no | — |
| 92 | C7×M16 | C2×D4 | 1 | 6 | yes | — |
| 93 | C2×C7×C8 | — | 2 | 3 | no | — |
| 94 | C2×C7×C8 | — | 2 | 3 | no | — |
| 95 | #1136 | — | 2 | 3 | no | — |
| 96 | #1136 | — | 2 | 3 | no | — |
| 97 | #1136 | — | 2 | 3 | no | — |
| 98 | #1136 | C2×D4 | 1 | 6 | yes | — |
| 99 | C2×C7⋊C8 | — | 2 | 3 | no | — |
| 100 | C2×C7⋊C8 | — | 2 | 3 | no | — |
| 101 | #1131 | — | 2 | 6 | no | — |
| 102 | #1131 | — | 2 | 6 | no | — |
| 103 | #1131 | — | 4 | 3 | no | — |
| 104 | #1131 | — | 4 | 3 | no | — |
| 105 | C8×D7 | — | 2 | 6 | no | — |
| 106 | C8×D7 | — | 2 | 6 | no | — |
| 107 | C8×D7 | — | 4 | 3 | no | — |
| 108 | C8×D7 | — | 4 | 3 | no | — |
| 109 | C22×#41 | — | 7 | 1 | no | — |
| 110 | C22×M16 | — | 7 | 3 | no | — |
| 111 | C22×M16 | — | 7 | 3 | no | — |
| 112 | #430 | — | 7 | 3 | no | — |
| 113 | #430 | — | 7 | 3 | no | — |
| 114 | #430 | — | 7 | 3 | no | — |
| 115 | #430 | — | 14 | 6 | no | — |
| 116 | #430 | C2×D7 | 1 | 3 | yes | — |
| 117 | C2×#108 | — | 14 | 6 | no | — |
| 118 | #413 | — | 14 | 6 | no | — |
| 119 | #413 | — | 14 | 6 | no | — |
| 120 | C2×#101 | — | 7 | 3 | no | — |
| 121 | C2×#101 | C2×D7 | 1 | 3 | yes | — |
| 122 | C2×#100 | — | 7 | 3 | no | — |
| 123 | C2×#100 | — | 7 | 3 | no | — |
| 124 | #338 | — | 7 | 3 | no | — |
| 125 | #338 | C2×D7 | 1 | 1 | yes | characteristic |
| 126 | C23×C7 | C22×Q8 | 1 | 1 | yes | characteristic |
| 127 | C22×D7 | — | 2 | 6 | no | — |
| 128 | C22×D7 | — | 8 | 1 | no | — |
| 129 | C22×D7 | — | 8 | 2 | no | — |
| 130 | C22×D7 | C22⋊Q8 | 1 | 3 | yes | — |
| 131 | C22×D7 | C22⋊Q8 | 1 | 3 | yes | — |
| 132 | C7×Q8 | — | 4 | 3 | no | — |
| 133 | C7×Q8 | — | 4 | 1 | no | — |
| 134 | C2×C4×C7 | — | 2 | 6 | no | — |
| 135 | C2×C4×C7 | C22×D4 | 1 | 3 | yes | — |
| 136 | C2×C4×C7 | C2×#41 | 1 | 3 | yes | — |
| 137 | C2×Dic7 | C22⋊Q8 | 1 | 3 | yes | — |
| 138 | C2×Dic7 | C22⋊Q8 | 1 | 3 | yes | — |
| 139 | D28 | — | 4 | 6 | no | — |
| 140 | D28 | — | 4 | 2 | no | — |
| 141 | D28 | — | 4 | 3 | no | — |
| 142 | D28 | — | 4 | 1 | no | — |
| 143 | C4×D7 | — | 4 | 2 | no | — |
| 144 | C4×D7 | — | 4 | 6 | no | — |
| 145 | C4×D7 | #121 | 1 | 3 | yes | — |
| 146 | C4×D7 | #121 | 1 | 3 | yes | — |
| 147 | C4×D7 | #121 | 1 | 3 | yes | — |
| 148 | C4×D7 | #121 | 1 | 3 | yes | — |
| 149 | C4×D7 | #126 | 1 | 1 | yes | characteristic |
| 150 | C4×D7 | #126 | 1 | 3 | yes | — |
| 151 | C7×C8 | — | 2 | 6 | no | — |
| 152 | C7×C8 | — | 4 | 3 | no | — |
| 153 | C7⋊C8 | — | 2 | 6 | no | — |
| 154 | C7⋊C8 | — | 4 | 3 | no | — |
| 155 | C2×#41 | — | 14 | 3 | no | — |
| 156 | C2×#41 | — | 14 | 3 | no | — |
| 157 | C2×#41 | — | 14 | 3 | no | — |
| 158 | C2×#41 | — | 14 | 3 | no | — |
| 159 | C22×Q8 | C22×D7 | 1 | 1 | yes | characteristic |
| 160 | C22×D4 | — | 7 | 2 | no | — |
| 161 | C22×D4 | — | 7 | 1 | no | — |
| 162 | C23×C4 | — | 7 | 1 | no | — |
| 163 | C23×C4 | — | 7 | 2 | no | — |
| 164 | C2×M16 | — | 7 | 3 | no | — |
| 165 | C2×M16 | — | 7 | 6 | no | — |
| 166 | C2×M16 | — | 7 | 6 | no | — |
| 167 | C2×M16 | — | 7 | 3 | no | — |
| 168 | C2×M16 | — | 14 | 3 | no | — |
| 169 | C2×M16 | — | 14 | 3 | no | — |
| 170 | C2×M16 | — | 14 | 3 | no | — |
| 171 | C2×M16 | — | 14 | 3 | no | — |
| 172 | C2×M16 | — | 14 | 3 | no | — |
| 173 | C2×M16 | — | 14 | 3 | no | — |
| 174 | C2×M16 | — | 14 | 3 | no | — |
| 175 | C2×M16 | — | 14 | 3 | no | — |
| 176 | C2×M16 | C22×D7 | 1 | 3 | yes | — |
| 177 | C2×M16 | C22×D7 | 1 | 3 | yes | — |
| 178 | C22×C8 | — | 14 | 3 | no | — |
| 179 | C22×C8 | — | 14 | 3 | no | — |
| 180 | #108 | — | 2 | 6 | no | — |
| 181 | #108 | — | 14 | 6 | no | — |
| 182 | #108 | — | 14 | 6 | no | — |
| 183 | #108 | — | 14 | 6 | no | — |
| 184 | #101 | — | 2 | 6 | no | — |
| 185 | #101 | — | 14 | 6 | no | — |
| 186 | #100 | — | 14 | 6 | no | — |
| 187 | #100 | — | 14 | 6 | no | — |
| 188 | C22×C7 | — | 4 | 1 | no | — |
| 189 | C22×C7 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 190 | C2×D7 | — | 4 | 3 | no | — |
| 191 | C2×D7 | — | 4 | 3 | no | — |
| 192 | C2×D7 | — | 4 | 2 | no | — |
| 193 | C2×D7 | — | 8 | 6 | no | — |
| 194 | C2×D7 | #396 | 1 | 3 | yes | — |
| 195 | C2×D7 | #396 | 1 | 1 | yes | characteristic |
| 196 | C4×C7 | — | 4 | 2 | no | — |
| 197 | C4×C7 | C2×#121 | 1 | 3 | yes | — |
| 198 | C4×C7 | C2×#126 | 1 | 1 | yes | characteristic |
| 199 | Dic7 | #396 | 1 | 1 | yes | characteristic |
| 200 | Dic7 | #396 | 1 | 3 | yes | — |
| 201 | C24 | — | 7 | 1 | no | — |
| 202 | C24 | — | 7 | 2 | no | — |
| 203 | #41 | — | 28 | 1 | no | — |
| 204 | #41 | — | 28 | 3 | no | — |
| 205 | #41 | — | 28 | 3 | no | — |
| 206 | #41 | — | 28 | 3 | no | — |
| 207 | #41 | — | 28 | 3 | no | — |
| 208 | #41 | — | 28 | 3 | no | — |
| 209 | C2×Q8 | — | 2 | 3 | no | — |
| 210 | C2×Q8 | — | 2 | 3 | no | — |
| 211 | C2×D4 | — | 14 | 6 | no | — |
| 212 | C2×D4 | — | 14 | 6 | no | — |
| 213 | C2×D4 | — | 14 | 3 | no | — |
| 214 | C2×D4 | — | 14 | 3 | no | — |
| 215 | C22×C4 | — | 7 | 1 | no | — |
| 216 | C22×C4 | — | 7 | 3 | no | — |
| 217 | C22×C4 | — | 7 | 3 | no | — |
| 218 | C22×C4 | — | 7 | 3 | no | — |
| 219 | C22×C4 | — | 7 | 3 | no | — |
| 220 | C22×C4 | — | 14 | 6 | no | — |
| 221 | C22×C4 | — | 14 | 6 | no | — |
| 222 | C22×C4 | Q8×D7 | 1 | 2 | yes | — |
| 223 | C22×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 224 | M16 | — | 2 | 3 | no | — |
| 225 | M16 | — | 2 | 3 | no | — |
| 226 | M16 | — | 2 | 3 | no | — |
| 227 | M16 | — | 7 | 6 | no | — |
| 228 | M16 | — | 14 | 6 | no | — |
| 229 | M16 | — | 14 | 6 | no | — |
| 230 | M16 | — | 14 | 3 | no | — |
| 231 | M16 | — | 14 | 3 | no | — |
| 232 | M16 | — | 14 | 3 | no | — |
| 233 | M16 | — | 28 | 3 | no | — |
| 234 | M16 | — | 28 | 3 | no | — |
| 235 | M16 | D4×D7 | 1 | 6 | yes | — |
| 236 | C2×C8 | — | 2 | 3 | no | — |
| 237 | C2×C8 | — | 2 | 3 | no | — |
| 238 | C2×C8 | — | 14 | 6 | no | — |
| 239 | C2×C8 | — | 14 | 6 | no | — |
| 240 | C2×C8 | — | 14 | 3 | no | — |
| 241 | C2×C8 | — | 14 | 3 | no | — |
| 242 | C2×C8 | — | 28 | 3 | no | — |
| 243 | C2×C8 | — | 28 | 3 | no | — |
| 244 | C2×C7 | — | 2 | 3 | no | — |
| 245 | C2×C7 | C2×#396 | 1 | 1 | yes | characteristic |
| 246 | D7 | — | 2 | 3 | no | — |
| 247 | D7 | — | 2 | 1 | no | — |
| 248 | D7 | — | 8 | 2 | no | — |
| 249 | C23 | — | 7 | 3 | no | — |
| 250 | C23 | — | 7 | 3 | no | — |
| 251 | C23 | — | 14 | 6 | no | — |
| 252 | C23 | — | 56 | 1 | no | — |
| 253 | C23 | — | 56 | 2 | no | — |
| 254 | C23 | C2×Q8×D7 | 1 | 1 | yes | characteristic |
| 255 | Q8 | — | 4 | 1 | no | — |
| 256 | Q8 | — | 4 | 3 | no | — |
| 257 | D4 | — | 28 | 6 | no | — |
| 258 | D4 | — | 28 | 2 | no | — |
| 259 | D4 | — | 28 | 1 | no | — |
| 260 | D4 | — | 28 | 3 | no | — |
| 261 | C2×C4 | — | 2 | 6 | no | — |
| 262 | C2×C4 | — | 7 | 3 | no | — |
| 263 | C2×C4 | — | 7 | 3 | no | — |
| 264 | C2×C4 | — | 7 | 1 | no | — |
| 265 | C2×C4 | — | 7 | 3 | no | — |
| 266 | C2×C4 | — | 7 | 3 | no | — |
| 267 | C2×C4 | — | 7 | 3 | no | — |
| 268 | C2×C4 | — | 7 | 3 | no | — |
| 269 | C2×C4 | — | 7 | 3 | no | — |
| 270 | C2×C4 | — | 28 | 2 | no | — |
| 271 | C2×C4 | — | 28 | 6 | no | — |
| 272 | C2×C4 | C2×D4×D7 | 1 | 3 | yes | — |
| 273 | C2×C4 | D7×#41 | 1 | 3 | yes | — |
| 274 | C8 | — | 2 | 6 | no | — |
| 275 | C8 | — | 4 | 3 | no | — |
| 276 | C8 | — | 14 | 6 | no | — |
| 277 | C8 | — | 28 | 3 | no | — |
| 278 | C7 | #23622 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 279 | C22 | — | 4 | 1 | no | — |
| 280 | C22 | — | 7 | 3 | no | — |
| 281 | C22 | — | 7 | 1 | no | — |
| 282 | C22 | — | 28 | 3 | no | — |
| 283 | C22 | — | 28 | 3 | no | — |
| 284 | C22 | — | 28 | 2 | no | — |
| 285 | C22 | — | 56 | 6 | no | — |
| 286 | C22 | D7×C22⋊Q8 | 1 | 3 | yes | — |
| 287 | C4 | — | 4 | 2 | no | — |
| 288 | C4 | — | 7 | 1 | no | — |
| 289 | C4 | — | 7 | 3 | no | — |
| 290 | C4 | D7×#121 | 1 | 3 | yes | — |
| 291 | C4 | D7×#126 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 2 | 3 | no | — |
| 293 | C2 | — | 14 | 3 | no | — |
| 294 | C2 | — | 14 | 1 | no | — |
| 295 | C2 | — | 56 | 2 | no | — |
| 296 | C2 | D7×#396 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D7×#430 | C2 | semidirect |
| 3. C7×#2070 | C2 | semidirect |
| 4. #11728484 | C2 | semidirect |
| 5. #11727865 | C2 | semidirect |
| 6. #11726476 | C2 | semidirect |
| 7. D7×#338 | C2 | semidirect |
| 62. #2070 | D7 | semidirect |
| 278. C7 | #23622 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7×#430 | 2 |
| C2×#89523 | 2 |
| C7×#2070 | 2 |
| #11728484 | 2 |
| #11727865 | 2 |
| #11726476 | 2 |
| D7×#338 | 2 |
| #23622 | 7 |