Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1916 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1916 conjugacy classes of subgroups — too many to draw. Only its 287 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 | C2×#90439 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #11741207 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C7×#338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11726481 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | C22×#1157 | C22 | 1 | 3 | yes | — |
| 6 | C22×C7×M16 | C22 | 1 | 3 | yes | — |
| 7 | C22×#1136 | C22 | 1 | 1 | yes | characteristic |
| 8 | #90439 | — | 2 | 12 | no | — |
| 9 | #90439 | — | 2 | 6 | no | — |
| 10 | #90439 | C22 | 1 | 12 | yes | complemented |
| 11 | C2×#6472 | — | 2 | 6 | no | — |
| 12 | C2×#6471 | — | 2 | 6 | no | — |
| 13 | C2×#6467 | — | 2 | 6 | no | — |
| 14 | #89901 | C22 | 1 | 12 | yes | complemented |
| 15 | #89898 | — | 2 | 6 | no | — |
| 16 | #89882 | — | 2 | 6 | no | — |
| 17 | C7×#338 | C22 | 1 | 4 | yes | complemented |
| 18 | #23434 | — | 7 | 1 | no | maximal, Sylow |
| 19 | C23×C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 20 | C22×C22⋊D7 | — | 2 | 3 | no | — |
| 21 | C2×#1157 | — | 2 | 3 | no | — |
| 22 | C2×#1157 | — | 2 | 3 | no | — |
| 23 | C2×#1157 | — | 2 | 6 | no | — |
| 24 | C2×#1157 | — | 2 | 6 | no | — |
| 25 | C2×#1157 | — | 2 | 6 | no | — |
| 26 | C2×#1157 | — | 2 | 6 | no | — |
| 27 | C2×#1157 | D4 | 1 | 6 | yes | — |
| 28 | C2×#1157 | C23 | 1 | 6 | yes | — |
| 29 | C22×D28 | D4 | 1 | 3 | yes | — |
| 30 | C22×C4×D7 | — | 2 | 3 | no | — |
| 31 | C22×Dic14 | D4 | 1 | 3 | yes | — |
| 32 | C2×C7×M16 | — | 2 | 6 | no | — |
| 33 | C2×C7×M16 | — | 2 | 6 | no | — |
| 34 | C2×C7×M16 | D4 | 1 | 6 | yes | — |
| 35 | C2×C7×M16 | C23 | 1 | 6 | yes | — |
| 36 | C22×C7×C8 | D4 | 1 | 6 | yes | — |
| 37 | C2×#1136 | — | 2 | 3 | no | — |
| 38 | C2×#1136 | — | 2 | 3 | no | — |
| 39 | C22×C7⋊C8 | C23 | 1 | 2 | yes | — |
| 40 | #6472 | — | 2 | 24 | no | — |
| 41 | #6472 | — | 2 | 24 | no | — |
| 42 | #6471 | — | 2 | 24 | no | — |
| 43 | #6471 | — | 2 | 24 | no | — |
| 44 | C2×Dic28 | — | 2 | 6 | no | — |
| 45 | #6467 | — | 2 | 24 | no | — |
| 46 | #6467 | — | 2 | 24 | no | — |
| 47 | C2×D56 | — | 2 | 6 | no | — |
| 48 | C2×#1132 | — | 2 | 12 | no | — |
| 49 | (C2×D7)⋊C8 | — | 2 | 24 | no | — |
| 50 | C2×#575 | — | 7 | 3 | no | — |
| 51 | #2879 | — | 7 | 3 | no | — |
| 52 | #2017 | — | 7 | 8 | no | — |
| 53 | C2×#338 | D7 | 1 | 1 | yes | characteristic, complemented |
| 54 | C24×C7 | #41 | 1 | 1 | yes | characteristic |
| 55 | C23×D7 | — | 2 | 3 | no | — |
| 56 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 57 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 58 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 59 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 60 | C22×C4×C7 | #41 | 1 | 1 | yes | characteristic |
| 61 | C22×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 62 | C2×C22⋊D7 | — | 2 | 6 | no | — |
| 63 | C2×C22⋊D7 | — | 2 | 6 | no | — |
| 64 | C2×C22⋊D7 | — | 4 | 6 | no | — |
| 65 | C2×C22⋊D7 | — | 4 | 6 | no | — |
| 66 | C22×Dic7 | — | 2 | 3 | no | — |
| 67 | #1157 | — | 2 | 12 | no | — |
| 68 | #1157 | — | 2 | 12 | no | — |
| 69 | #1157 | — | 2 | 12 | no | — |
| 70 | #1157 | — | 2 | 12 | no | — |
| 71 | #1157 | — | 4 | 3 | no | — |
| 72 | #1157 | — | 4 | 3 | no | — |
| 73 | #1157 | — | 4 | 6 | no | — |
| 74 | #1157 | — | 4 | 6 | no | — |
| 75 | #1157 | — | 4 | 3 | no | — |
| 76 | #1157 | — | 4 | 3 | no | — |
| 77 | C2×D28 | — | 2 | 6 | no | — |
| 78 | C2×D28 | — | 2 | 6 | no | — |
| 79 | C2×D28 | — | 2 | 3 | no | — |
| 80 | C2×D28 | C2×D4 | 1 | 6 | yes | — |
| 81 | C2×C4×D7 | — | 2 | 6 | no | — |
| 82 | C2×C4×D7 | — | 2 | 6 | no | — |
| 83 | C2×C4×D7 | — | 4 | 6 | no | — |
| 84 | C2×C4×D7 | — | 4 | 6 | no | — |
| 85 | C2×Dic14 | — | 2 | 6 | no | — |
| 86 | C2×Dic14 | — | 2 | 6 | no | — |
| 87 | C2×Dic14 | — | 2 | 3 | no | — |
| 88 | C2×Dic14 | C2×D4 | 1 | 6 | yes | — |
| 89 | C7×M16 | — | 2 | 12 | no | — |
| 90 | C7×M16 | — | 2 | 12 | no | — |
| 91 | C2×C7×C8 | — | 2 | 6 | no | — |
| 92 | C2×C7×C8 | — | 2 | 6 | no | — |
| 93 | C2×C7×C8 | C2×D4 | 1 | 12 | yes | — |
| 94 | #1136 | — | 4 | 3 | no | — |
| 95 | #1136 | — | 4 | 1 | no | — |
| 96 | C2×C7⋊C8 | — | 2 | 6 | no | — |
| 97 | Dic28 | — | 2 | 24 | no | — |
| 98 | D56 | — | 2 | 24 | no | — |
| 99 | #1132 | — | 2 | 48 | no | — |
| 100 | C22×#41 | — | 7 | 3 | no | — |
| 101 | #575 | — | 7 | 12 | no | — |
| 102 | #575 | — | 14 | 12 | no | — |
| 103 | #575 | — | 14 | 6 | no | — |
| 104 | C2×#137 | — | 14 | 6 | no | — |
| 105 | C2×AGL(1,ℤ/8) | — | 14 | 6 | no | — |
| 106 | C2×#135 | — | 14 | 6 | no | — |
| 107 | C22×M16 | — | 7 | 1 | no | — |
| 108 | C22×M16 | C2×D7 | 1 | 3 | yes | — |
| 109 | #408 | — | 7 | 12 | no | — |
| 110 | #408 | — | 14 | 6 | no | — |
| 111 | #407 | — | 14 | 6 | no | — |
| 112 | #338 | C2×D7 | 1 | 4 | yes | complemented |
| 113 | C23×C7 | — | 8 | 1 | no | — |
| 114 | C23×C7 | #121 | 1 | 3 | yes | — |
| 115 | C23×C7 | #121 | 1 | 3 | yes | — |
| 116 | C23×C7 | C2×#41 | 1 | 1 | yes | characteristic |
| 117 | C22×D7 | — | 2 | 6 | no | — |
| 118 | C22×D7 | — | 4 | 6 | no | — |
| 119 | C22×D7 | — | 8 | 6 | no | — |
| 120 | C2×C4×C7 | #120 | 1 | 3 | yes | — |
| 121 | C2×C4×C7 | #120 | 1 | 3 | yes | — |
| 122 | C2×C4×C7 | #120 | 1 | 6 | yes | — |
| 123 | C2×C4×C7 | #121 | 1 | 3 | yes | — |
| 124 | C2×C4×C7 | #121 | 1 | 3 | yes | — |
| 125 | C2×C4×C7 | #127 | 1 | 3 | yes | — |
| 126 | C2×C4×C7 | #127 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C7 | C22×D4 | 1 | 3 | yes | — |
| 128 | C2×C4×C7 | C22×D4 | 1 | 3 | yes | — |
| 129 | C22⋊D7 | — | 4 | 12 | no | — |
| 130 | C22⋊D7 | — | 4 | 6 | no | — |
| 131 | C22⋊D7 | — | 4 | 6 | no | — |
| 132 | C2×Dic7 | — | 2 | 6 | no | — |
| 133 | C2×Dic7 | — | 4 | 6 | no | — |
| 134 | D28 | — | 2 | 12 | no | — |
| 135 | D28 | — | 4 | 3 | no | — |
| 136 | D28 | — | 4 | 3 | no | — |
| 137 | C4×D7 | — | 4 | 6 | no | — |
| 138 | C4×D7 | — | 4 | 6 | no | — |
| 139 | C4×D7 | — | 4 | 12 | no | — |
| 140 | Dic14 | — | 2 | 12 | no | — |
| 141 | Dic14 | — | 4 | 3 | no | — |
| 142 | Dic14 | — | 4 | 3 | no | — |
| 143 | C7×C8 | — | 2 | 12 | no | — |
| 144 | C7⋊C8 | — | 4 | 2 | no | — |
| 145 | C2×#41 | — | 7 | 6 | no | — |
| 146 | C2×#41 | — | 7 | 6 | no | — |
| 147 | C2×#41 | — | 14 | 3 | no | — |
| 148 | C2×#41 | — | 14 | 3 | no | — |
| 149 | C2×#41 | — | 14 | 6 | no | — |
| 150 | C2×#41 | — | 14 | 6 | no | — |
| 151 | C2×#41 | — | 14 | 6 | no | — |
| 152 | C2×#41 | — | 14 | 6 | no | — |
| 153 | C22×Q8 | — | 7 | 3 | no | — |
| 154 | C22×D4 | — | 7 | 3 | no | — |
| 155 | C22×D4 | — | 14 | 3 | no | — |
| 156 | C23×C4 | — | 14 | 3 | no | — |
| 157 | C23×C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 158 | #137 | — | 14 | 24 | no | — |
| 159 | #137 | — | 14 | 24 | no | — |
| 160 | AGL(1,ℤ/8) | — | 14 | 24 | no | — |
| 161 | AGL(1,ℤ/8) | — | 14 | 24 | no | — |
| 162 | #135 | — | 14 | 24 | no | — |
| 163 | #135 | — | 14 | 24 | no | — |
| 164 | C2×Q16 | — | 14 | 6 | no | — |
| 165 | C2×SD16 | — | 14 | 12 | no | — |
| 166 | C2×D8 | — | 14 | 6 | no | — |
| 167 | C2×M16 | — | 2 | 6 | no | — |
| 168 | C2×M16 | — | 2 | 6 | no | — |
| 169 | C2×M16 | — | 14 | 3 | no | — |
| 170 | C2×M16 | — | 14 | 3 | no | — |
| 171 | C2×M16 | D28 | 1 | 6 | yes | — |
| 172 | C2×M16 | C22×D7 | 1 | 6 | yes | — |
| 173 | C22×C8 | — | 7 | 2 | no | — |
| 174 | C22×C8 | D28 | 1 | 6 | yes | — |
| 175 | C22⋊C8 | — | 14 | 24 | no | — |
| 176 | C22×C7 | — | 4 | 3 | no | — |
| 177 | C22×C7 | — | 4 | 3 | no | — |
| 178 | C22×C7 | — | 4 | 1 | no | — |
| 179 | C22×C7 | #392 | 1 | 3 | yes | — |
| 180 | C22×C7 | #392 | 1 | 1 | yes | characteristic |
| 181 | C22×C7 | C2×#121 | 1 | 3 | yes | — |
| 182 | C2×D7 | — | 4 | 12 | no | — |
| 183 | C2×D7 | — | 4 | 6 | no | — |
| 184 | C2×D7 | — | 8 | 12 | no | — |
| 185 | C4×C7 | #392 | 1 | 1 | yes | characteristic |
| 186 | C4×C7 | #392 | 1 | 3 | yes | — |
| 187 | C4×C7 | C2×#120 | 1 | 3 | yes | — |
| 188 | C4×C7 | C2×#127 | 1 | 1 | yes | characteristic |
| 189 | Dic7 | — | 4 | 6 | no | — |
| 190 | C24 | — | 14 | 3 | no | — |
| 191 | C24 | #1161 | 1 | 1 | yes | characteristic |
| 192 | #41 | — | 14 | 12 | no | — |
| 193 | #41 | — | 14 | 12 | no | — |
| 194 | #41 | — | 14 | 12 | no | — |
| 195 | #41 | — | 14 | 12 | no | — |
| 196 | #41 | — | 28 | 3 | no | — |
| 197 | #41 | — | 28 | 3 | no | — |
| 198 | #41 | — | 28 | 3 | no | — |
| 199 | #41 | — | 28 | 3 | no | — |
| 200 | #41 | — | 28 | 6 | no | — |
| 201 | #41 | — | 28 | 6 | no | — |
| 202 | C2×Q8 | — | 7 | 6 | no | — |
| 203 | C2×Q8 | — | 14 | 3 | no | — |
| 204 | C2×Q8 | — | 14 | 6 | no | — |
| 205 | C2×Q8 | — | 14 | 6 | no | — |
| 206 | C2×D4 | — | 7 | 6 | no | — |
| 207 | C2×D4 | — | 14 | 6 | no | — |
| 208 | C2×D4 | — | 14 | 6 | no | — |
| 209 | C2×D4 | — | 14 | 6 | no | — |
| 210 | C2×D4 | — | 14 | 6 | no | — |
| 211 | C2×D4 | — | 14 | 3 | no | — |
| 212 | C2×D4 | — | 28 | 6 | no | — |
| 213 | C2×D4 | — | 28 | 6 | no | — |
| 214 | C22×C4 | — | 14 | 3 | no | — |
| 215 | C22×C4 | — | 14 | 6 | no | — |
| 216 | C22×C4 | — | 14 | 6 | no | — |
| 217 | C22×C4 | — | 28 | 6 | no | — |
| 218 | C22×C4 | — | 28 | 6 | no | — |
| 219 | C22×C4 | C2×D28 | 1 | 3 | yes | — |
| 220 | C22×C4 | C2×D28 | 1 | 3 | yes | — |
| 221 | C22×C4 | D4×D7 | 1 | 3 | yes | — |
| 222 | C22×C4 | D4×D7 | 1 | 3 | yes | — |
| 223 | C22×C4 | #1161 | 1 | 1 | yes | characteristic |
| 224 | C22×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 225 | Q16 | — | 14 | 24 | no | — |
| 226 | SD16 | — | 14 | 48 | no | — |
| 227 | D8 | — | 14 | 24 | no | — |
| 228 | M16 | — | 2 | 12 | no | — |
| 229 | M16 | — | 2 | 12 | no | — |
| 230 | M16 | — | 28 | 3 | no | — |
| 231 | M16 | — | 28 | 1 | no | — |
| 232 | C2×C8 | — | 2 | 6 | no | — |
| 233 | C2×C8 | — | 2 | 6 | no | — |
| 234 | C2×C8 | — | 14 | 6 | no | — |
| 235 | C2×C8 | C2×D28 | 1 | 12 | yes | — |
| 236 | C2×C7 | — | 2 | 3 | no | — |
| 237 | C2×C7 | — | 2 | 3 | no | — |
| 238 | C2×C7 | — | 2 | 1 | no | — |
| 239 | C2×C7 | C2×#392 | 1 | 1 | yes | characteristic |
| 240 | D7 | — | 8 | 6 | no | — |
| 241 | C23 | — | 8 | 1 | no | — |
| 242 | C23 | — | 14 | 6 | no | — |
| 243 | C23 | — | 28 | 6 | no | — |
| 244 | C23 | — | 56 | 6 | no | — |
| 245 | C23 | #6458 | 1 | 3 | yes | — |
| 246 | C23 | #6458 | 1 | 3 | yes | — |
| 247 | C23 | C2×#1161 | 1 | 1 | yes | characteristic |
| 248 | Q8 | — | 14 | 12 | no | — |
| 249 | Q8 | — | 28 | 3 | no | — |
| 250 | Q8 | — | 28 | 3 | no | — |
| 251 | D4 | — | 14 | 12 | no | — |
| 252 | D4 | — | 28 | 12 | no | — |
| 253 | D4 | — | 28 | 6 | no | — |
| 254 | D4 | — | 28 | 6 | no | — |
| 255 | D4 | — | 28 | 3 | no | — |
| 256 | D4 | — | 28 | 3 | no | — |
| 257 | C2×C4 | — | 14 | 6 | no | — |
| 258 | C2×C4 | — | 28 | 6 | no | — |
| 259 | C2×C4 | — | 28 | 6 | no | — |
| 260 | C2×C4 | — | 28 | 6 | no | — |
| 261 | C2×C4 | — | 28 | 12 | no | — |
| 262 | C2×C4 | #6437 | 1 | 3 | yes | — |
| 263 | C2×C4 | #6437 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | #6445 | 1 | 3 | yes | — |
| 265 | C2×C4 | #6445 | 1 | 3 | yes | — |
| 266 | C2×C4 | #6445 | 1 | 6 | yes | — |
| 267 | C2×C4 | #6458 | 1 | 3 | yes | — |
| 268 | C2×C4 | #6458 | 1 | 3 | yes | — |
| 269 | C2×C4 | C22×D28 | 1 | 3 | yes | — |
| 270 | C2×C4 | C2×D4×D7 | 1 | 3 | yes | — |
| 271 | C8 | — | 2 | 12 | no | — |
| 272 | C8 | — | 28 | 2 | no | — |
| 273 | C7 | #23434 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 274 | C22 | — | 4 | 3 | no | — |
| 275 | C22 | — | 4 | 3 | no | — |
| 276 | C22 | — | 4 | 1 | no | — |
| 277 | C22 | — | 28 | 12 | no | — |
| 278 | C22 | — | 28 | 6 | no | — |
| 279 | C22 | — | 56 | 12 | no | — |
| 280 | C22 | #89443 | 1 | 3 | yes | — |
| 281 | C22 | #89443 | 1 | 1 | yes | characteristic |
| 282 | C22 | C2×#6458 | 1 | 3 | yes | — |
| 283 | C4 | — | 28 | 6 | no | — |
| 284 | C4 | #89443 | 1 | 3 | yes | — |
| 285 | C4 | #89443 | 1 | 1 | yes | characteristic |
| 286 | C4 | C2×#6437 | 1 | 1 | yes | characteristic |
| 287 | C4 | C2×#6445 | 1 | 3 | yes | — |
| 288 | C2 | — | 2 | 3 | no | — |
| 289 | C2 | — | 2 | 3 | no | — |
| 290 | C2 | — | 2 | 1 | no | — |
| 291 | C2 | — | 56 | 6 | no | — |
| 292 | C2 | C2×#89443 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#90439 | C2 | semidirect |
| 2. #11741207 | C2 | semidirect |
| 3. C2×C7×#338 | C2 | semidirect |
| 4. #11726481 | C2 | semidirect |
| 10. #90439 | C22 | semidirect |
| 14. #89901 | C22 | semidirect |
| 17. C7×#338 | C22 | semidirect |
| 53. C2×#338 | D7 | semidirect |
| 112. #338 | C2×D7 | semidirect |
| 273. C7 | #23434 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#90439 | 2 |
| #11741207 | 2 |
| C2×C7×#338 | 2 |
| #11726481 | 2 |
| #23434 | 7 |