Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
482 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 71 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 | #11740767 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#2132 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11729559 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11725356 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #90616 | C4 | 1 | 1 | yes | characteristic |
| 6 | C7×#532 | C22 | 1 | 1 | yes | characteristic |
| 7 | #90513 | — | 2 | 1 | no | — |
| 8 | C7×#413 | — | 2 | 1 | no | — |
| 9 | #90009 | C4 | 1 | 1 | yes | characteristic |
| 10 | #89867 | C22 | 1 | 2 | yes | — |
| 11 | #89849 | — | 2 | 1 | no | — |
| 12 | #89834 | C4 | 1 | 2 | yes | — |
| 13 | #89833 | — | 2 | 1 | no | — |
| 14 | C2×#6379 | — | 2 | 1 | no | — |
| 15 | #89697 | — | 2 | 1 | no | — |
| 16 | C7×#356 | C22 | 1 | 2 | yes | complemented |
| 17 | C7×#355 | — | 2 | 1 | no | — |
| 18 | #89341 | C22 | 1 | 2 | yes | complemented |
| 19 | #89339 | — | 2 | 1 | no | — |
| 20 | #13572 | — | 7 | 1 | no | maximal, Sylow |
| 21 | C2×C7×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C7×#41 | D4 | 1 | 1 | yes | characteristic |
| 23 | #6554 | — | 4 | 1 | no | — |
| 24 | #6553 | — | 4 | 1 | no | — |
| 25 | D7×#41 | — | 4 | 2 | no | — |
| 26 | #6551 | C2×C4 | 1 | 2 | yes | — |
| 27 | #6548 | — | 2 | 1 | no | — |
| 28 | #6548 | — | 4 | 1 | no | — |
| 29 | C2×#1157 | — | 2 | 1 | no | — |
| 30 | C2×#1157 | — | 4 | 1 | no | — |
| 31 | C2×C7×M16 | — | 4 | 1 | no | — |
| 32 | C7×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C7×#127 | D4 | 1 | 1 | yes | characteristic |
| 34 | C7×#124 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C7×#124 | D4 | 1 | 1 | yes | characteristic |
| 36 | C7×#121 | — | 4 | 1 | no | — |
| 37 | C4×C7×D4 | — | 4 | 1 | no | — |
| 38 | C2×C42×C7 | D4 | 1 | 1 | yes | characteristic |
| 39 | #6514 | — | 4 | 1 | no | — |
| 40 | #6513 | — | 4 | 2 | no | — |
| 41 | #6512 | — | 4 | 1 | no | — |
| 42 | #6505 | — | 2 | 2 | no | — |
| 43 | #6495 | — | 2 | 2 | no | — |
| 44 | C2×#1136 | — | 2 | 1 | no | — |
| 45 | C7×#101 | — | 2 | 2 | no | — |
| 46 | C7×#100 | — | 4 | 1 | no | — |
| 47 | #6410 | C2×C4 | 1 | 2 | yes | — |
| 48 | #6407 | — | 2 | 1 | no | — |
| 49 | C22⋊(C7⋊C8) | — | 4 | 1 | no | — |
| 50 | #6379 | — | 2 | 2 | no | — |
| 51 | #6379 | — | 2 | 2 | no | — |
| 52 | C4⋊(C7⋊C8) | — | 2 | 2 | no | — |
| 53 | C4×C7⋊C8 | — | 4 | 1 | no | — |
| 54 | #3018 | — | 7 | 1 | no | — |
| 55 | #2132 | D7 | 1 | 1 | yes | characteristic, complemented |
| 56 | #1543 | — | 7 | 1 | no | — |
| 57 | #1405 | — | 7 | 4 | no | — |
| 58 | C7×#41 | — | 4 | 1 | no | — |
| 59 | C7×#41 | — | 4 | 1 | no | — |
| 60 | C7×#41 | — | 4 | 1 | no | — |
| 61 | C7×#41 | — | 4 | 1 | no | — |
| 62 | C2×C7×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C7×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C2×C7×D4 | — | 2 | 1 | no | — |
| 65 | C2×C7×D4 | — | 2 | 1 | no | — |
| 66 | C2×C7×D4 | — | 4 | 1 | no | — |
| 67 | C2×C7×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C7×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C22×C4×C7 | — | 2 | 1 | no | — |
| 70 | C22×C4×C7 | — | 2 | 1 | no | — |
| 71 | C22×C4×C7 | — | 2 | 1 | no | — |
| 72 | C22×C4×C7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 74 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 75 | #1161 | — | 4 | 2 | no | — |
| 76 | Q8×D7 | — | 4 | 2 | no | — |
| 77 | #1159 | — | 4 | 2 | no | — |
| 78 | #1159 | — | 4 | 4 | no | — |
| 79 | D4×D7 | — | 4 | 2 | no | — |
| 80 | D4×D7 | — | 4 | 4 | no | — |
| 81 | #1157 | — | 2 | 2 | no | — |
| 82 | #1157 | — | 2 | 2 | no | — |
| 83 | #1157 | — | 4 | 1 | no | — |
| 84 | #1157 | — | 4 | 2 | no | — |
| 85 | #1157 | — | 4 | 1 | no | — |
| 86 | #1157 | — | 4 | 2 | no | — |
| 87 | #1157 | — | 4 | 2 | no | — |
| 88 | C2×D28 | — | 2 | 1 | no | — |
| 89 | C2×D28 | — | 4 | 1 | no | — |
| 90 | C2×C4×D7 | — | 4 | 1 | no | — |
| 91 | C2×C4×D7 | — | 4 | 2 | no | — |
| 92 | C2×Dic14 | — | 2 | 1 | no | — |
| 93 | C2×Dic14 | — | 4 | 1 | no | — |
| 94 | C7×M16 | — | 4 | 2 | no | — |
| 95 | C7×M16 | — | 8 | 1 | no | — |
| 96 | C2×C7×C8 | — | 8 | 1 | no | — |
| 97 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 98 | C7×C22⋊C4 | — | 4 | 1 | no | — |
| 99 | C7×C22⋊C4 | — | 4 | 1 | no | — |
| 100 | C42×C7 | — | 2 | 1 | no | — |
| 101 | C42×C7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | #1144 | — | 4 | 2 | no | — |
| 104 | #1143 | — | 4 | 2 | no | — |
| 105 | #1142 | — | 4 | 2 | no | — |
| 106 | #1141 | — | 4 | 2 | no | — |
| 107 | #1136 | — | 2 | 2 | no | — |
| 108 | #1136 | — | 4 | 1 | no | — |
| 109 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 110 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 111 | #585 | — | 7 | 1 | no | — |
| 112 | #575 | — | 14 | 1 | no | — |
| 113 | #532 | C2×D7 | 1 | 1 | yes | characteristic |
| 114 | #456 | — | 7 | 2 | no | — |
| 115 | #455 | — | 14 | 1 | no | — |
| 116 | #454 | — | 7 | 2 | no | — |
| 117 | #453 | — | 14 | 1 | no | — |
| 118 | #431 | — | 14 | 1 | no | — |
| 119 | C2×#104 | — | 14 | 1 | no | — |
| 120 | #413 | — | 2 | 1 | no | — |
| 121 | #413 | — | 7 | 1 | no | — |
| 122 | #356 | C2×D7 | 1 | 2 | yes | complemented |
| 123 | #355 | — | 2 | 1 | no | — |
| 124 | #333 | — | 7 | 2 | no | — |
| 125 | #331 | — | 14 | 1 | no | — |
| 126 | C23×C7 | — | 2 | 1 | no | — |
| 127 | C23×C7 | — | 2 | 1 | no | — |
| 128 | C23×C7 | #99 | 1 | 1 | yes | characteristic |
| 129 | C22×D7 | — | 4 | 1 | no | — |
| 130 | C22×D7 | — | 4 | 2 | no | — |
| 131 | C7×Q8 | — | 4 | 1 | no | — |
| 132 | C7×Q8 | — | 4 | 1 | no | — |
| 133 | C7×D4 | — | 4 | 1 | no | — |
| 134 | C7×D4 | — | 4 | 1 | no | — |
| 135 | C7×D4 | — | 4 | 2 | no | — |
| 136 | C7×D4 | — | 8 | 2 | no | — |
| 137 | C7×D4 | — | 8 | 1 | no | — |
| 138 | C2×C4×C7 | — | 2 | 1 | no | — |
| 139 | C2×C4×C7 | — | 2 | 1 | no | — |
| 140 | C2×C4×C7 | — | 2 | 1 | no | — |
| 141 | C2×C4×C7 | — | 2 | 1 | no | — |
| 142 | C2×C4×C7 | — | 2 | 1 | no | — |
| 143 | C2×C4×C7 | — | 2 | 1 | no | — |
| 144 | C2×C4×C7 | — | 4 | 2 | no | — |
| 145 | C2×C4×C7 | — | 4 | 1 | no | — |
| 146 | C2×C4×C7 | — | 4 | 1 | no | — |
| 147 | C2×C4×C7 | — | 4 | 1 | no | — |
| 148 | C2×C4×C7 | — | 4 | 1 | no | — |
| 149 | C2×C4×C7 | #99 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 151 | C22⋊D7 | — | 4 | 2 | no | — |
| 152 | C22⋊D7 | — | 4 | 4 | no | — |
| 153 | C22⋊D7 | — | 4 | 2 | no | — |
| 154 | C2×Dic7 | — | 4 | 1 | no | — |
| 155 | C2×Dic7 | — | 4 | 2 | no | — |
| 156 | D28 | — | 2 | 2 | no | — |
| 157 | D28 | — | 4 | 2 | no | — |
| 158 | D28 | — | 4 | 1 | no | — |
| 159 | C4×D7 | — | 4 | 2 | no | — |
| 160 | C4×D7 | — | 4 | 4 | no | — |
| 161 | C4×D7 | — | 4 | 2 | no | — |
| 162 | Dic14 | — | 2 | 2 | no | — |
| 163 | Dic14 | — | 4 | 1 | no | — |
| 164 | Dic14 | — | 4 | 2 | no | — |
| 165 | C7×C8 | — | 8 | 2 | no | — |
| 166 | C7⋊C8 | — | 4 | 2 | no | — |
| 167 | C7⋊C8 | — | 8 | 2 | no | — |
| 168 | 2−1+4 | — | 7 | 2 | no | — |
| 169 | 2−1+4 | — | 28 | 1 | no | — |
| 170 | H(5,𝔽2) | — | 14 | 1 | no | — |
| 171 | H(5,𝔽2) | — | 28 | 1 | no | — |
| 172 | H(5,𝔽2) | — | 28 | 1 | no | — |
| 173 | C2×#41 | — | 14 | 1 | no | — |
| 174 | C2×#41 | — | 28 | 1 | no | — |
| 175 | C2×#41 | — | 28 | 2 | no | — |
| 176 | C2×#41 | C4×D7 | 1 | 1 | yes | characteristic |
| 177 | C2×#41 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 178 | #137 | — | 14 | 2 | no | — |
| 179 | #137 | — | 28 | 1 | no | — |
| 180 | AGL(1,ℤ/8) | — | 14 | 2 | no | — |
| 181 | AGL(1,ℤ/8) | — | 28 | 1 | no | — |
| 182 | #135 | — | 28 | 2 | no | — |
| 183 | C2×M16 | — | 4 | 1 | no | — |
| 184 | C2×M16 | — | 14 | 1 | no | — |
| 185 | C4⋊Q8 | C22×D7 | 1 | 1 | yes | characteristic |
| 186 | #127 | D28 | 1 | 1 | yes | characteristic |
| 187 | #124 | C4×D7 | 1 | 1 | yes | characteristic |
| 188 | #124 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 189 | #121 | — | 4 | 1 | no | — |
| 190 | C4×D4 | — | 4 | 1 | no | — |
| 191 | C2×C42 | D28 | 1 | 1 | yes | characteristic |
| 192 | C4⋊C8 | — | 14 | 2 | no | — |
| 193 | #104 | — | 14 | 2 | no | — |
| 194 | #104 | — | 14 | 2 | no | — |
| 195 | #101 | — | 2 | 2 | no | — |
| 196 | #101 | — | 7 | 2 | no | — |
| 197 | #100 | — | 4 | 1 | no | — |
| 198 | #100 | — | 14 | 1 | no | — |
| 199 | C22⋊C8 | — | 28 | 1 | no | — |
| 200 | C4×C8 | — | 28 | 1 | no | — |
| 201 | C22×C7 | — | 2 | 1 | no | — |
| 202 | C22×C7 | — | 4 | 1 | no | — |
| 203 | C22×C7 | — | 4 | 1 | no | — |
| 204 | C22×C7 | — | 4 | 1 | no | — |
| 205 | C22×C7 | — | 8 | 1 | no | — |
| 206 | C22×C7 | — | 8 | 1 | no | — |
| 207 | C22×C7 | C2×#99 | 1 | 1 | yes | characteristic |
| 208 | C2×D7 | — | 4 | 2 | no | — |
| 209 | C2×D7 | — | 16 | 1 | no | — |
| 210 | C2×D7 | — | 16 | 2 | no | — |
| 211 | C4×C7 | — | 2 | 2 | no | — |
| 212 | C4×C7 | — | 2 | 1 | no | — |
| 213 | C4×C7 | — | 4 | 1 | no | — |
| 214 | C4×C7 | — | 4 | 1 | no | — |
| 215 | C4×C7 | — | 4 | 1 | no | — |
| 216 | C4×C7 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 217 | C4×C7 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 218 | Dic7 | — | 4 | 2 | no | — |
| 219 | #41 | — | 4 | 1 | no | — |
| 220 | #41 | — | 4 | 1 | no | — |
| 221 | #41 | — | 4 | 1 | no | — |
| 222 | #41 | — | 4 | 1 | no | — |
| 223 | #41 | — | 14 | 2 | no | — |
| 224 | #41 | — | 14 | 2 | no | — |
| 225 | #41 | — | 28 | 1 | no | — |
| 226 | #41 | — | 28 | 2 | no | — |
| 227 | #41 | — | 28 | 2 | no | — |
| 228 | #41 | — | 28 | 2 | no | — |
| 229 | #41 | — | 28 | 2 | no | — |
| 230 | #41 | — | 28 | 4 | no | — |
| 231 | #41 | — | 28 | 2 | no | — |
| 232 | #41 | — | 28 | 1 | no | — |
| 233 | C2×Q8 | — | 14 | 1 | no | — |
| 234 | C2×Q8 | — | 28 | 1 | no | — |
| 235 | C2×Q8 | — | 28 | 2 | no | — |
| 236 | C2×Q8 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 237 | C2×Q8 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 238 | C2×D4 | — | 2 | 1 | no | — |
| 239 | C2×D4 | — | 2 | 1 | no | — |
| 240 | C2×D4 | — | 4 | 1 | no | — |
| 241 | C2×D4 | — | 14 | 1 | no | — |
| 242 | C2×D4 | — | 28 | 1 | no | — |
| 243 | C2×D4 | — | 28 | 2 | no | — |
| 244 | C2×D4 | — | 28 | 4 | no | — |
| 245 | C2×D4 | — | 28 | 2 | no | — |
| 246 | C2×D4 | — | 28 | 1 | no | — |
| 247 | C2×D4 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 248 | C2×D4 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 249 | C22×C4 | — | 2 | 1 | no | — |
| 250 | C22×C4 | — | 2 | 1 | no | — |
| 251 | C22×C4 | — | 2 | 1 | no | — |
| 252 | C22×C4 | — | 28 | 1 | no | — |
| 253 | C22×C4 | — | 28 | 2 | no | — |
| 254 | C22×C4 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 255 | Q16 | — | 28 | 2 | no | — |
| 256 | SD16 | — | 28 | 2 | no | — |
| 257 | SD16 | — | 28 | 2 | no | — |
| 258 | D8 | — | 28 | 2 | no | — |
| 259 | M16 | — | 4 | 2 | no | — |
| 260 | M16 | — | 8 | 1 | no | — |
| 261 | M16 | — | 14 | 2 | no | — |
| 262 | M16 | — | 28 | 1 | no | — |
| 263 | C2×C8 | — | 8 | 1 | no | — |
| 264 | C2×C8 | — | 28 | 1 | no | — |
| 265 | C2×C8 | — | 28 | 2 | no | — |
| 266 | C4⋊C4 | — | 4 | 1 | no | — |
| 267 | C22⋊C4 | — | 4 | 1 | no | — |
| 268 | C22⋊C4 | — | 4 | 1 | no | — |
| 269 | C42 | — | 2 | 1 | no | — |
| 270 | C42 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 271 | C42 | C2×D28 | 1 | 1 | yes | characteristic |
| 272 | C2×C7 | — | 2 | 1 | no | — |
| 273 | C2×C7 | — | 4 | 1 | no | — |
| 274 | C2×C7 | — | 8 | 1 | no | — |
| 275 | C2×C7 | — | 8 | 1 | no | — |
| 276 | C2×C7 | C2×AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 277 | D7 | — | 8 | 2 | no | — |
| 278 | C23 | — | 2 | 1 | no | — |
| 279 | C23 | — | 2 | 1 | no | — |
| 280 | C23 | — | 28 | 1 | no | — |
| 281 | C23 | — | 28 | 2 | no | — |
| 282 | C23 | #6380 | 1 | 1 | yes | characteristic |
| 283 | Q8 | — | 4 | 1 | no | — |
| 284 | Q8 | — | 4 | 1 | no | — |
| 285 | Q8 | — | 14 | 2 | no | — |
| 286 | Q8 | — | 28 | 2 | no | — |
| 287 | Q8 | — | 28 | 1 | no | — |
| 288 | D4 | — | 4 | 1 | no | — |
| 289 | D4 | — | 4 | 1 | no | — |
| 290 | D4 | — | 4 | 2 | no | — |
| 291 | D4 | — | 8 | 2 | no | — |
| 292 | D4 | — | 8 | 1 | no | — |
| 293 | D4 | — | 14 | 2 | no | — |
| 294 | D4 | — | 28 | 2 | no | — |
| 295 | D4 | — | 28 | 2 | no | — |
| 296 | D4 | — | 28 | 4 | no | — |
| 297 | D4 | — | 28 | 2 | no | — |
| 298 | D4 | — | 28 | 1 | no | — |
| 299 | C2×C4 | — | 2 | 1 | no | — |
| 300 | C2×C4 | — | 2 | 1 | no | — |
| 301 | C2×C4 | — | 2 | 1 | no | — |
| 302 | C2×C4 | — | 2 | 1 | no | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 1 | no | — |
| 305 | C2×C4 | — | 4 | 2 | no | — |
| 306 | C2×C4 | — | 4 | 1 | no | — |
| 307 | C2×C4 | — | 4 | 1 | no | — |
| 308 | C2×C4 | — | 4 | 1 | no | — |
| 309 | C2×C4 | — | 4 | 1 | no | — |
| 310 | C2×C4 | — | 28 | 2 | no | — |
| 311 | C2×C4 | — | 28 | 4 | no | — |
| 312 | C2×C4 | — | 28 | 1 | no | — |
| 313 | C2×C4 | — | 28 | 2 | no | — |
| 314 | C2×C4 | — | 28 | 2 | no | — |
| 315 | C2×C4 | #6380 | 1 | 1 | yes | characteristic |
| 316 | C2×C4 | C2×(C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 317 | C8 | — | 8 | 2 | no | — |
| 318 | C8 | — | 28 | 2 | no | — |
| 319 | C8 | — | 56 | 2 | no | — |
| 320 | C7 | #13572 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 321 | C22 | — | 2 | 1 | no | — |
| 322 | C22 | — | 4 | 1 | no | — |
| 323 | C22 | — | 4 | 1 | no | — |
| 324 | C22 | — | 4 | 1 | no | — |
| 325 | C22 | — | 8 | 1 | no | — |
| 326 | C22 | — | 8 | 1 | no | — |
| 327 | C22 | — | 28 | 2 | no | — |
| 328 | C22 | — | 112 | 1 | no | — |
| 329 | C22 | — | 112 | 2 | no | — |
| 330 | C22 | C2×#6380 | 1 | 1 | yes | characteristic |
| 331 | C4 | — | 2 | 2 | no | — |
| 332 | C4 | — | 2 | 1 | no | — |
| 333 | C4 | — | 4 | 1 | no | — |
| 334 | C4 | — | 4 | 1 | no | — |
| 335 | C4 | — | 4 | 1 | no | — |
| 336 | C4 | — | 28 | 2 | no | — |
| 337 | C4 | #89266 | 1 | 1 | yes | characteristic |
| 338 | C4 | #89266 | 1 | 1 | yes | characteristic |
| 339 | C2 | — | 2 | 1 | no | — |
| 340 | C2 | — | 4 | 1 | no | — |
| 341 | C2 | — | 8 | 1 | no | — |
| 342 | C2 | — | 8 | 1 | no | — |
| 343 | C2 | — | 56 | 2 | no | — |
| 344 | C2 | C2×#89266 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2132 | C2 | semidirect |
| 3. #11729559 | C2 | semidirect |
| 4. #11725356 | C2 | semidirect |
| 16. C7×#356 | C22 | semidirect |
| 18. #89341 | C22 | semidirect |
| 55. #2132 | D7 | semidirect |
| 122. #356 | C2×D7 | semidirect |
| 320. C7 | #13572 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11740767 | 2 |
| C7×#2132 | 2 |
| #11729559 | 2 |
| #11725356 | 2 |
| #13572 | 7 |