Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
588 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #11731813 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C7×#2039 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11729632 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11725441 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11725299 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C7×#537 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#6458 | — | 2 | 2 | no | — |
| 8 | C2×C4×D28 | — | 2 | 1 | no | — |
| 9 | C2×C7×#104 | — | 2 | 1 | no | — |
| 10 | C7×#417 | — | 2 | 1 | no | — |
| 11 | C7×#384 | C22 | 1 | 1 | yes | characteristic |
| 12 | #89808 | — | 2 | 2 | no | — |
| 13 | #89763 | — | 2 | 1 | no | — |
| 14 | #89763 | C22 | 1 | 2 | yes | — |
| 15 | #89759 | — | 2 | 2 | no | — |
| 16 | C7×#343 | C22 | 1 | 1 | yes | characteristic |
| 17 | #89332 | — | 2 | 2 | no | — |
| 18 | #89323 | C22 | 1 | 2 | yes | — |
| 19 | #12152 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C2×C7×#41 | — | 2 | 1 | no | — |
| 21 | C22×D28 | — | 2 | 1 | no | — |
| 22 | C22×D28 | — | 2 | 1 | no | — |
| 23 | C22×D28 | D4 | 1 | 2 | yes | — |
| 24 | C22×C4×D7 | — | 2 | 2 | no | — |
| 25 | C2×C7×M16 | — | 2 | 1 | no | — |
| 26 | C7×C22⋊Q8 | — | 4 | 1 | no | — |
| 27 | C7×#121 | — | 2 | 2 | no | — |
| 28 | C7×#121 | — | 4 | 1 | no | — |
| 29 | C2×C7×C4⋊C4 | — | 2 | 2 | no | — |
| 30 | C2×C7×C4⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C7×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×C42×C7 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×#1141 | — | 4 | 2 | no | — |
| 34 | C2×(C2×D7)⋊C4 | — | 2 | 2 | no | — |
| 35 | C2×(C2×D7)⋊C4 | — | 2 | 2 | no | — |
| 36 | C2×(C2×D7)⋊C4 | — | 2 | 2 | no | — |
| 37 | C2×C4⋊Dic7 | — | 2 | 1 | no | — |
| 38 | #6458 | — | 4 | 8 | no | — |
| 39 | C4×D28 | — | 4 | 2 | no | — |
| 40 | C4×D28 | — | 4 | 2 | no | — |
| 41 | C7×#104 | — | 4 | 2 | no | — |
| 42 | C7×#103 | — | 4 | 1 | no | — |
| 43 | C7×#102 | — | 4 | 1 | no | — |
| 44 | C7⋊#95 | — | 2 | 1 | no | — |
| 45 | C7⋊#95 | D4 | 1 | 2 | yes | — |
| 46 | C22⋊(C7⋊C8) | — | 2 | 2 | no | — |
| 47 | #6383 | — | 4 | 2 | no | — |
| 48 | #2537 | — | 7 | 1 | no | — |
| 49 | #2039 | D7 | 1 | 1 | yes | characteristic, complemented |
| 50 | #1598 | — | 7 | 2 | no | — |
| 51 | #1345 | — | 7 | 2 | no | — |
| 52 | #1325 | — | 7 | 1 | no | — |
| 53 | C23×D7 | — | 2 | 2 | no | — |
| 54 | C7×#41 | — | 4 | 1 | no | — |
| 55 | C7×#41 | — | 4 | 1 | no | — |
| 56 | C7×#41 | — | 4 | 1 | no | — |
| 57 | C7×#41 | — | 4 | 1 | no | — |
| 58 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 59 | C2×C7×D4 | — | 2 | 1 | no | — |
| 60 | C2×C7×D4 | — | 2 | 2 | no | — |
| 61 | C2×C7×D4 | — | 4 | 1 | no | — |
| 62 | C22×C4×C7 | — | 2 | 1 | no | — |
| 63 | C22×C4×C7 | — | 2 | 1 | no | — |
| 64 | C22×C4×C7 | — | 4 | 1 | no | — |
| 65 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C22×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 67 | C22×Dic7 | — | 2 | 2 | no | — |
| 68 | C2×D28 | — | 2 | 4 | no | — |
| 69 | C2×D28 | — | 4 | 2 | no | — |
| 70 | C2×D28 | — | 4 | 2 | no | — |
| 71 | C2×D28 | — | 4 | 2 | no | — |
| 72 | C2×D28 | — | 4 | 4 | no | — |
| 73 | C2×D28 | — | 4 | 4 | no | — |
| 74 | C2×D28 | — | 8 | 1 | no | — |
| 75 | C2×C4×D7 | — | 4 | 4 | no | — |
| 76 | C2×C4×D7 | — | 4 | 4 | no | — |
| 77 | C2×C4×D7 | — | 4 | 4 | no | — |
| 78 | C7×M16 | — | 4 | 1 | no | — |
| 79 | C7×M16 | — | 4 | 1 | no | — |
| 80 | C2×C7×C8 | — | 4 | 1 | no | — |
| 81 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 82 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 83 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 84 | C7×C4⋊C4 | — | 4 | 4 | no | — |
| 85 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 86 | C42×C7 | — | 2 | 1 | no | — |
| 87 | C42×C7 | D8 | 1 | 2 | yes | complemented |
| 88 | #1141 | — | 8 | 4 | no | — |
| 89 | (C2×D7)⋊C4 | — | 4 | 4 | no | — |
| 90 | (C2×D7)⋊C4 | — | 4 | 4 | no | — |
| 91 | (C2×D7)⋊C4 | — | 4 | 2 | no | — |
| 92 | (C2×D7)⋊C4 | — | 4 | 2 | no | — |
| 93 | C4⋊Dic7 | — | 4 | 2 | no | — |
| 94 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 95 | #537 | C2×D7 | 1 | 1 | yes | characteristic |
| 96 | C2×#121 | — | 14 | 2 | no | — |
| 97 | C2×C4×D4 | — | 14 | 1 | no | — |
| 98 | #447 | — | 14 | 2 | no | — |
| 99 | C2×#104 | — | 2 | 1 | no | — |
| 100 | #417 | — | 2 | 1 | no | — |
| 101 | #394 | — | 7 | 2 | no | — |
| 102 | #394 | — | 14 | 1 | no | — |
| 103 | #384 | C2×D7 | 1 | 1 | yes | characteristic |
| 104 | #380 | — | 14 | 2 | no | — |
| 105 | #343 | C2×D7 | 1 | 1 | yes | characteristic |
| 106 | #327 | — | 14 | 2 | no | — |
| 107 | #324 | — | 7 | 2 | no | — |
| 108 | C23×C7 | — | 4 | 1 | no | — |
| 109 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 110 | C22×D7 | — | 2 | 4 | no | — |
| 111 | C22×D7 | — | 4 | 2 | no | — |
| 112 | C22×D7 | — | 4 | 2 | no | — |
| 113 | C22×D7 | — | 4 | 2 | no | — |
| 114 | C22×D7 | — | 4 | 2 | no | — |
| 115 | C22×D7 | — | 4 | 2 | no | — |
| 116 | C22×D7 | — | 4 | 2 | no | — |
| 117 | C7×Q8 | — | 4 | 1 | no | — |
| 118 | C7×Q8 | — | 4 | 1 | no | — |
| 119 | C7×D4 | — | 4 | 1 | no | — |
| 120 | C7×D4 | — | 4 | 1 | no | — |
| 121 | C7×D4 | — | 8 | 2 | no | — |
| 122 | C7×D4 | — | 8 | 1 | no | — |
| 123 | C7×D4 | — | 8 | 1 | no | — |
| 124 | C2×C4×C7 | — | 2 | 2 | no | — |
| 125 | C2×C4×C7 | — | 2 | 2 | no | — |
| 126 | C2×C4×C7 | — | 2 | 2 | no | — |
| 127 | C2×C4×C7 | — | 2 | 1 | no | — |
| 128 | C2×C4×C7 | — | 2 | 1 | no | — |
| 129 | C2×C4×C7 | — | 2 | 2 | no | — |
| 130 | C2×C4×C7 | — | 4 | 2 | no | — |
| 131 | C2×C4×C7 | — | 4 | 2 | no | — |
| 132 | C2×C4×C7 | — | 4 | 1 | no | — |
| 133 | C2×C4×C7 | — | 4 | 1 | no | — |
| 134 | C2×C4×C7 | — | 8 | 1 | no | — |
| 135 | C2×C4×C7 | — | 8 | 1 | no | — |
| 136 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 138 | C2×Dic7 | — | 4 | 2 | no | — |
| 139 | C2×Dic7 | — | 4 | 2 | no | — |
| 140 | C2×Dic7 | — | 4 | 2 | no | — |
| 141 | D28 | — | 4 | 8 | no | — |
| 142 | D28 | — | 4 | 4 | no | — |
| 143 | D28 | — | 8 | 2 | no | — |
| 144 | D28 | — | 8 | 4 | no | — |
| 145 | C4×D7 | — | 8 | 8 | no | — |
| 146 | C7×C8 | — | 8 | 1 | no | — |
| 147 | C7⋊C8 | — | 8 | 2 | no | — |
| 148 | C2×#41 | — | 2 | 1 | no | — |
| 149 | C22×D4 | — | 7 | 2 | no | — |
| 150 | C22×D4 | — | 14 | 1 | no | — |
| 151 | C22×D4 | — | 14 | 1 | no | — |
| 152 | C23×C4 | — | 14 | 2 | no | — |
| 153 | C2×D8 | — | 28 | 2 | no | — |
| 154 | C2×M16 | — | 2 | 1 | no | — |
| 155 | C22⋊Q8 | — | 4 | 1 | no | — |
| 156 | #121 | — | 2 | 2 | no | — |
| 157 | #121 | — | 4 | 1 | no | — |
| 158 | #121 | — | 28 | 8 | no | — |
| 159 | C4×D4 | — | 28 | 2 | no | — |
| 160 | C4×D4 | — | 28 | 2 | no | — |
| 161 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 162 | C2×C4⋊C4 | — | 14 | 1 | no | — |
| 163 | C2×C4⋊C4 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 164 | C2×C4⋊C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 165 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 166 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 167 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 168 | C2×C42 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 169 | #104 | — | 4 | 2 | no | — |
| 170 | #103 | — | 4 | 1 | no | — |
| 171 | #102 | — | 4 | 1 | no | — |
| 172 | #102 | — | 28 | 2 | no | — |
| 173 | C22⋊C8 | — | 14 | 2 | no | — |
| 174 | #95 | — | 7 | 2 | no | — |
| 175 | #95 | — | 14 | 1 | no | — |
| 176 | C22×C7 | — | 2 | 1 | no | — |
| 177 | C22×C7 | — | 2 | 1 | no | — |
| 178 | C22×C7 | — | 8 | 1 | no | — |
| 179 | C22×C7 | — | 8 | 1 | no | — |
| 180 | C22×C7 | — | 8 | 1 | no | — |
| 181 | C22×C7 | #447 | 1 | 1 | yes | characteristic |
| 182 | C22×C7 | #454 | 1 | 1 | yes | characteristic |
| 183 | C22×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 184 | C2×D7 | — | 4 | 4 | no | — |
| 185 | C2×D7 | — | 4 | 4 | no | — |
| 186 | C2×D7 | — | 4 | 4 | no | — |
| 187 | C2×D7 | — | 8 | 2 | no | — |
| 188 | C2×D7 | — | 8 | 2 | no | — |
| 189 | C2×D7 | — | 8 | 2 | no | — |
| 190 | C2×D7 | — | 8 | 2 | no | — |
| 191 | C4×C7 | — | 2 | 4 | no | — |
| 192 | C4×C7 | — | 2 | 1 | no | — |
| 193 | C4×C7 | — | 2 | 1 | no | — |
| 194 | C4×C7 | — | 4 | 2 | no | — |
| 195 | C4×C7 | — | 8 | 1 | no | — |
| 196 | C4×C7 | — | 8 | 1 | no | — |
| 197 | Dic7 | — | 8 | 2 | no | — |
| 198 | C24 | — | 14 | 2 | no | — |
| 199 | #41 | — | 4 | 1 | no | — |
| 200 | #41 | — | 4 | 1 | no | — |
| 201 | #41 | — | 4 | 1 | no | — |
| 202 | #41 | — | 4 | 1 | no | — |
| 203 | C2×Q8 | — | 2 | 1 | no | — |
| 204 | C2×D4 | — | 2 | 1 | no | — |
| 205 | C2×D4 | — | 2 | 2 | no | — |
| 206 | C2×D4 | — | 4 | 1 | no | — |
| 207 | C2×D4 | — | 14 | 4 | no | — |
| 208 | C2×D4 | — | 28 | 2 | no | — |
| 209 | C2×D4 | — | 28 | 2 | no | — |
| 210 | C2×D4 | — | 28 | 2 | no | — |
| 211 | C2×D4 | — | 28 | 4 | no | — |
| 212 | C2×D4 | — | 28 | 4 | no | — |
| 213 | C2×D4 | — | 56 | 1 | no | — |
| 214 | C22×C4 | — | 2 | 1 | no | — |
| 215 | C22×C4 | — | 2 | 1 | no | — |
| 216 | C22×C4 | — | 4 | 1 | no | — |
| 217 | C22×C4 | — | 14 | 2 | no | — |
| 218 | C22×C4 | — | 28 | 4 | no | — |
| 219 | C22×C4 | — | 28 | 4 | no | — |
| 220 | C22×C4 | — | 28 | 4 | no | — |
| 221 | C22×C4 | D4×D7 | 1 | 2 | yes | — |
| 222 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 223 | D8 | — | 56 | 4 | no | — |
| 224 | M16 | — | 4 | 1 | no | — |
| 225 | M16 | — | 4 | 1 | no | — |
| 226 | C2×C8 | — | 4 | 1 | no | — |
| 227 | C2×C8 | — | 28 | 2 | no | — |
| 228 | C4⋊C4 | — | 2 | 2 | no | — |
| 229 | C4⋊C4 | — | 4 | 1 | no | — |
| 230 | C4⋊C4 | — | 4 | 1 | no | — |
| 231 | C4⋊C4 | — | 4 | 4 | no | — |
| 232 | C4⋊C4 | — | 28 | 2 | no | — |
| 233 | C22⋊C4 | — | 4 | 2 | no | — |
| 234 | C22⋊C4 | — | 28 | 4 | no | — |
| 235 | C22⋊C4 | — | 28 | 4 | no | — |
| 236 | C22⋊C4 | — | 28 | 2 | no | — |
| 237 | C22⋊C4 | — | 28 | 2 | no | — |
| 238 | C42 | — | 2 | 1 | no | — |
| 239 | C42 | #1141 | 1 | 2 | yes | complemented |
| 240 | C2×C7 | — | 2 | 1 | no | — |
| 241 | C2×C7 | — | 2 | 1 | no | — |
| 242 | C2×C7 | — | 16 | 1 | no | — |
| 243 | C2×C7 | #1598 | 1 | 1 | yes | characteristic |
| 244 | C2×C7 | #2196 | 1 | 1 | yes | characteristic |
| 245 | C2×C7 | #2204 | 1 | 1 | yes | characteristic |
| 246 | D7 | — | 8 | 4 | no | — |
| 247 | C23 | — | 4 | 1 | no | — |
| 248 | C23 | — | 14 | 4 | no | — |
| 249 | C23 | — | 28 | 2 | no | — |
| 250 | C23 | — | 28 | 2 | no | — |
| 251 | C23 | — | 28 | 2 | no | — |
| 252 | C23 | — | 28 | 2 | no | — |
| 253 | C23 | — | 28 | 2 | no | — |
| 254 | C23 | — | 28 | 2 | no | — |
| 255 | C23 | #6500 | 1 | 1 | yes | characteristic |
| 256 | Q8 | — | 4 | 1 | no | — |
| 257 | Q8 | — | 4 | 1 | no | — |
| 258 | D4 | — | 4 | 1 | no | — |
| 259 | D4 | — | 4 | 1 | no | — |
| 260 | D4 | — | 8 | 2 | no | — |
| 261 | D4 | — | 8 | 1 | no | — |
| 262 | D4 | — | 8 | 1 | no | — |
| 263 | D4 | — | 28 | 8 | no | — |
| 264 | D4 | — | 28 | 4 | no | — |
| 265 | D4 | — | 56 | 2 | no | — |
| 266 | D4 | — | 56 | 4 | no | — |
| 267 | C2×C4 | — | 2 | 2 | no | — |
| 268 | C2×C4 | — | 2 | 2 | no | — |
| 269 | C2×C4 | — | 2 | 1 | no | — |
| 270 | C2×C4 | — | 2 | 2 | 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 | 1 | no | — |
| 276 | C2×C4 | — | 4 | 1 | no | — |
| 277 | C2×C4 | — | 8 | 1 | no | — |
| 278 | C2×C4 | — | 8 | 1 | no | — |
| 279 | C2×C4 | — | 28 | 2 | no | — |
| 280 | C2×C4 | — | 28 | 2 | no | — |
| 281 | C2×C4 | — | 28 | 2 | no | — |
| 282 | C2×C4 | — | 56 | 8 | no | — |
| 283 | C2×C4 | C2×#1141 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #6512 | 1 | 1 | yes | characteristic |
| 285 | C8 | — | 8 | 1 | no | — |
| 286 | C8 | — | 56 | 2 | no | — |
| 287 | C7 | #12152 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 2 | 1 | no | — |
| 290 | C22 | — | 8 | 1 | no | — |
| 291 | C22 | — | 8 | 1 | no | — |
| 292 | C22 | — | 8 | 1 | no | — |
| 293 | C22 | — | 28 | 4 | no | — |
| 294 | C22 | — | 28 | 4 | no | — |
| 295 | C22 | — | 28 | 4 | no | — |
| 296 | C22 | — | 56 | 2 | no | — |
| 297 | C22 | — | 56 | 2 | no | — |
| 298 | C22 | — | 56 | 2 | no | — |
| 299 | C22 | — | 56 | 2 | no | — |
| 300 | C22 | #89804 | 1 | 1 | yes | characteristic |
| 301 | C22 | #89808 | 1 | 1 | yes | characteristic |
| 302 | C22 | #89834 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 2 | 4 | no | — |
| 304 | C4 | — | 2 | 1 | no | — |
| 305 | C4 | — | 2 | 1 | no | — |
| 306 | C4 | — | 4 | 2 | no | — |
| 307 | C4 | — | 8 | 1 | no | — |
| 308 | C4 | — | 8 | 1 | no | — |
| 309 | C4 | — | 56 | 2 | no | — |
| 310 | C2 | — | 2 | 1 | no | — |
| 311 | C2 | — | 2 | 1 | no | — |
| 312 | C2 | — | 16 | 1 | no | — |
| 313 | C2 | — | 56 | 4 | no | — |
| 314 | C2 | #11727457 | 1 | 1 | yes | characteristic |
| 315 | C2 | #11728738 | 1 | 1 | yes | characteristic |
| 316 | C2 | #11728748 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11731813 | C2 | semidirect |
| 2. C7×#2039 | C2 | semidirect |
| 3. #11729632 | C2 | semidirect |
| 4. #11725441 | C2 | semidirect |
| 5. #11725299 | C2 | semidirect |
| 49. #2039 | D7 | semidirect |
| 87. C42×C7 | D8 | semidirect |
| 239. C42 | #1141 | semidirect |
| 287. C7 | #12152 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11731813 | 2 |
| C7×#2039 | 2 |
| #11729632 | 2 |
| #11725441 | 2 |
| #11725299 | 2 |
| #12152 | 7 |