← #31010123
Subgroups — #31010123
Order 17152.
806 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16863831 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#16863773 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22562249 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #22562244 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #22561965 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22561924 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C4×#135×C67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×D536 | — | 2 | 1 | no | — |
| 9 | C22×D536 | C22 | 1 | 2 | yes | — |
| 10 | C2×#14524040 | — | 2 | 1 | no | — |
| 11 | #16863831 | — | 2 | 8 | no | — |
| 12 | C2×#14523986 | C22 | 1 | 4 | yes | complemented |
| 13 | C2×#14523984 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×#14523983 | C22 | 1 | 2 | yes | — |
| 15 | #16863773 | — | 2 | 2 | no | — |
| 16 | #16863773 | C22 | 1 | 4 | yes | — |
| 17 | C4⋊(C67⋊C16) | C22 | 1 | 4 | yes | complemented |
| 18 | #16863196 | C22 | 1 | 8 | yes | complemented |
| 19 | C2×#135×C67 | C22 | 1 | 1 | yes | characteristic |
| 20 | C4×#41×C67 | — | 2 | 1 | no | — |
| 21 | C4×Q16×C67 | C22 | 1 | 2 | yes | complemented |
| 22 | C4×SD16×C67 | — | 2 | 2 | no | — |
| 23 | C4×D8×C67 | C22 | 1 | 2 | yes | complemented |
| 24 | #427×C67 | C22 | 1 | 1 | yes | characteristic |
| 25 | #416×C67 | — | 2 | 1 | no | — |
| 26 | C2×C4×C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 27 | C22×D268 | — | 2 | 2 | no | — |
| 28 | C22×D268 | — | 4 | 2 | no | — |
| 29 | C2×D536 | — | 2 | 4 | no | — |
| 30 | C2×D536 | — | 2 | 2 | no | — |
| 31 | C2×D536 | — | 2 | 4 | no | — |
| 32 | C2×D536 | — | 2 | 4 | no | — |
| 33 | C2×D536 | — | 4 | 4 | no | — |
| 34 | C2×D536 | C23 | 1 | 4 | yes | — |
| 35 | #14524040 | — | 2 | 4 | no | — |
| 36 | #14524040 | — | 4 | 1 | no | — |
| 37 | #14524040 | — | 4 | 1 | no | — |
| 38 | #14523986 | — | 2 | 8 | no | — |
| 39 | #14523984 | — | 2 | 8 | no | — |
| 40 | #14523983 | — | 2 | 4 | no | — |
| 41 | C2×C67⋊C16 | C23 | 1 | 4 | yes | — |
| 42 | C2×#41×C67 | — | 2 | 1 | no | — |
| 43 | #135×C67 | — | 2 | 2 | no | — |
| 44 | #135×C67 | D4 | 1 | 4 | yes | complemented |
| 45 | C2×Q16×C67 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×SD16×C67 | — | 2 | 1 | no | — |
| 47 | C2×D8×C67 | C23 | 1 | 1 | yes | characteristic |
| 48 | C22×C8×C67 | D4 | 1 | 1 | yes | characteristic |
| 49 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 50 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 51 | C4×D4×C67 | — | 2 | 2 | no | — |
| 52 | C4×D4×C67 | — | 4 | 2 | no | — |
| 53 | #117×C67 | — | 2 | 1 | no | — |
| 54 | #117×C67 | — | 4 | 1 | no | — |
| 55 | C2×C42×C67 | — | 4 | 1 | no | — |
| 56 | C2×C42×C67 | D4 | 1 | 1 | yes | characteristic |
| 57 | #107×C67 | C23 | 1 | 2 | yes | — |
| 58 | #106×C67 | — | 2 | 1 | no | — |
| 59 | #103×C67 | — | 2 | 2 | no | — |
| 60 | #102×C67 | — | 2 | 2 | no | — |
| 61 | C4×C8×C67 | D4 | 1 | 2 | yes | — |
| 62 | C4×C8×C67 | C23 | 1 | 2 | yes | — |
| 63 | C23×D67 | — | 4 | 2 | no | — |
| 64 | C2×D268 | — | 2 | 4 | no | — |
| 65 | C2×D268 | — | 4 | 4 | no | — |
| 66 | C2×D268 | — | 4 | 4 | no | — |
| 67 | C2×D268 | — | 4 | 4 | no | — |
| 68 | C2×D268 | — | 4 | 4 | no | — |
| 69 | C2×D268 | — | 4 | 2 | no | — |
| 70 | C2×D268 | — | 8 | 8 | no | — |
| 71 | D536 | — | 2 | 4 | no | — |
| 72 | D536 | — | 2 | 4 | no | — |
| 73 | D536 | — | 4 | 8 | no | — |
| 74 | D536 | — | 4 | 4 | no | — |
| 75 | C67⋊C16 | — | 2 | 4 | no | — |
| 76 | #41×C67 | — | 2 | 4 | no | — |
| 77 | #41×C67 | — | 4 | 2 | no | — |
| 78 | C2×Q8×C67 | — | 2 | 1 | no | — |
| 79 | C2×D4×C67 | — | 2 | 1 | no | — |
| 80 | C2×D4×C67 | — | 4 | 1 | no | — |
| 81 | C22×C4×C67 | — | 4 | 1 | no | — |
| 82 | C22×C4×C67 | — | 4 | 2 | no | — |
| 83 | C22×C4×C67 | D8 | 1 | 2 | yes | — |
| 84 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | Q16×C67 | — | 2 | 1 | no | — |
| 86 | Q16×C67 | C2×D4 | 1 | 2 | yes | complemented |
| 87 | SD16×C67 | — | 2 | 4 | no | — |
| 88 | D8×C67 | — | 2 | 1 | no | — |
| 89 | D8×C67 | C2×D4 | 1 | 2 | yes | complemented |
| 90 | C2×C8×C67 | — | 2 | 2 | no | — |
| 91 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 92 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 94 | C2×C8×C67 | #41 | 1 | 2 | yes | — |
| 95 | C2×C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 96 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 97 | C4⋊C4×C67 | — | 4 | 2 | no | — |
| 98 | C22⋊C4×C67 | — | 4 | 1 | no | — |
| 99 | C22⋊C4×C67 | — | 4 | 2 | no | — |
| 100 | C42×C67 | — | 4 | 1 | no | — |
| 101 | C42×C67 | — | 4 | 2 | no | — |
| 102 | C42×C67 | D8 | 1 | 1 | yes | characteristic |
| 103 | C42×C67 | D8 | 1 | 1 | yes | characteristic |
| 104 | C42×C67 | C2×D4 | 1 | 2 | yes | — |
| 105 | C23×C67 | — | 4 | 1 | no | — |
| 106 | C23×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 107 | C22×D67 | — | 4 | 4 | no | — |
| 108 | C22×D67 | — | 8 | 8 | no | — |
| 109 | C22×D67 | — | 8 | 4 | no | — |
| 110 | Q8×C67 | — | 2 | 2 | no | — |
| 111 | Q8×C67 | — | 4 | 1 | no | — |
| 112 | D4×C67 | — | 2 | 2 | no | — |
| 113 | D4×C67 | — | 4 | 4 | no | — |
| 114 | D4×C67 | — | 4 | 1 | no | — |
| 115 | C2×C4×C67 | — | 2 | 4 | no | — |
| 116 | C2×C4×C67 | — | 4 | 1 | no | — |
| 117 | C2×C4×C67 | — | 4 | 4 | no | — |
| 118 | C2×C4×C67 | — | 4 | 2 | no | — |
| 119 | C2×C4×C67 | — | 8 | 4 | no | — |
| 120 | C2×C4×C67 | #121 | 1 | 2 | yes | — |
| 121 | C2×C4×C67 | C2×D8 | 1 | 2 | yes | — |
| 122 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C67 | C2×D8 | 1 | 2 | yes | — |
| 124 | C2×C4×C67 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 125 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 126 | D268 | — | 4 | 4 | no | — |
| 127 | D268 | — | 4 | 4 | no | — |
| 128 | D268 | — | 8 | 8 | no | — |
| 129 | D268 | — | 8 | 4 | no | — |
| 130 | D268 | — | 8 | 8 | no | — |
| 131 | C8×C67 | — | 2 | 2 | no | — |
| 132 | C8×C67 | #121 | 1 | 2 | yes | — |
| 133 | C8×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 134 | C8×C67 | C2×#41 | 1 | 1 | yes | characteristic |
| 135 | C22×C67 | — | 2 | 2 | no | — |
| 136 | C22×C67 | — | 4 | 2 | no | — |
| 137 | C22×C67 | — | 8 | 2 | no | — |
| 138 | C22×C67 | #459 | 1 | 2 | yes | — |
| 139 | C22×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 140 | C2×D67 | — | 8 | 8 | no | — |
| 141 | C2×D67 | — | 8 | 4 | no | — |
| 142 | C2×D67 | — | 16 | 8 | no | — |
| 143 | C4×C67 | — | 2 | 2 | no | — |
| 144 | C4×C67 | — | 4 | 2 | no | — |
| 145 | C4×C67 | — | 8 | 2 | no | — |
| 146 | C4×C67 | #459 | 1 | 2 | yes | — |
| 147 | C4×C67 | #509 | 1 | 4 | yes | complemented |
| 148 | C4×C67 | C2×#121 | 1 | 1 | yes | characteristic |
| 149 | C4×C67 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 150 | #33582 | — | 67 | 1 | no | maximal, Sylow |
| 151 | C2×C67 | — | 2 | 2 | no | — |
| 152 | C2×C67 | — | 8 | 2 | no | — |
| 153 | C2×C67 | C2×#459 | 1 | 1 | yes | characteristic |
| 154 | C2×C67 | C2×#509 | 1 | 2 | yes | — |
| 155 | D67 | — | 16 | 4 | no | — |
| 156 | C2×#509 | — | 67 | 2 | no | — |
| 157 | C2×#493 | — | 67 | 1 | no | — |
| 158 | C4×#135 | D67 | 1 | 1 | yes | characteristic, complemented |
| 159 | #2207 | — | 67 | 4 | no | — |
| 160 | #2206 | — | 67 | 4 | no | — |
| 161 | #2150 | — | 67 | 1 | no | — |
| 162 | #2140 | — | 67 | 2 | no | — |
| 163 | C67 | #33582 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 164 | C2×#135 | C2×D67 | 1 | 1 | yes | characteristic |
| 165 | C22×D8 | — | 67 | 2 | no | — |
| 166 | C22×D8 | — | 134 | 1 | no | — |
| 167 | C2×#127 | — | 134 | 1 | no | — |
| 168 | C4×#41 | — | 2 | 1 | no | — |
| 169 | #509 | — | 134 | 8 | no | — |
| 170 | C2×SD32 | — | 67 | 4 | no | — |
| 171 | C2×D16 | — | 67 | 4 | no | — |
| 172 | C2×M32 | — | 67 | 2 | no | — |
| 173 | #493 | — | 67 | 4 | no | — |
| 174 | #493 | — | 134 | 2 | no | — |
| 175 | C4×Q16 | C2×D67 | 1 | 2 | yes | complemented |
| 176 | C4×SD16 | — | 2 | 2 | no | — |
| 177 | C4×D8 | C2×D67 | 1 | 2 | yes | complemented |
| 178 | #427 | C2×D67 | 1 | 1 | yes | characteristic |
| 179 | #416 | — | 2 | 1 | no | — |
| 180 | C2×C4×C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 181 | C4⋊C16 | — | 67 | 4 | no | — |
| 182 | #357 | — | 67 | 8 | no | — |
| 183 | C2×#41 | — | 2 | 1 | no | — |
| 184 | C22×D4 | — | 268 | 2 | no | — |
| 185 | C22×D4 | — | 134 | 2 | no | — |
| 186 | #135 | — | 2 | 2 | no | — |
| 187 | #135 | D268 | 1 | 4 | yes | complemented |
| 188 | C2×Q16 | C22×D67 | 1 | 1 | yes | characteristic |
| 189 | C2×SD16 | — | 2 | 1 | no | — |
| 190 | C2×D8 | — | 268 | 4 | no | — |
| 191 | C2×D8 | — | 67 | 4 | no | — |
| 192 | C2×D8 | — | 134 | 2 | no | — |
| 193 | C2×D8 | — | 134 | 4 | no | — |
| 194 | C2×D8 | — | 134 | 4 | no | — |
| 195 | C2×D8 | — | 134 | 4 | no | — |
| 196 | C2×D8 | C22×D67 | 1 | 1 | yes | characteristic |
| 197 | C22×C8 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 198 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 199 | #127 | — | 268 | 1 | no | — |
| 200 | #127 | — | 268 | 1 | no | — |
| 201 | #127 | — | 134 | 4 | no | — |
| 202 | C4×Q8 | — | 2 | 2 | no | — |
| 203 | C4×D4 | — | 2 | 2 | no | — |
| 204 | C4×D4 | — | 4 | 2 | no | — |
| 205 | #117 | — | 2 | 1 | no | — |
| 206 | #117 | — | 4 | 1 | no | — |
| 207 | C2×C42 | — | 4 | 1 | no | — |
| 208 | C2×C42 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 209 | SD32 | — | 134 | 8 | no | — |
| 210 | D16 | — | 134 | 8 | no | — |
| 211 | M32 | — | 134 | 4 | no | — |
| 212 | C2×C16 | — | 67 | 4 | no | — |
| 213 | #107 | C22×D67 | 1 | 2 | yes | — |
| 214 | #106 | — | 2 | 1 | no | — |
| 215 | #103 | — | 2 | 2 | no | — |
| 216 | #102 | — | 2 | 2 | no | — |
| 217 | C4×C8 | C22⋊D67 | 1 | 2 | yes | — |
| 218 | C4×C8 | C22×D67 | 1 | 2 | yes | — |
| 219 | C24 | — | 268 | 2 | no | — |
| 220 | #41 | — | 2 | 4 | no | — |
| 221 | #41 | — | 4 | 2 | no | — |
| 222 | C2×Q8 | — | 2 | 1 | no | — |
| 223 | C2×D4 | — | 2 | 1 | no | — |
| 224 | C2×D4 | — | 4 | 1 | no | — |
| 225 | C2×D4 | — | 268 | 4 | no | — |
| 226 | C2×D4 | — | 268 | 2 | no | — |
| 227 | C2×D4 | — | 268 | 4 | no | — |
| 228 | C2×D4 | — | 268 | 4 | no | — |
| 229 | C2×D4 | — | 268 | 4 | no | — |
| 230 | C2×D4 | — | 536 | 8 | no | — |
| 231 | C2×D4 | — | 134 | 4 | no | — |
| 232 | C22×C4 | — | 4 | 1 | no | — |
| 233 | C22×C4 | — | 4 | 2 | no | — |
| 234 | C22×C4 | #11761613 | 1 | 2 | yes | — |
| 235 | C22×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 236 | Q16 | — | 2 | 1 | no | — |
| 237 | Q16 | C2×D268 | 1 | 2 | yes | complemented |
| 238 | SD16 | — | 2 | 4 | no | — |
| 239 | D8 | — | 2 | 1 | no | — |
| 240 | D8 | — | 268 | 8 | no | — |
| 241 | D8 | — | 268 | 4 | no | — |
| 242 | D8 | — | 134 | 4 | no | — |
| 243 | D8 | — | 134 | 4 | no | — |
| 244 | D8 | C2×D268 | 1 | 2 | yes | complemented |
| 245 | C2×C8 | — | 2 | 2 | no | — |
| 246 | C2×C8 | C2×C22⋊D67 | 1 | 2 | yes | — |
| 247 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C2×D268 | 1 | 2 | yes | — |
| 249 | C2×C8 | #11761628 | 1 | 2 | yes | — |
| 250 | C2×C8 | C23×D67 | 1 | 1 | yes | characteristic |
| 251 | C4⋊C4 | — | 2 | 2 | no | — |
| 252 | C4⋊C4 | — | 4 | 2 | no | — |
| 253 | C22⋊C4 | — | 4 | 1 | no | — |
| 254 | C22⋊C4 | — | 4 | 2 | no | — |
| 255 | C42 | — | 4 | 1 | no | — |
| 256 | C42 | — | 4 | 2 | no | — |
| 257 | C42 | #11761613 | 1 | 1 | yes | characteristic |
| 258 | C42 | #11761613 | 1 | 1 | yes | characteristic |
| 259 | C42 | C2×C22⋊D67 | 1 | 2 | yes | — |
| 260 | C16 | — | 134 | 4 | no | — |
| 261 | C23 | — | 4 | 1 | no | — |
| 262 | C23 | — | 268 | 4 | no | — |
| 263 | C23 | — | 536 | 8 | no | — |
| 264 | C23 | — | 536 | 4 | no | — |
| 265 | C23 | C2×#11761613 | 1 | 1 | yes | characteristic |
| 266 | Q8 | — | 2 | 2 | no | — |
| 267 | Q8 | — | 4 | 1 | no | — |
| 268 | D4 | — | 2 | 2 | no | — |
| 269 | D4 | — | 4 | 4 | no | — |
| 270 | D4 | — | 4 | 1 | no | — |
| 271 | D4 | — | 268 | 4 | no | — |
| 272 | D4 | — | 268 | 4 | no | — |
| 273 | D4 | — | 536 | 8 | no | — |
| 274 | D4 | — | 536 | 8 | no | — |
| 275 | D4 | — | 536 | 4 | no | — |
| 276 | C2×C4 | — | 2 | 4 | no | — |
| 277 | C2×C4 | — | 4 | 1 | no | — |
| 278 | C2×C4 | — | 4 | 4 | no | — |
| 279 | C2×C4 | — | 4 | 2 | no | — |
| 280 | C2×C4 | — | 8 | 4 | no | — |
| 281 | C2×C4 | #14524017 | 1 | 2 | yes | — |
| 282 | C2×C4 | C2×#11761613 | 1 | 2 | yes | — |
| 283 | C2×C4 | C2×#11761613 | 1 | 2 | yes | — |
| 284 | C2×C4 | C2×#11761613 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #14524068 | 1 | 2 | yes | — |
| 286 | C2×C4 | C22×C22⋊D67 | 1 | 1 | yes | characteristic |
| 287 | C8 | — | 2 | 2 | no | — |
| 288 | C8 | #14524017 | 1 | 2 | yes | — |
| 289 | C8 | C22×D268 | 1 | 1 | yes | characteristic |
| 290 | C8 | C2×#11761628 | 1 | 1 | yes | characteristic |
| 291 | C22 | — | 2 | 2 | no | — |
| 292 | C22 | — | 4 | 2 | no | — |
| 293 | C22 | — | 8 | 2 | no | — |
| 294 | C22 | — | 536 | 8 | no | — |
| 295 | C22 | — | 536 | 4 | no | — |
| 296 | C22 | — | 1072 | 8 | no | — |
| 297 | C22 | #16863591 | 1 | 2 | yes | — |
| 298 | C22 | C22×#11761613 | 1 | 1 | yes | characteristic |
| 299 | C4 | — | 2 | 2 | no | — |
| 300 | C4 | — | 4 | 2 | no | — |
| 301 | C4 | — | 8 | 2 | no | — |
| 302 | C4 | #16863591 | 1 | 2 | yes | — |
| 303 | C4 | #16863831 | 1 | 4 | yes | complemented |
| 304 | C4 | C2×#14524017 | 1 | 1 | yes | characteristic |
| 305 | C4 | C2×#14524068 | 1 | 1 | yes | characteristic |
| 306 | C2 | — | 2 | 2 | no | — |
| 307 | C2 | — | 8 | 2 | no | — |
| 308 | C2 | — | 1072 | 4 | no | — |
| 309 | C2 | C2×#16863591 | 1 | 1 | yes | characteristic |
| 310 | C2 | C2×#16863831 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta