← #137×Dic251 #44559186
Subgroups — #137×Dic251 #44559186
Order 32128.
484 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #41×Dic251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×Q8×Dic251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #30631794 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #30631788 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | Q16×Dic251 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | SD16×Dic251 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #30631729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #30631696 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #30631688 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | M16×Dic251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C2×#137×C251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×#16860794 | C22 | 1 | 1 | yes | characteristic |
| 13 | Q8×Dic251 | — | 2 | 1 | no | — |
| 14 | Q8×Dic251 | — | 2 | 1 | no | — |
| 15 | Q8×Dic251 | C22 | 1 | 2 | yes | — |
| 16 | Q8×Dic251 | C22 | 1 | 1 | yes | characteristic |
| 17 | Q8×Dic251 | C22 | 1 | 2 | yes | — |
| 18 | Q8×Dic251 | C22 | 1 | 1 | yes | characteristic |
| 19 | D4×Dic251 | — | 2 | 1 | no | — |
| 20 | D4×Dic251 | — | 2 | 1 | no | — |
| 21 | D4×Dic251 | C22 | 1 | 1 | yes | characteristic |
| 22 | D4×Dic251 | C22 | 1 | 1 | yes | characteristic |
| 23 | #22548674 | — | 2 | 1 | no | — |
| 24 | #22548674 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×C4⋊Dic251 | — | 2 | 1 | no | — |
| 26 | C2×C4⋊Dic251 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×Dic251 | — | 2 | 1 | no | — |
| 28 | C2×C4×Dic251 | — | 2 | 1 | no | — |
| 29 | C2×C4×Dic251 | C22 | 1 | 1 | yes | characteristic |
| 30 | #22548649 | C22 | 1 | 2 | yes | — |
| 31 | #22548647 | C22 | 1 | 2 | yes | — |
| 32 | #22548639 | C22 | 1 | 2 | yes | — |
| 33 | #22548639 | C22 | 1 | 2 | yes | — |
| 34 | #22548639 | C22 | 1 | 2 | yes | — |
| 35 | #22548636 | C22 | 1 | 2 | yes | — |
| 36 | #22548625 | C22 | 1 | 2 | yes | — |
| 37 | C8×Dic251 | C22 | 1 | 2 | yes | — |
| 38 | C2×#41×C251 | C22 | 1 | 1 | yes | characteristic |
| 39 | C22×Q8×C251 | C22 | 1 | 1 | yes | characteristic |
| 40 | #137×C251 | C4 | 1 | 8 | yes | complemented |
| 41 | C2×Q16×C251 | C22 | 1 | 2 | yes | — |
| 42 | C2×SD16×C251 | C22 | 1 | 2 | yes | — |
| 43 | C2×M16×C251 | C22 | 1 | 1 | yes | characteristic |
| 44 | C22×Dic251 | — | 2 | 1 | no | — |
| 45 | C22×Dic251 | — | 2 | 1 | no | — |
| 46 | C22×Dic251 | — | 2 | 1 | no | — |
| 47 | C22×Dic251 | D4 | 1 | 1 | yes | characteristic |
| 48 | C22×Dic251 | D4 | 1 | 1 | yes | characteristic |
| 49 | #16860794 | — | 2 | 2 | no | — |
| 50 | C2×C251⋊C8 | C23 | 1 | 2 | yes | — |
| 51 | C4⋊Dic251 | — | 2 | 2 | no | — |
| 52 | C4⋊Dic251 | — | 2 | 2 | no | — |
| 53 | C4⋊Dic251 | — | 2 | 1 | no | — |
| 54 | C4⋊Dic251 | — | 2 | 1 | no | — |
| 55 | C4⋊Dic251 | — | 2 | 1 | no | — |
| 56 | C4⋊Dic251 | C23 | 1 | 2 | yes | — |
| 57 | C4⋊Dic251 | C23 | 1 | 1 | yes | characteristic |
| 58 | C4⋊Dic251 | C23 | 1 | 1 | yes | characteristic |
| 59 | C22⋊Dic251 | — | 2 | 1 | no | — |
| 60 | C22⋊Dic251 | — | 2 | 1 | no | — |
| 61 | C22⋊Dic251 | — | 2 | 1 | no | — |
| 62 | C4×Dic251 | — | 2 | 1 | no | — |
| 63 | C4×Dic251 | — | 2 | 1 | no | — |
| 64 | C4×Dic251 | — | 2 | 2 | no | — |
| 65 | C4×Dic251 | — | 2 | 2 | no | — |
| 66 | C4×Dic251 | — | 2 | 1 | no | — |
| 67 | C4×Dic251 | D4 | 1 | 1 | yes | characteristic |
| 68 | C4×Dic251 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×Dic251 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4×Dic251 | C23 | 1 | 1 | yes | characteristic |
| 71 | #41×C251 | — | 2 | 2 | no | — |
| 72 | #41×C251 | C2×C4 | 1 | 4 | yes | — |
| 73 | C2×Q8×C251 | — | 2 | 1 | no | — |
| 74 | C2×Q8×C251 | — | 2 | 2 | no | — |
| 75 | C2×Q8×C251 | C2×C4 | 1 | 4 | yes | complemented |
| 76 | C2×Q8×C251 | C23 | 1 | 2 | yes | — |
| 77 | C2×Q8×C251 | C23 | 1 | 1 | yes | characteristic |
| 78 | C2×D4×C251 | — | 2 | 1 | no | — |
| 79 | C2×D4×C251 | C23 | 1 | 1 | yes | characteristic |
| 80 | C22×C4×C251 | — | 2 | 1 | no | — |
| 81 | C22×C4×C251 | — | 2 | 1 | no | — |
| 82 | C22×C4×C251 | C23 | 1 | 1 | yes | characteristic |
| 83 | Q16×C251 | C2×C4 | 1 | 8 | yes | complemented |
| 84 | SD16×C251 | C2×C4 | 1 | 8 | yes | complemented |
| 85 | M16×C251 | C2×C4 | 1 | 4 | yes | complemented |
| 86 | C2×C8×C251 | C23 | 1 | 2 | yes | — |
| 87 | C2×Dic251 | — | 2 | 2 | no | — |
| 88 | C2×Dic251 | — | 2 | 1 | no | — |
| 89 | C2×Dic251 | — | 2 | 1 | no | — |
| 90 | C2×Dic251 | — | 2 | 2 | no | — |
| 91 | C2×Dic251 | — | 4 | 1 | no | — |
| 92 | C2×Dic251 | — | 4 | 2 | no | — |
| 93 | C2×Dic251 | — | 4 | 2 | no | — |
| 94 | C2×Dic251 | — | 4 | 2 | no | — |
| 95 | C2×Dic251 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×Dic251 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×Dic251 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×Dic251 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C251⋊C8 | — | 2 | 2 | no | — |
| 100 | C23×C251 | — | 2 | 1 | no | — |
| 101 | C23×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | Q8×C251 | — | 2 | 4 | no | — |
| 103 | Q8×C251 | — | 2 | 2 | no | — |
| 104 | Q8×C251 | — | 2 | 1 | no | — |
| 105 | Q8×C251 | C22×C4 | 1 | 4 | yes | — |
| 106 | Q8×C251 | C22×C4 | 1 | 2 | yes | — |
| 107 | D4×C251 | — | 2 | 4 | no | — |
| 108 | D4×C251 | — | 2 | 1 | no | — |
| 109 | D4×C251 | C22×C4 | 1 | 2 | yes | — |
| 110 | C2×C4×C251 | — | 2 | 4 | no | — |
| 111 | C2×C4×C251 | — | 2 | 1 | no | — |
| 112 | C2×C4×C251 | — | 2 | 2 | no | — |
| 113 | C2×C4×C251 | — | 2 | 4 | no | — |
| 114 | C2×C4×C251 | C22×C4 | 1 | 2 | yes | — |
| 115 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C251 | #41 | 1 | 2 | yes | — |
| 117 | C2×C4×C251 | C24 | 1 | 1 | yes | characteristic |
| 118 | C8×C251 | C22×C4 | 1 | 4 | yes | — |
| 119 | C22×C251 | — | 2 | 1 | no | — |
| 120 | C22×C251 | — | 2 | 1 | no | — |
| 121 | C22×C251 | — | 2 | 2 | no | — |
| 122 | C22×C251 | — | 4 | 1 | no | — |
| 123 | C22×C251 | — | 4 | 1 | no | — |
| 124 | C22×C251 | C4×D4 | 1 | 2 | yes | — |
| 125 | C22×C251 | C22×D4 | 1 | 1 | yes | characteristic |
| 126 | C4×C251 | — | 2 | 2 | no | — |
| 127 | C4×C251 | — | 2 | 4 | no | — |
| 128 | C4×C251 | C4×D4 | 1 | 2 | yes | — |
| 129 | C4×C251 | C23×C4 | 1 | 1 | yes | characteristic |
| 130 | C4×C251 | C2×#41 | 1 | 1 | yes | characteristic |
| 131 | Dic251 | — | 2 | 1 | no | — |
| 132 | Dic251 | — | 2 | 1 | no | — |
| 133 | Dic251 | — | 2 | 1 | no | — |
| 134 | Dic251 | — | 4 | 1 | no | — |
| 135 | Dic251 | — | 4 | 1 | no | — |
| 136 | Dic251 | — | 4 | 2 | no | — |
| 137 | Dic251 | #137 | 1 | 2 | yes | direct product factor |
| 138 | C2×C251 | — | 2 | 2 | no | — |
| 139 | C2×C251 | — | 4 | 2 | no | — |
| 140 | C2×C251 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 141 | C2×C251 | C2×#137 | 1 | 1 | yes | characteristic |
| 142 | C2×C251 | #575 | 1 | 1 | yes | characteristic |
| 143 | C251 | C4×#137 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 144 | C4×#137 | — | 251 | 1 | no | maximal, Sylow |
| 145 | C2×#137 | D251 | 1 | 1 | yes | characteristic |
| 146 | C4×#41 | — | 251 | 1 | no | — |
| 147 | C2×C4×Q8 | — | 251 | 1 | no | — |
| 148 | #441 | — | 251 | 2 | no | — |
| 149 | #440 | — | 251 | 2 | no | — |
| 150 | C4×Q16 | — | 251 | 2 | no | — |
| 151 | C4×SD16 | — | 251 | 2 | no | — |
| 152 | #428 | — | 251 | 1 | no | — |
| 153 | #419 | — | 251 | 1 | no | — |
| 154 | #417 | — | 251 | 1 | no | — |
| 155 | C4×M16 | — | 251 | 1 | no | — |
| 156 | C2×#41 | C2×D251 | 1 | 1 | yes | characteristic |
| 157 | C22×Q8 | C2×D251 | 1 | 1 | yes | characteristic |
| 158 | #137 | Dic251 | 1 | 8 | yes | direct product factor |
| 159 | C2×Q16 | C2×D251 | 1 | 2 | yes | — |
| 160 | C2×SD16 | C2×D251 | 1 | 2 | yes | — |
| 161 | C2×M16 | — | 251 | 1 | no | — |
| 162 | C2×M16 | C2×D251 | 1 | 1 | yes | characteristic |
| 163 | C4×Q8 | — | 502 | 1 | no | — |
| 164 | C4×Q8 | — | 502 | 1 | no | — |
| 165 | C4×Q8 | — | 251 | 2 | no | — |
| 166 | C4×Q8 | — | 251 | 2 | no | — |
| 167 | C4×Q8 | — | 251 | 1 | no | — |
| 168 | C4×Q8 | — | 251 | 1 | no | — |
| 169 | C4×D4 | — | 502 | 1 | no | — |
| 170 | C4×D4 | — | 502 | 1 | no | — |
| 171 | C4×D4 | — | 251 | 1 | no | — |
| 172 | C4×D4 | — | 251 | 1 | no | — |
| 173 | #117 | — | 502 | 1 | no | — |
| 174 | #117 | — | 251 | 1 | no | — |
| 175 | C2×C4⋊C4 | — | 502 | 1 | no | — |
| 176 | C2×C4⋊C4 | — | 251 | 1 | no | — |
| 177 | C2×C42 | — | 502 | 1 | no | — |
| 178 | C2×C42 | — | 502 | 1 | no | — |
| 179 | C2×C42 | — | 251 | 1 | no | — |
| 180 | #107 | — | 251 | 2 | no | — |
| 181 | #106 | — | 251 | 2 | no | — |
| 182 | #103 | — | 251 | 2 | no | — |
| 183 | #103 | — | 251 | 2 | no | — |
| 184 | #103 | — | 251 | 2 | no | — |
| 185 | #102 | — | 251 | 2 | no | — |
| 186 | #97 | — | 251 | 2 | no | — |
| 187 | C4×C8 | — | 251 | 2 | no | — |
| 188 | #41 | — | 2 | 2 | no | — |
| 189 | #41 | C2×Dic251 | 1 | 4 | yes | — |
| 190 | C2×Q8 | — | 2 | 1 | no | — |
| 191 | C2×Q8 | — | 2 | 2 | no | — |
| 192 | C2×Q8 | C2×Dic251 | 1 | 4 | yes | complemented |
| 193 | C2×Q8 | C22×D251 | 1 | 2 | yes | — |
| 194 | C2×Q8 | C22×D251 | 1 | 1 | yes | characteristic |
| 195 | C2×D4 | — | 2 | 1 | no | — |
| 196 | C2×D4 | C22×D251 | 1 | 1 | yes | characteristic |
| 197 | C22×C4 | — | 2 | 1 | no | — |
| 198 | C22×C4 | — | 2 | 1 | no | — |
| 199 | C22×C4 | — | 502 | 1 | no | — |
| 200 | C22×C4 | — | 502 | 1 | no | — |
| 201 | C22×C4 | — | 502 | 1 | no | — |
| 202 | C22×C4 | — | 251 | 1 | no | — |
| 203 | C22×C4 | — | 251 | 1 | no | — |
| 204 | C22×C4 | C22×D251 | 1 | 1 | yes | characteristic |
| 205 | Q16 | C2×Dic251 | 1 | 8 | yes | complemented |
| 206 | SD16 | C2×Dic251 | 1 | 8 | yes | complemented |
| 207 | M16 | — | 502 | 2 | no | — |
| 208 | M16 | C2×Dic251 | 1 | 4 | yes | complemented |
| 209 | C2×C8 | — | 251 | 2 | no | — |
| 210 | C2×C8 | C22×D251 | 1 | 2 | yes | — |
| 211 | C4⋊C4 | — | 502 | 2 | no | — |
| 212 | C4⋊C4 | — | 502 | 2 | no | — |
| 213 | C4⋊C4 | — | 502 | 1 | no | — |
| 214 | C4⋊C4 | — | 502 | 1 | no | — |
| 215 | C4⋊C4 | — | 502 | 1 | no | — |
| 216 | C4⋊C4 | — | 251 | 2 | no | — |
| 217 | C4⋊C4 | — | 251 | 1 | no | — |
| 218 | C4⋊C4 | — | 251 | 1 | no | — |
| 219 | C22⋊C4 | — | 502 | 1 | no | — |
| 220 | C22⋊C4 | — | 502 | 1 | no | — |
| 221 | C22⋊C4 | — | 502 | 1 | no | — |
| 222 | C42 | — | 502 | 1 | no | — |
| 223 | C42 | — | 502 | 1 | no | — |
| 224 | C42 | — | 502 | 2 | no | — |
| 225 | C42 | — | 502 | 2 | no | — |
| 226 | C42 | — | 502 | 1 | no | — |
| 227 | C42 | — | 251 | 1 | no | — |
| 228 | C42 | — | 251 | 1 | no | — |
| 229 | C42 | — | 251 | 1 | no | — |
| 230 | C42 | — | 251 | 1 | no | — |
| 231 | C23 | — | 2 | 1 | no | — |
| 232 | C23 | D4×D251 | 1 | 1 | yes | characteristic |
| 233 | Q8 | — | 2 | 2 | no | — |
| 234 | Q8 | — | 2 | 1 | no | — |
| 235 | Q8 | — | 2 | 4 | no | — |
| 236 | Q8 | C22×Dic251 | 1 | 4 | yes | — |
| 237 | Q8 | C22×Dic251 | 1 | 2 | yes | — |
| 238 | D4 | — | 2 | 4 | no | — |
| 239 | D4 | — | 2 | 1 | no | — |
| 240 | D4 | C22×Dic251 | 1 | 2 | yes | — |
| 241 | C2×C4 | — | 2 | 1 | no | — |
| 242 | C2×C4 | — | 2 | 2 | no | — |
| 243 | C2×C4 | — | 2 | 4 | no | — |
| 244 | C2×C4 | — | 2 | 4 | no | — |
| 245 | C2×C4 | — | 1004 | 2 | no | — |
| 246 | C2×C4 | — | 1004 | 2 | no | — |
| 247 | C2×C4 | — | 1004 | 2 | no | — |
| 248 | C2×C4 | — | 1004 | 1 | no | — |
| 249 | C2×C4 | — | 502 | 1 | no | — |
| 250 | C2×C4 | — | 502 | 1 | no | — |
| 251 | C2×C4 | — | 502 | 2 | no | — |
| 252 | C2×C4 | — | 502 | 2 | no | — |
| 253 | C2×C4 | — | 251 | 1 | no | — |
| 254 | C2×C4 | — | 251 | 1 | no | — |
| 255 | C2×C4 | — | 251 | 1 | no | — |
| 256 | C2×C4 | — | 251 | 1 | no | — |
| 257 | C2×C4 | C22×Dic251 | 1 | 2 | yes | — |
| 258 | C2×C4 | D4×D251 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | #16860809 | 1 | 2 | yes | — |
| 260 | C2×C4 | C23×D251 | 1 | 1 | yes | characteristic |
| 261 | C8 | — | 502 | 2 | no | — |
| 262 | C8 | C22×Dic251 | 1 | 4 | yes | — |
| 263 | C22 | — | 2 | 1 | no | — |
| 264 | C22 | — | 2 | 1 | no | — |
| 265 | C22 | — | 2 | 2 | no | — |
| 266 | C22 | — | 4 | 1 | no | — |
| 267 | C22 | — | 4 | 1 | no | — |
| 268 | C22 | D4×Dic251 | 1 | 2 | yes | — |
| 269 | C22 | C2×D4×D251 | 1 | 1 | yes | characteristic |
| 270 | C4 | — | 2 | 2 | no | — |
| 271 | C4 | — | 2 | 4 | no | — |
| 272 | C4 | — | 1004 | 1 | no | — |
| 273 | C4 | — | 1004 | 2 | no | — |
| 274 | C4 | — | 1004 | 1 | no | — |
| 275 | C4 | — | 502 | 1 | no | — |
| 276 | C4 | — | 502 | 1 | no | — |
| 277 | C4 | — | 502 | 1 | no | — |
| 278 | C4 | — | 251 | 2 | no | — |
| 279 | C4 | D4×Dic251 | 1 | 2 | yes | — |
| 280 | C4 | C23×Dic251 | 1 | 1 | yes | characteristic |
| 281 | C4 | C2×#16860809 | 1 | 1 | yes | characteristic |
| 282 | C2 | — | 2 | 2 | no | — |
| 283 | C2 | — | 4 | 2 | no | — |
| 284 | C2 | C2×D4×Dic251 | 1 | 1 | yes | characteristic |
| 285 | C2 | #137×D251 | 1 | 1 | yes | characteristic |
| 286 | C2 | #30632486 | 1 | 1 | yes | characteristic |
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