← #44564761
Subgroups — #44564761
Order 32128.
1114 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #30632435 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #30632431 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #30632391 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #30632355 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #30632335 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #30632289 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #30632267 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #30632264 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #30632243 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #30632236 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #562×C251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#16860811 | C22 | 1 | 2 | yes | — |
| 13 | C2×#16860810 | C22 | 1 | 4 | yes | — |
| 14 | C2×#16860809 | C22 | 1 | 2 | yes | — |
| 15 | C2×#16860809 | C22 | 1 | 4 | yes | — |
| 16 | #22548714 | C22 | 1 | 2 | yes | — |
| 17 | #22548713 | C22 | 1 | 2 | yes | — |
| 18 | #22548711 | C22 | 1 | 2 | yes | — |
| 19 | #22548711 | C22 | 1 | 4 | yes | — |
| 20 | #22548707 | C22 | 1 | 1 | yes | characteristic |
| 21 | #22548705 | C22 | 1 | 4 | yes | — |
| 22 | #22548703 | C22 | 1 | 2 | yes | — |
| 23 | #22548700 | C22 | 1 | 4 | yes | — |
| 24 | #22548699 | C22 | 1 | 4 | yes | — |
| 25 | #22548697 | C22 | 1 | 4 | yes | — |
| 26 | #22548693 | C22 | 1 | 4 | yes | — |
| 27 | #22548693 | C22 | 1 | 4 | yes | — |
| 28 | #22548693 | C22 | 1 | 4 | yes | — |
| 29 | #22548692 | C22 | 1 | 4 | yes | — |
| 30 | #22548692 | C22 | 1 | 4 | yes | — |
| 31 | #22548691 | C22 | 1 | 4 | yes | — |
| 32 | #22548690 | C22 | 1 | 4 | yes | — |
| 33 | #22548690 | C22 | 1 | 4 | yes | — |
| 34 | #22548689 | C22 | 1 | 4 | yes | — |
| 35 | #22548688 | C22 | 1 | 4 | yes | — |
| 36 | #22548688 | C22 | 1 | 4 | yes | — |
| 37 | #22548688 | C22 | 1 | 4 | yes | — |
| 38 | #22548688 | C22 | 1 | 4 | yes | — |
| 39 | #22548688 | C22 | 1 | 4 | yes | — |
| 40 | #22548687 | C22 | 1 | 4 | yes | — |
| 41 | #22548687 | C22 | 1 | 4 | yes | — |
| 42 | #22548687 | C22 | 1 | 4 | yes | — |
| 43 | #22548680 | C22 | 1 | 4 | yes | — |
| 44 | C4×C22⋊D251 | C22 | 1 | 4 | yes | — |
| 45 | (C22⋊D251)⋊C4 | C22 | 1 | 4 | yes | — |
| 46 | (C22⋊D251)⋊C4 | C22 | 1 | 4 | yes | — |
| 47 | (C22⋊D251)⋊C4 | C22 | 1 | 4 | yes | — |
| 48 | #22548673 | C22 | 1 | 2 | yes | — |
| 49 | C2×Dic251⋊C4 | C22 | 1 | 4 | yes | — |
| 50 | C4⋊C4×D251 | C22 | 1 | 4 | yes | — |
| 51 | C42×D251 | C22 | 1 | 1 | yes | characteristic |
| 52 | #125×C251 | C22 | 1 | 1 | yes | characteristic |
| 53 | #124×C251 | C22 | 1 | 2 | yes | — |
| 54 | #123×C251 | C22 | 1 | 4 | yes | — |
| 55 | C22⋊Q8×C251 | C22 | 1 | 4 | yes | — |
| 56 | #121×C251 | C22 | 1 | 4 | yes | — |
| 57 | #16860811 | — | 4 | 4 | no | — |
| 58 | #16860810 | — | 4 | 8 | no | — |
| 59 | #16860809 | — | 4 | 4 | no | — |
| 60 | #16860809 | — | 4 | 8 | no | — |
| 61 | C2×Dic502 | C23 | 1 | 4 | yes | — |
| 62 | C2×Dic502 | C23 | 1 | 2 | yes | — |
| 63 | C2×D1004 | C23 | 1 | 4 | yes | — |
| 64 | C2×D1004 | C23 | 1 | 4 | yes | — |
| 65 | C2×D1004 | C23 | 1 | 2 | yes | — |
| 66 | C2×C22⋊D251 | C23 | 1 | 4 | yes | — |
| 67 | C2×C22⋊D251 | C23 | 1 | 4 | yes | — |
| 68 | C2×C22⋊D251 | C23 | 1 | 4 | yes | — |
| 69 | C2×C22⋊D251 | C23 | 1 | 4 | yes | — |
| 70 | C2×C22⋊D251 | C23 | 1 | 4 | yes | — |
| 71 | C22×Dic251 | C23 | 1 | 4 | yes | — |
| 72 | C22×Dic251 | C23 | 1 | 4 | yes | — |
| 73 | C2×C4×D251 | C23 | 1 | 2 | yes | — |
| 74 | C2×C4×D251 | C23 | 1 | 4 | yes | — |
| 75 | C2×C4×D251 | C23 | 1 | 1 | yes | characteristic |
| 76 | C2×C4×D251 | C23 | 1 | 4 | yes | — |
| 77 | C2×C4×D251 | C23 | 1 | 4 | yes | — |
| 78 | C4⋊Dic251 | C23 | 1 | 4 | yes | — |
| 79 | C4⋊Dic251 | C23 | 1 | 2 | yes | — |
| 80 | Dic251⋊C4 | C23 | 1 | 4 | yes | — |
| 81 | Dic251⋊C4 | C23 | 1 | 4 | yes | — |
| 82 | Dic251⋊C4 | C23 | 1 | 4 | yes | — |
| 83 | Dic251⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | Dic251⋊C4 | C23 | 1 | 4 | yes | — |
| 85 | Dic251⋊C4 | C23 | 1 | 4 | yes | — |
| 86 | C22⋊Dic251 | C23 | 1 | 4 | yes | — |
| 87 | (C2×D251)⋊C4 | C23 | 1 | 4 | yes | — |
| 88 | (C2×D251)⋊C4 | C23 | 1 | 4 | yes | — |
| 89 | (C2×D251)⋊C4 | C23 | 1 | 4 | yes | — |
| 90 | (C2×D251)⋊C4 | C23 | 1 | 4 | yes | — |
| 91 | (C2×D251)⋊C4 | C23 | 1 | 4 | yes | — |
| 92 | C4×Dic251 | C23 | 1 | 1 | yes | characteristic |
| 93 | C4×Dic251 | C23 | 1 | 2 | yes | — |
| 94 | C4×Dic251 | C23 | 1 | 4 | yes | — |
| 95 | C2×Q8×C251 | C23 | 1 | 2 | yes | — |
| 96 | C2×D4×C251 | C23 | 1 | 4 | yes | — |
| 97 | C2×D4×C251 | C23 | 1 | 2 | yes | — |
| 98 | C22×C4×C251 | C23 | 1 | 4 | yes | — |
| 99 | C4⋊C4×C251 | C23 | 1 | 4 | yes | — |
| 100 | C4⋊C4×C251 | C23 | 1 | 4 | yes | — |
| 101 | C4⋊C4×C251 | C23 | 1 | 2 | yes | — |
| 102 | C22⋊C4×C251 | C23 | 1 | 4 | yes | — |
| 103 | C22⋊C4×C251 | C23 | 1 | 4 | yes | — |
| 104 | C22⋊C4×C251 | C23 | 1 | 4 | yes | — |
| 105 | C42×C251 | C23 | 1 | 1 | yes | characteristic |
| 106 | C22×D251 | C24 | 1 | 1 | yes | characteristic |
| 107 | C22×D251 | C24 | 1 | 4 | yes | — |
| 108 | Dic502 | — | 2 | 8 | no | — |
| 109 | Dic502 | — | 4 | 2 | no | — |
| 110 | D1004 | — | 4 | 4 | no | — |
| 111 | D1004 | — | 4 | 4 | no | — |
| 112 | D1004 | — | 4 | 2 | no | — |
| 113 | C22⋊D251 | — | 2 | 8 | no | — |
| 114 | C22⋊D251 | — | 2 | 8 | no | — |
| 115 | C22⋊D251 | — | 4 | 4 | no | — |
| 116 | C22⋊D251 | — | 4 | 4 | no | — |
| 117 | C22⋊D251 | — | 4 | 4 | no | — |
| 118 | C2×Dic251 | — | 2 | 8 | no | — |
| 119 | C2×Dic251 | — | 4 | 4 | no | — |
| 120 | C2×Dic251 | C24 | 1 | 4 | yes | — |
| 121 | C2×Dic251 | C24 | 1 | 4 | yes | — |
| 122 | C2×Dic251 | C24 | 1 | 2 | yes | — |
| 123 | C2×Dic251 | C24 | 1 | 1 | yes | characteristic |
| 124 | C4×D251 | — | 2 | 4 | no | — |
| 125 | C4×D251 | — | 2 | 8 | no | — |
| 126 | C4×D251 | — | 2 | 2 | no | — |
| 127 | C4×D251 | — | 4 | 4 | no | — |
| 128 | C4×D251 | — | 4 | 4 | no | — |
| 129 | C23×C251 | C24 | 1 | 4 | yes | — |
| 130 | Q8×C251 | — | 4 | 2 | no | — |
| 131 | D4×C251 | — | 4 | 4 | no | — |
| 132 | D4×C251 | — | 4 | 2 | no | — |
| 133 | C2×C4×C251 | — | 4 | 4 | no | — |
| 134 | C2×C4×C251 | C24 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C251 | C24 | 1 | 2 | yes | — |
| 136 | C2×C4×C251 | C24 | 1 | 4 | yes | — |
| 137 | C2×C4×C251 | C24 | 1 | 4 | yes | — |
| 138 | C22×C251 | — | 2 | 4 | no | — |
| 139 | C22×C251 | — | 2 | 4 | no | — |
| 140 | C22×C251 | — | 2 | 4 | no | — |
| 141 | C22×C251 | C25 | 1 | 1 | yes | characteristic |
| 142 | C2×D251 | — | 2 | 2 | no | — |
| 143 | C2×D251 | — | 2 | 4 | no | — |
| 144 | C2×D251 | — | 2 | 4 | no | — |
| 145 | C2×D251 | — | 2 | 1 | no | — |
| 146 | C2×D251 | — | 2 | 4 | no | — |
| 147 | C4×C251 | — | 2 | 4 | no | — |
| 148 | C4×C251 | — | 2 | 4 | no | — |
| 149 | C4×C251 | — | 2 | 2 | no | — |
| 150 | C4×C251 | — | 2 | 1 | no | — |
| 151 | Dic251 | — | 2 | 4 | no | — |
| 152 | Dic251 | — | 2 | 4 | no | — |
| 153 | Dic251 | H(5,𝔽2) | 1 | 4 | yes | — |
| 154 | Dic251 | 2−1+4 | 1 | 2 | yes | — |
| 155 | C2×C251 | — | 4 | 4 | no | — |
| 156 | C2×C251 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 157 | C2×C251 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 158 | D251 | — | 4 | 1 | no | — |
| 159 | D251 | — | 4 | 4 | no | — |
| 160 | C251 | #3548 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 161 | #3548 | — | 251 | 1 | no | maximal, Sylow |
| 162 | #562 | — | 251 | 1 | no | — |
| 163 | #562 | — | 251 | 2 | no | — |
| 164 | #562 | D251 | 1 | 1 | yes | characteristic, complemented |
| 165 | #554 | — | 251 | 4 | no | — |
| 166 | #550 | — | 251 | 4 | no | — |
| 167 | #547 | — | 251 | 4 | no | — |
| 168 | #540 | — | 251 | 4 | no | — |
| 169 | #537 | — | 251 | 4 | no | — |
| 170 | #537 | — | 251 | 4 | no | — |
| 171 | #533 | — | 251 | 1 | no | — |
| 172 | #532 | — | 251 | 2 | no | — |
| 173 | C2×#41 | — | 251 | 2 | no | — |
| 174 | C2×#41 | — | 251 | 2 | no | — |
| 175 | C2×#41 | — | 251 | 4 | no | — |
| 176 | C2×#41 | — | 251 | 4 | no | — |
| 177 | C4⋊Q8 | — | 251 | 2 | no | — |
| 178 | C4⋊Q8 | — | 251 | 2 | no | — |
| 179 | #127 | — | 251 | 2 | no | — |
| 180 | #127 | — | 251 | 4 | no | — |
| 181 | #125 | — | 251 | 4 | no | — |
| 182 | #125 | — | 251 | 1 | no | — |
| 183 | #125 | C2×D251 | 1 | 1 | yes | characteristic |
| 184 | #124 | — | 251 | 2 | no | — |
| 185 | #124 | C2×D251 | 1 | 2 | yes | — |
| 186 | #123 | — | 251 | 4 | no | — |
| 187 | #123 | — | 251 | 4 | no | — |
| 188 | #123 | — | 251 | 4 | no | — |
| 189 | #123 | C2×D251 | 1 | 4 | yes | — |
| 190 | C22⋊Q8 | — | 251 | 4 | no | — |
| 191 | C22⋊Q8 | — | 251 | 4 | no | — |
| 192 | C22⋊Q8 | — | 251 | 4 | no | — |
| 193 | C22⋊Q8 | — | 251 | 4 | no | — |
| 194 | C22⋊Q8 | — | 251 | 4 | no | — |
| 195 | C22⋊Q8 | — | 251 | 4 | no | — |
| 196 | C22⋊Q8 | C2×D251 | 1 | 4 | yes | — |
| 197 | #121 | — | 251 | 4 | no | — |
| 198 | #121 | — | 251 | 4 | no | — |
| 199 | #121 | — | 251 | 4 | no | — |
| 200 | #121 | — | 251 | 4 | no | — |
| 201 | #121 | — | 251 | 4 | no | — |
| 202 | #121 | — | 251 | 4 | no | — |
| 203 | #121 | — | 251 | 4 | no | — |
| 204 | #121 | — | 251 | 4 | no | — |
| 205 | #121 | — | 251 | 4 | no | — |
| 206 | #121 | — | 251 | 4 | no | — |
| 207 | #121 | — | 251 | 4 | no | — |
| 208 | #121 | C2×D251 | 1 | 4 | yes | — |
| 209 | C4×Q8 | — | 251 | 4 | no | — |
| 210 | C4×D4 | — | 251 | 4 | no | — |
| 211 | C4×D4 | — | 251 | 4 | no | — |
| 212 | C4×D4 | — | 251 | 4 | no | — |
| 213 | C4×D4 | — | 251 | 4 | no | — |
| 214 | #117 | — | 251 | 2 | no | — |
| 215 | C2×C4⋊C4 | — | 251 | 4 | no | — |
| 216 | C2×C4⋊C4 | — | 251 | 4 | no | — |
| 217 | C2×C42 | — | 251 | 1 | no | — |
| 218 | #41 | — | 1004 | 8 | no | — |
| 219 | #41 | — | 1004 | 8 | no | — |
| 220 | #41 | — | 1004 | 4 | no | — |
| 221 | #41 | — | 1004 | 4 | no | — |
| 222 | C2×Q8 | — | 251 | 2 | no | — |
| 223 | C2×Q8 | — | 251 | 4 | no | — |
| 224 | C2×Q8 | C22×D251 | 1 | 2 | yes | — |
| 225 | C2×D4 | — | 251 | 4 | no | — |
| 226 | C2×D4 | — | 251 | 4 | no | — |
| 227 | C2×D4 | — | 251 | 4 | no | — |
| 228 | C2×D4 | — | 251 | 4 | no | — |
| 229 | C2×D4 | — | 251 | 4 | no | — |
| 230 | C2×D4 | — | 251 | 4 | no | — |
| 231 | C2×D4 | — | 251 | 4 | no | — |
| 232 | C2×D4 | — | 251 | 2 | no | — |
| 233 | C2×D4 | C22×D251 | 1 | 4 | yes | — |
| 234 | C2×D4 | C22×D251 | 1 | 2 | yes | — |
| 235 | C22×C4 | — | 251 | 4 | no | — |
| 236 | C22×C4 | — | 251 | 4 | no | — |
| 237 | C22×C4 | — | 251 | 4 | no | — |
| 238 | C22×C4 | — | 251 | 4 | no | — |
| 239 | C22×C4 | — | 251 | 1 | no | — |
| 240 | C22×C4 | — | 251 | 2 | no | — |
| 241 | C22×C4 | — | 251 | 4 | no | — |
| 242 | C22×C4 | C22×D251 | 1 | 4 | yes | — |
| 243 | C4⋊C4 | — | 251 | 4 | no | — |
| 244 | C4⋊C4 | — | 251 | 4 | no | — |
| 245 | C4⋊C4 | — | 251 | 4 | no | — |
| 246 | C4⋊C4 | — | 251 | 4 | no | — |
| 247 | C4⋊C4 | — | 251 | 4 | no | — |
| 248 | C4⋊C4 | — | 251 | 2 | no | — |
| 249 | C4⋊C4 | — | 251 | 4 | no | — |
| 250 | C4⋊C4 | — | 251 | 4 | no | — |
| 251 | C4⋊C4 | C22×D251 | 1 | 4 | yes | — |
| 252 | C4⋊C4 | C22×D251 | 1 | 2 | yes | — |
| 253 | C4⋊C4 | C22×D251 | 1 | 4 | yes | — |
| 254 | C22⋊C4 | — | 251 | 4 | no | — |
| 255 | C22⋊C4 | — | 251 | 4 | no | — |
| 256 | C22⋊C4 | — | 251 | 4 | no | — |
| 257 | C22⋊C4 | — | 251 | 4 | no | — |
| 258 | C22⋊C4 | — | 251 | 4 | no | — |
| 259 | C22⋊C4 | — | 251 | 4 | no | — |
| 260 | C22⋊C4 | C22×D251 | 1 | 4 | yes | — |
| 261 | C22⋊C4 | C22×D251 | 1 | 4 | yes | — |
| 262 | C22⋊C4 | C22×D251 | 1 | 4 | yes | — |
| 263 | C42 | — | 251 | 1 | no | — |
| 264 | C42 | — | 251 | 2 | no | — |
| 265 | C42 | — | 251 | 4 | no | — |
| 266 | C42 | C22×D251 | 1 | 1 | yes | characteristic |
| 267 | C23 | — | 251 | 4 | no | — |
| 268 | C23 | — | 251 | 1 | no | — |
| 269 | C23 | C23×D251 | 1 | 4 | yes | — |
| 270 | Q8 | — | 4 | 2 | no | — |
| 271 | Q8 | — | 1004 | 2 | no | — |
| 272 | Q8 | — | 502 | 8 | no | — |
| 273 | D4 | — | 4 | 4 | no | — |
| 274 | D4 | — | 4 | 2 | no | — |
| 275 | D4 | — | 1004 | 4 | no | — |
| 276 | D4 | — | 1004 | 4 | no | — |
| 277 | D4 | — | 1004 | 2 | no | — |
| 278 | D4 | — | 1004 | 4 | no | — |
| 279 | D4 | — | 1004 | 4 | no | — |
| 280 | D4 | — | 1004 | 4 | no | — |
| 281 | D4 | — | 502 | 8 | no | — |
| 282 | D4 | — | 502 | 8 | no | — |
| 283 | C2×C4 | — | 4 | 4 | no | — |
| 284 | C2×C4 | — | 1004 | 4 | no | — |
| 285 | C2×C4 | — | 1004 | 4 | no | — |
| 286 | C2×C4 | — | 1004 | 4 | no | — |
| 287 | C2×C4 | — | 502 | 8 | no | — |
| 288 | C2×C4 | — | 502 | 4 | no | — |
| 289 | C2×C4 | — | 502 | 2 | no | — |
| 290 | C2×C4 | — | 502 | 8 | no | — |
| 291 | C2×C4 | — | 251 | 4 | no | — |
| 292 | C2×C4 | — | 251 | 4 | no | — |
| 293 | C2×C4 | — | 251 | 1 | no | — |
| 294 | C2×C4 | — | 251 | 2 | no | — |
| 295 | C2×C4 | C23×D251 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C23×D251 | 1 | 2 | yes | — |
| 297 | C2×C4 | C23×D251 | 1 | 4 | yes | — |
| 298 | C2×C4 | C23×D251 | 1 | 4 | yes | — |
| 299 | C22 | — | 2 | 4 | no | — |
| 300 | C22 | — | 2 | 4 | no | — |
| 301 | C22 | — | 2 | 4 | no | — |
| 302 | C22 | — | 502 | 4 | no | — |
| 303 | C22 | — | 502 | 4 | no | — |
| 304 | C22 | — | 502 | 2 | no | — |
| 305 | C22 | — | 502 | 1 | no | — |
| 306 | C22 | — | 502 | 4 | no | — |
| 307 | C22 | C24×D251 | 1 | 1 | yes | characteristic |
| 308 | C4 | — | 2 | 4 | no | — |
| 309 | C4 | — | 2 | 4 | no | — |
| 310 | C4 | — | 2 | 2 | no | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 502 | 4 | no | — |
| 313 | C4 | — | 502 | 4 | no | — |
| 314 | C4 | — | 251 | 4 | no | — |
| 315 | C4 | — | 251 | 2 | no | — |
| 316 | C2 | — | 4 | 4 | no | — |
| 317 | C2 | — | 1004 | 4 | no | — |
| 318 | C2 | — | 1004 | 1 | no | — |
| 319 | C2 | H(5,𝔽2)×D251 | 1 | 2 | yes | — |
| 320 | C2 | 2−1+4×D251 | 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