← #30606723
Subgroups — #30606723
Order 15616.
318 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 | #21462152 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16840653 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#16840652 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #21461613 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #21460859 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #21460859 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #21447412 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#14519852 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14519852 | C22 | 1 | 1 | yes | characteristic |
| 10 | D61⋊(C4⋊C8) | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#14519831 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#14519831 | C22 | 1 | 1 | yes | characteristic |
| 13 | C4×C4⋊#6923 | C4 | 1 | 1 | yes | characteristic |
| 14 | #16840683 | C4 | 1 | 2 | yes | complemented |
| 15 | C4⋊#11758187 | C4 | 1 | 2 | yes | complemented |
| 16 | #16840653 | — | 2 | 2 | no | — |
| 17 | #16840652 | — | 2 | 2 | no | — |
| 18 | (C2×#6923)⋊C8 | C4 | 1 | 1 | yes | characteristic |
| 19 | (C2×#6923)⋊C8 | C4 | 1 | 1 | yes | characteristic |
| 20 | C2×#14519716 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#14519716 | C22 | 1 | 1 | yes | characteristic |
| 22 | C4×C8×D61 | C22 | 1 | 1 | yes | characteristic |
| 23 | #16839538 | C4 | 1 | 1 | yes | characteristic, complemented |
| 24 | C16×Dic61 | C4 | 1 | 1 | yes | characteristic, complemented |
| 25 | C16×Dic61 | C4 | 1 | 1 | yes | characteristic, complemented |
| 26 | C61×#346 | C4 | 1 | 1 | yes | characteristic, complemented |
| 27 | C2×D61⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×D61⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×C4⋊#6923 | — | 2 | 1 | no | — |
| 30 | C2×C4×#6923 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C2×C4×#6923 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | #14519852 | — | 2 | 1 | no | — |
| 33 | #14519852 | — | 2 | 1 | no | — |
| 34 | #14519852 | — | 2 | 1 | no | — |
| 35 | #14519852 | — | 2 | 1 | no | — |
| 36 | C2×#11758187 | C2×C4 | 1 | 2 | yes | — |
| 37 | C2×#11758187 | C2×C4 | 1 | 2 | yes | — |
| 38 | #14519844 | — | 2 | 1 | no | — |
| 39 | C4⋊#92420 | — | 2 | 1 | no | — |
| 40 | #14519831 | — | 2 | 1 | no | — |
| 41 | #14519831 | — | 2 | 1 | no | — |
| 42 | #14519831 | — | 2 | 1 | no | — |
| 43 | #14519831 | — | 2 | 1 | no | — |
| 44 | (C2×#6923)⋊C4 | — | 2 | 1 | no | — |
| 45 | C2×C8×D61 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C2×C8×D61 | C23 | 1 | 1 | yes | characteristic |
| 47 | C42×D61 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | #14519716 | — | 2 | 1 | no | — |
| 49 | #14519716 | — | 2 | 1 | no | — |
| 50 | #14519716 | D4 | 1 | 2 | yes | — |
| 51 | #14519716 | Q8 | 1 | 2 | yes | — |
| 52 | C2×#11758185 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C2×#11758185 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C4×#92418 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C8×Dic61 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C8×Dic61 | C2×C4 | 1 | 1 | yes | characteristic |
| 57 | C2×C16×C61 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | C2×C16×C61 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C4×C8×C61 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | C22×#6923 | — | 2 | 1 | no | — |
| 61 | C22×#6923 | — | 2 | 1 | no | — |
| 62 | C2×C4×D61 | M16 | 1 | 1 | yes | characteristic |
| 63 | C2×C4×D61 | M16 | 1 | 1 | yes | characteristic |
| 64 | C2×C4×D61 | C22×C4 | 1 | 1 | yes | characteristic |
| 65 | C2×#92420 | — | 2 | 1 | no | — |
| 66 | C2×#92420 | — | 2 | 1 | no | — |
| 67 | C4⋊#6923 | — | 4 | 2 | no | — |
| 68 | C4×#6923 | — | 2 | 1 | no | — |
| 69 | C4×#6923 | — | 2 | 1 | no | — |
| 70 | C4×#6923 | — | 2 | 1 | no | — |
| 71 | C4×#6923 | — | 2 | 1 | no | — |
| 72 | D61⋊C8 | — | 2 | 1 | no | — |
| 73 | D61⋊C8 | — | 2 | 1 | no | — |
| 74 | D61⋊C8 | — | 2 | 1 | no | — |
| 75 | D61⋊C8 | — | 2 | 1 | no | — |
| 76 | C2×C8×C61 | C42 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C61 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C42×C61 | C42 | 1 | 1 | yes | characteristic |
| 79 | C4×Dic61 | C42 | 1 | 1 | yes | characteristic |
| 80 | C4×Dic61 | M16 | 1 | 1 | yes | characteristic |
| 81 | C4×Dic61 | M16 | 1 | 1 | yes | characteristic |
| 82 | C2×#92418 | C42 | 1 | 1 | yes | characteristic |
| 83 | C2×#92418 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C8×D61 | — | 2 | 1 | no | — |
| 85 | C8×D61 | — | 2 | 1 | no | — |
| 86 | C8×D61 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C8×D61 | C2×Q8 | 1 | 1 | yes | characteristic |
| 88 | C8×D61 | #41 | 1 | 1 | yes | characteristic |
| 89 | C8×D61 | #41 | 1 | 1 | yes | characteristic |
| 90 | #11758187 | — | 2 | 2 | no | — |
| 91 | #11758187 | — | 2 | 2 | no | — |
| 92 | C16×C61 | — | 2 | 1 | no | — |
| 93 | C16×C61 | C4⋊C4 | 1 | 2 | yes | complemented |
| 94 | #11758185 | — | 2 | 1 | no | — |
| 95 | #11758185 | C4⋊C4 | 1 | 2 | yes | complemented |
| 96 | C22×D61 | C2×M16 | 1 | 1 | yes | characteristic |
| 97 | C2×#6923 | — | 4 | 1 | no | — |
| 98 | C2×#6923 | — | 4 | 1 | no | — |
| 99 | C2×#6923 | — | 4 | 1 | no | — |
| 100 | C2×#6923 | — | 4 | 1 | no | — |
| 101 | C2×#6923 | — | 4 | 1 | no | — |
| 102 | C2×#6923 | — | 4 | 1 | no | — |
| 103 | C2×C4×C61 | #97 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C61 | #97 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C61 | C2×C42 | 1 | 1 | yes | characteristic |
| 106 | C2×Dic61 | #97 | 1 | 1 | yes | characteristic |
| 107 | C2×Dic61 | #97 | 1 | 1 | yes | characteristic |
| 108 | C2×Dic61 | C2×M16 | 1 | 1 | yes | characteristic |
| 109 | C4×D61 | — | 2 | 2 | no | — |
| 110 | C4×D61 | — | 4 | 1 | no | — |
| 111 | C4×D61 | C4×D4 | 1 | 1 | yes | characteristic |
| 112 | C4×D61 | C4×Q8 | 1 | 1 | yes | characteristic |
| 113 | C4×D61 | #131 | 1 | 1 | yes | characteristic |
| 114 | C4×D61 | #131 | 1 | 1 | yes | characteristic |
| 115 | #92420 | — | 4 | 1 | no | — |
| 116 | #92420 | — | 4 | 1 | no | — |
| 117 | C8×C61 | — | 2 | 1 | no | — |
| 118 | C8×C61 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 119 | C8×C61 | #117 | 1 | 1 | yes | characteristic |
| 120 | #92418 | — | 2 | 1 | no | — |
| 121 | #92418 | C4×D4 | 1 | 1 | yes | characteristic |
| 122 | #92418 | C4×Q8 | 1 | 1 | yes | characteristic |
| 123 | #12666 | — | 61 | 1 | no | maximal, Sylow |
| 124 | C22×C61 | C2×#97 | 1 | 1 | yes | characteristic |
| 125 | C2×D61 | — | 2 | 1 | no | — |
| 126 | C2×D61 | — | 2 | 1 | no | — |
| 127 | C2×D61 | #436 | 1 | 1 | yes | characteristic |
| 128 | C2×D61 | #446 | 1 | 1 | yes | characteristic |
| 129 | #6923 | — | 8 | 1 | no | — |
| 130 | #6923 | — | 8 | 1 | no | — |
| 131 | C4×C61 | — | 2 | 1 | no | — |
| 132 | C4×C61 | — | 2 | 1 | no | — |
| 133 | C4×C61 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 134 | C4×C61 | #405 | 1 | 1 | yes | characteristic |
| 135 | Dic61 | — | 2 | 1 | no | — |
| 136 | Dic61 | #347 | 1 | 2 | yes | complemented |
| 137 | Dic61 | #436 | 1 | 1 | yes | characteristic |
| 138 | Dic61 | #446 | 1 | 1 | yes | characteristic |
| 139 | #2150 | — | 61 | 1 | no | — |
| 140 | C2×#347 | — | 61 | 1 | no | — |
| 141 | C2×#347 | — | 61 | 1 | no | — |
| 142 | C4×M32 | — | 61 | 1 | no | — |
| 143 | #1770 | — | 61 | 1 | no | — |
| 144 | #1375 | — | 61 | 1 | no | — |
| 145 | #1375 | — | 61 | 1 | no | — |
| 146 | C2×C61 | #1770 | 1 | 1 | yes | characteristic |
| 147 | C2×C61 | C2×#347 | 1 | 1 | yes | characteristic |
| 148 | C2×C61 | #2110 | 1 | 1 | yes | characteristic |
| 149 | D61 | — | 2 | 1 | no | — |
| 150 | D61 | — | 2 | 1 | no | — |
| 151 | C2×M32 | — | 61 | 1 | no | — |
| 152 | C2×M32 | — | 61 | 1 | no | — |
| 153 | C2×M32 | — | 61 | 1 | no | — |
| 154 | C2×M32 | — | 61 | 1 | no | — |
| 155 | C2×C4⋊C8 | — | 61 | 1 | no | — |
| 156 | C2×#97 | — | 61 | 1 | no | — |
| 157 | C2×#97 | — | 61 | 1 | no | — |
| 158 | C2×C4×C8 | — | 61 | 1 | no | — |
| 159 | C4×C4⋊C4 | — | 61 | 1 | no | — |
| 160 | C4⋊C16 | — | 61 | 2 | no | — |
| 161 | C4⋊C16 | — | 61 | 2 | no | — |
| 162 | #347 | — | 122 | 2 | no | — |
| 163 | #347 | — | 122 | 2 | no | — |
| 164 | #346 | — | 61 | 1 | no | — |
| 165 | #346 | #6923 | 1 | 1 | yes | characteristic, complemented |
| 166 | C4×C16 | — | 61 | 1 | no | — |
| 167 | C4×C16 | — | 61 | 1 | no | — |
| 168 | #336 | — | 61 | 1 | no | — |
| 169 | #336 | — | 61 | 1 | no | — |
| 170 | C61 | #12666 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | C22×C8 | — | 61 | 1 | no | — |
| 172 | C22×C8 | — | 61 | 1 | no | — |
| 173 | C22×C8 | — | 61 | 1 | no | — |
| 174 | C22×C8 | — | 61 | 1 | no | — |
| 175 | C2×C4⋊C4 | — | 122 | 1 | no | — |
| 176 | C2×C42 | — | 61 | 1 | no | — |
| 177 | C2×C42 | — | 61 | 1 | no | — |
| 178 | C2×C42 | — | 61 | 1 | no | — |
| 179 | M32 | — | 61 | 2 | no | — |
| 180 | M32 | — | 61 | 2 | no | — |
| 181 | M32 | — | 122 | 1 | no | — |
| 182 | M32 | — | 122 | 1 | no | — |
| 183 | M32 | — | 122 | 1 | no | — |
| 184 | M32 | — | 122 | 1 | no | — |
| 185 | M32 | — | 122 | 1 | no | — |
| 186 | M32 | — | 122 | 1 | no | — |
| 187 | C2×C16 | — | 61 | 1 | no | — |
| 188 | C2×C16 | — | 61 | 2 | no | — |
| 189 | C2×C16 | — | 61 | 2 | no | — |
| 190 | C2×C16 | — | 61 | 1 | no | — |
| 191 | C2×C16 | C2×#6923 | 1 | 1 | yes | characteristic |
| 192 | C2×C16 | C2×#6923 | 1 | 1 | yes | characteristic |
| 193 | C4⋊C8 | — | 122 | 1 | no | — |
| 194 | C4⋊C8 | — | 122 | 1 | no | — |
| 195 | #97 | — | 122 | 1 | no | — |
| 196 | #97 | — | 122 | 1 | no | — |
| 197 | #97 | — | 122 | 1 | no | — |
| 198 | #97 | — | 122 | 1 | no | — |
| 199 | C4×C8 | — | 61 | 1 | no | — |
| 200 | C4×C8 | — | 61 | 1 | no | — |
| 201 | C4×C8 | — | 61 | 1 | no | — |
| 202 | C4×C8 | C2×#6923 | 1 | 1 | yes | characteristic |
| 203 | #95 | — | 122 | 1 | no | — |
| 204 | C22×C4 | — | 61 | 1 | no | — |
| 205 | C22×C4 | — | 61 | 1 | no | — |
| 206 | C22×C4 | — | 61 | 1 | no | — |
| 207 | C22×C4 | — | 122 | 1 | no | — |
| 208 | C22×C4 | — | 122 | 1 | no | — |
| 209 | C2×C8 | — | 61 | 1 | no | — |
| 210 | C2×C8 | — | 61 | 1 | no | — |
| 211 | C2×C8 | — | 61 | 1 | no | — |
| 212 | C2×C8 | — | 61 | 1 | no | — |
| 213 | C2×C8 | — | 61 | 1 | no | — |
| 214 | C2×C8 | — | 61 | 1 | no | — |
| 215 | C2×C8 | — | 122 | 1 | no | — |
| 216 | C2×C8 | — | 122 | 1 | no | — |
| 217 | C2×C8 | — | 122 | 1 | no | — |
| 218 | C2×C8 | — | 122 | 1 | no | — |
| 219 | C2×C8 | — | 122 | 1 | no | — |
| 220 | C2×C8 | — | 122 | 1 | no | — |
| 221 | C2×C8 | — | 122 | 1 | no | — |
| 222 | C2×C8 | — | 122 | 1 | no | — |
| 223 | C2×C8 | C4×#6923 | 1 | 1 | yes | characteristic |
| 224 | C2×C8 | C22×#6923 | 1 | 1 | yes | characteristic |
| 225 | C4⋊C4 | — | 244 | 2 | no | — |
| 226 | C42 | — | 61 | 1 | no | — |
| 227 | C42 | — | 61 | 1 | no | — |
| 228 | C42 | — | 61 | 1 | no | — |
| 229 | C42 | — | 122 | 1 | no | — |
| 230 | C42 | — | 122 | 1 | no | — |
| 231 | C42 | — | 122 | 1 | no | — |
| 232 | C42 | — | 122 | 1 | no | — |
| 233 | C42 | C4×#6923 | 1 | 1 | yes | characteristic |
| 234 | C16 | — | 2 | 1 | no | — |
| 235 | C16 | — | 61 | 2 | no | — |
| 236 | C16 | — | 122 | 2 | no | — |
| 237 | C16 | — | 122 | 2 | no | — |
| 238 | C16 | — | 122 | 1 | no | — |
| 239 | C16 | C4⋊#6923 | 1 | 2 | yes | complemented |
| 240 | C23 | — | 61 | 1 | no | — |
| 241 | C2×C4 | — | 61 | 1 | no | — |
| 242 | C2×C4 | — | 61 | 1 | no | — |
| 243 | C2×C4 | — | 61 | 1 | no | — |
| 244 | C2×C4 | — | 61 | 1 | no | — |
| 245 | C2×C4 | — | 61 | 1 | no | — |
| 246 | C2×C4 | — | 61 | 1 | no | — |
| 247 | C2×C4 | — | 61 | 1 | no | — |
| 248 | C2×C4 | — | 122 | 2 | no | — |
| 249 | C2×C4 | — | 244 | 1 | no | — |
| 250 | C2×C4 | — | 244 | 1 | no | — |
| 251 | C2×C4 | — | 244 | 1 | no | — |
| 252 | C2×C4 | — | 244 | 1 | no | — |
| 253 | C2×C4 | — | 244 | 1 | no | — |
| 254 | C2×C4 | — | 244 | 1 | no | — |
| 255 | C2×C4 | — | 244 | 1 | no | — |
| 256 | C2×C4 | #14519831 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | #14519831 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | C2×C4×#6923 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 2 | 1 | no | — |
| 260 | C8 | — | 61 | 1 | no | — |
| 261 | C8 | — | 61 | 1 | no | — |
| 262 | C8 | — | 122 | 1 | no | — |
| 263 | C8 | — | 244 | 1 | no | — |
| 264 | C8 | — | 244 | 1 | no | — |
| 265 | C8 | C2×C4⋊#6923 | 1 | 1 | yes | characteristic |
| 266 | C8 | #14519856 | 1 | 1 | yes | characteristic |
| 267 | C22 | — | 61 | 1 | no | — |
| 268 | C22 | — | 61 | 1 | no | — |
| 269 | C22 | — | 122 | 1 | no | — |
| 270 | C22 | — | 122 | 1 | no | — |
| 271 | C22 | C2×#14519831 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 2 | 1 | no | — |
| 274 | C4 | — | 61 | 2 | no | — |
| 275 | C4 | — | 61 | 1 | no | — |
| 276 | C4 | — | 61 | 1 | no | — |
| 277 | C4 | — | 122 | 1 | no | — |
| 278 | C4 | — | 488 | 1 | no | — |
| 279 | C4 | — | 488 | 1 | no | — |
| 280 | C4 | C4×C4⋊#6923 | 1 | 1 | yes | characteristic |
| 281 | C4 | #16840725 | 1 | 1 | yes | characteristic |
| 282 | C2 | — | 122 | 1 | no | — |
| 283 | C2 | — | 122 | 1 | no | — |
| 284 | C2 | #21461613 | 1 | 1 | yes | characteristic |
| 285 | C2 | #21462034 | 1 | 1 | yes | characteristic |
| 286 | C2 | #21462039 | 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