← #14418516
Subgroups — #14418516
Order 1920.
728 conjugacy classes of subgroups.
The subgroup lattice has 272 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#5470 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11751025 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C3×C4×#168 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11748439 | C2 | 1 | 2 | yes | maximal |
| 5 | #11748323 | C2 | 1 | 2 | yes | maximal |
| 6 | #11746863 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | Dic6×#168 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #10615059 | — | 3 | 1 | no | maximal, Hall |
| 9 | C2×C3×C42×C5 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#6710 | C22 | 1 | 2 | yes | — |
| 11 | C5×#926 | C22 | 1 | 2 | yes | — |
| 12 | C5×#924 | C4 | 1 | 4 | yes | — |
| 13 | C5×#870 | C4 | 1 | 2 | yes | — |
| 14 | C5×#869 | C4 | 1 | 2 | yes | — |
| 15 | C4×C5×Dic6 | C22 | 1 | 4 | yes | — |
| 16 | C22×C3×#168 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×C3×#168 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4⋊(C3⋊#168) | C22 | 1 | 4 | yes | — |
| 19 | C3×C4×#168 | C22 | 1 | 2 | yes | — |
| 20 | C3×C4×#168 | C22 | 1 | 2 | yes | — |
| 21 | Dic3⋊#168 | C22 | 1 | 8 | yes | — |
| 22 | #90953 | C22 | 1 | 4 | yes | — |
| 23 | Dic3×#168 | C22 | 1 | 4 | yes | — |
| 24 | #78445 | — | 5 | 1 | no | maximal, Hall |
| 25 | C5×#533 | — | 3 | 1 | no | — |
| 26 | #65313 | — | 3 | 4 | no | — |
| 27 | Q8×#168 | — | 3 | 4 | no | — |
| 28 | #65288 | — | 3 | 2 | no | — |
| 29 | #65286 | — | 3 | 2 | no | — |
| 30 | #65213 | — | 3 | 1 | no | — |
| 31 | C2×C4×#168 | D3 | 1 | 1 | yes | characteristic |
| 32 | C22×C3×C4×C5 | C2×C4 | 1 | 2 | yes | — |
| 33 | C22×C3×C4×C5 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C5×Dic6 | C2×C4 | 1 | 4 | yes | — |
| 35 | C3×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C3×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C3×C42×C5 | C23 | 1 | 2 | yes | — |
| 38 | #6710 | — | 2 | 4 | no | — |
| 39 | C2×C3⋊#168 | C23 | 1 | 4 | yes | — |
| 40 | C5×#217 | C2×C4 | 1 | 4 | yes | — |
| 41 | C5×C4⋊Dic3 | C2×C4 | 1 | 4 | yes | — |
| 42 | C5×C4⋊Dic3 | C2×C4 | 1 | 4 | yes | — |
| 43 | C5×Dic3⋊C4 | C2×C4 | 1 | 8 | yes | — |
| 44 | C4×C5×Dic3 | C23 | 1 | 4 | yes | — |
| 45 | C2×C3×#168 | — | 2 | 2 | no | — |
| 46 | C2×C3×#168 | Q8 | 1 | 4 | yes | complemented |
| 47 | C2×C3×#168 | C23 | 1 | 2 | yes | — |
| 48 | C2×C3×#168 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C3×#168 | C23 | 1 | 1 | yes | characteristic |
| 50 | #5470 | D5 | 1 | 1 | yes | characteristic |
| 51 | C2×C3×C4×C8 | — | 5 | 1 | no | — |
| 52 | #5102 | — | 5 | 2 | no | — |
| 53 | #5101 | — | 5 | 2 | no | — |
| 54 | #4924 | — | 5 | 1 | no | — |
| 55 | #4679 | — | 5 | 4 | no | — |
| 56 | C8×Dic6 | — | 5 | 4 | no | — |
| 57 | C5×C4⋊Q8 | — | 3 | 2 | no | — |
| 58 | C5×#125 | — | 3 | 2 | no | — |
| 59 | C5×C22⋊Q8 | — | 3 | 4 | no | — |
| 60 | C4×C5×Q8 | — | 3 | 4 | no | — |
| 61 | C5×#117 | — | 3 | 2 | no | — |
| 62 | C2×C42×C5 | C2×D3 | 1 | 1 | yes | characteristic |
| 63 | C2×#681 | — | 3 | 2 | no | — |
| 64 | C22×#168 | C2×D3 | 1 | 1 | yes | characteristic |
| 65 | C22×#168 | C2×D3 | 1 | 1 | yes | characteristic |
| 66 | C4⋊#168 | — | 3 | 4 | no | — |
| 67 | C4⋊#168 | — | 3 | 8 | no | — |
| 68 | #3935 | — | 3 | 4 | no | — |
| 69 | C4×#168 | — | 3 | 4 | no | — |
| 70 | C4×#168 | C2×D3 | 1 | 2 | yes | — |
| 71 | C4×#168 | C2×D3 | 1 | 2 | yes | — |
| 72 | #2960 | — | 15 | 1 | no | Sylow |
| 73 | C23×C3×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C2×C3×C4×C5 | — | 2 | 4 | no | — |
| 75 | C2×C3×C4×C5 | C22×C4 | 1 | 2 | yes | — |
| 76 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×C4×C5 | C22×C4 | 1 | 2 | yes | — |
| 78 | C2×C3×C4×C5 | C2×Q8 | 1 | 2 | yes | — |
| 79 | C2×C3×C4×C5 | #41 | 1 | 2 | yes | — |
| 80 | C2×C3×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 81 | C2×C5×Dic3 | C22×C4 | 1 | 8 | yes | — |
| 82 | C5×Dic6 | — | 2 | 8 | no | — |
| 83 | C3⋊#168 | — | 2 | 4 | no | — |
| 84 | C3×#168 | — | 2 | 2 | no | — |
| 85 | C3×#168 | C2×Q8 | 1 | 2 | yes | — |
| 86 | C3×#168 | C2×Q8 | 1 | 2 | yes | — |
| 87 | C22×C3×C8 | — | 5 | 1 | no | — |
| 88 | C22×C3×C8 | — | 5 | 1 | no | — |
| 89 | C2×C3×C42 | C2×D5 | 1 | 1 | yes | characteristic |
| 90 | #926 | C2×D5 | 1 | 2 | yes | — |
| 91 | #924 | Dic5 | 1 | 4 | yes | — |
| 92 | C2×#208 | — | 5 | 2 | no | — |
| 93 | #870 | Dic5 | 1 | 2 | yes | — |
| 94 | #869 | Dic5 | 1 | 2 | yes | — |
| 95 | C4×Dic6 | C2×D5 | 1 | 4 | yes | — |
| 96 | C3×C4×C8 | — | 5 | 2 | no | — |
| 97 | C3×C4×C8 | — | 5 | 2 | no | — |
| 98 | #815 | — | 5 | 4 | no | — |
| 99 | Dic3⋊C8 | — | 5 | 8 | no | — |
| 100 | C8×Dic3 | — | 5 | 4 | no | — |
| 101 | C4⋊(C3⋊C8) | — | 5 | 4 | no | — |
| 102 | C2×C5×Q8 | — | 3 | 4 | no | — |
| 103 | C22×C4×C5 | C4×D3 | 1 | 2 | yes | — |
| 104 | C22×C4×C5 | C22×D3 | 1 | 1 | yes | characteristic |
| 105 | C5×C4⋊C4 | — | 3 | 4 | no | — |
| 106 | C5×C4⋊C4 | — | 3 | 8 | no | — |
| 107 | C5×C4⋊C4 | — | 3 | 4 | no | — |
| 108 | C5×C22⋊C4 | — | 3 | 4 | no | — |
| 109 | C42×C5 | — | 3 | 4 | no | — |
| 110 | C42×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 111 | C42×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 112 | C42×C5 | C22×D3 | 1 | 2 | yes | — |
| 113 | #681 | — | 6 | 4 | no | — |
| 114 | C2×#168 | — | 2 | 2 | no | — |
| 115 | C2×#168 | — | 3 | 4 | no | — |
| 116 | C2×#168 | Dic6 | 1 | 4 | yes | complemented |
| 117 | C2×#168 | C22×D3 | 1 | 2 | yes | — |
| 118 | C2×#168 | C22×D3 | 1 | 1 | yes | characteristic |
| 119 | C2×#168 | C22×D3 | 1 | 1 | yes | characteristic |
| 120 | #533 | — | 3 | 1 | no | — |
| 121 | #446 | — | 15 | 4 | no | — |
| 122 | C8×Q8 | — | 15 | 4 | no | — |
| 123 | #424 | — | 15 | 2 | no | — |
| 124 | #423 | — | 15 | 1 | no | — |
| 125 | #405 | — | 15 | 2 | no | — |
| 126 | C2×C4×C8 | — | 5 | 1 | no | — |
| 127 | C22×C3×C5 | — | 2 | 2 | no | — |
| 128 | C22×C3×C5 | C4×Q8 | 1 | 2 | yes | — |
| 129 | C22×C3×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 130 | C3×C4×C5 | — | 2 | 2 | no | — |
| 131 | C3×C4×C5 | C4×Q8 | 1 | 2 | yes | — |
| 132 | C3×C4×C5 | #131 | 1 | 4 | yes | — |
| 133 | C3×C4×C5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 134 | C3×C4×C5 | C2×#41 | 1 | 1 | yes | characteristic |
| 135 | C5×Dic3 | — | 2 | 8 | no | — |
| 136 | C22×C3×C4 | C2×Dic5 | 1 | 2 | yes | — |
| 137 | C22×C3×C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 138 | C2×Dic6 | C2×Dic5 | 1 | 4 | yes | — |
| 139 | C2×C3×C8 | — | 5 | 4 | no | — |
| 140 | C2×C3×C8 | — | 5 | 1 | no | — |
| 141 | C2×C3×C8 | — | 5 | 1 | no | — |
| 142 | C2×C3×C8 | — | 5 | 2 | no | — |
| 143 | C2×C3×C8 | — | 10 | 2 | no | — |
| 144 | C3×C42 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 145 | C3×C42 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 146 | C3×C42 | C22×D5 | 1 | 2 | yes | — |
| 147 | #217 | C2×Dic5 | 1 | 4 | yes | — |
| 148 | C4⋊Dic3 | C2×Dic5 | 1 | 4 | yes | — |
| 149 | C4⋊Dic3 | C2×Dic5 | 1 | 4 | yes | — |
| 150 | Dic3⋊C4 | C2×Dic5 | 1 | 8 | yes | — |
| 151 | C4×Dic3 | C22×D5 | 1 | 4 | yes | — |
| 152 | #208 | — | 10 | 4 | no | — |
| 153 | C2×C3⋊C8 | — | 5 | 4 | no | — |
| 154 | C23×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 155 | C5×Q8 | — | 6 | 8 | no | — |
| 156 | C2×C4×C5 | — | 2 | 4 | no | — |
| 157 | C2×C4×C5 | — | 3 | 8 | no | — |
| 158 | C2×C4×C5 | C2×Dic6 | 1 | 2 | yes | — |
| 159 | C2×C4×C5 | C2×C4×D3 | 1 | 2 | yes | — |
| 160 | C2×C4×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 161 | C2×C4×C5 | C2×C4×D3 | 1 | 2 | yes | — |
| 162 | C2×C4×C5 | #235 | 1 | 2 | yes | — |
| 163 | C2×C4×C5 | C23×D3 | 1 | 1 | yes | characteristic |
| 164 | #168 | — | 2 | 2 | no | — |
| 165 | #168 | — | 6 | 4 | no | — |
| 166 | #168 | C2×Dic6 | 1 | 2 | yes | — |
| 167 | #168 | C2×Dic6 | 1 | 2 | yes | — |
| 168 | C2×M16 | — | 15 | 2 | no | — |
| 169 | C22×C8 | — | 5 | 1 | no | — |
| 170 | C22×C8 | — | 5 | 1 | no | — |
| 171 | C4⋊Q8 | — | 3 | 2 | no | — |
| 172 | #125 | — | 3 | 2 | no | — |
| 173 | C22⋊Q8 | — | 3 | 4 | no | — |
| 174 | C4×Q8 | — | 3 | 4 | no | — |
| 175 | #117 | — | 3 | 2 | no | — |
| 176 | C2×C42 | D3×D5 | 1 | 1 | yes | characteristic |
| 177 | C4⋊C8 | — | 15 | 8 | no | — |
| 178 | C4⋊C8 | — | 15 | 4 | no | — |
| 179 | #97 | — | 15 | 4 | no | — |
| 180 | C4×C8 | — | 5 | 2 | no | — |
| 181 | C4×C8 | — | 5 | 2 | no | — |
| 182 | C4×C8 | — | 15 | 4 | no | — |
| 183 | C2×C3×C5 | — | 2 | 2 | no | — |
| 184 | C2×C3×C5 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 185 | C2×C3×C5 | C2×#131 | 1 | 2 | yes | — |
| 186 | C23×C3 | C22×Dic5 | 1 | 1 | yes | characteristic |
| 187 | C2×C3×C4 | — | 2 | 4 | no | — |
| 188 | C2×C3×C4 | Q8×D5 | 1 | 2 | yes | — |
| 189 | C2×C3×C4 | #713 | 1 | 2 | yes | — |
| 190 | C2×C3×C4 | C22×Dic5 | 1 | 2 | yes | — |
| 191 | C2×C3×C4 | C22×Dic5 | 1 | 1 | yes | characteristic |
| 192 | C2×C3×C4 | C22×Dic5 | 1 | 2 | yes | — |
| 193 | C2×C3×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 194 | C2×Dic3 | C22×Dic5 | 1 | 8 | yes | — |
| 195 | Dic6 | — | 2 | 8 | no | — |
| 196 | C3×C8 | — | 5 | 2 | no | — |
| 197 | C3×C8 | — | 5 | 2 | no | — |
| 198 | C3×C8 | — | 10 | 2 | no | — |
| 199 | C3⋊C8 | — | 10 | 4 | no | — |
| 200 | C22×C5 | — | 2 | 2 | no | — |
| 201 | C22×C5 | C4×Dic6 | 1 | 2 | yes | — |
| 202 | C22×C5 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 203 | C4×C5 | — | 2 | 2 | no | — |
| 204 | C4×C5 | — | 6 | 8 | no | — |
| 205 | C4×C5 | C4×Dic6 | 1 | 2 | yes | — |
| 206 | C4×C5 | #901 | 1 | 4 | yes | — |
| 207 | C4×C5 | C22×Dic6 | 1 | 1 | yes | characteristic |
| 208 | C4×C5 | C2×#235 | 1 | 1 | yes | characteristic |
| 209 | C2×Q8 | — | 3 | 4 | no | — |
| 210 | C22×C4 | D3×Dic5 | 1 | 2 | yes | — |
| 211 | C22×C4 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 212 | M16 | — | 30 | 4 | no | — |
| 213 | C2×C8 | — | 5 | 4 | no | — |
| 214 | C2×C8 | — | 5 | 1 | no | — |
| 215 | C2×C8 | — | 5 | 1 | no | — |
| 216 | C2×C8 | — | 5 | 2 | no | — |
| 217 | C2×C8 | — | 10 | 2 | no | — |
| 218 | C2×C8 | — | 15 | 4 | no | — |
| 219 | C4⋊C4 | — | 3 | 4 | no | — |
| 220 | C4⋊C4 | — | 3 | 8 | no | — |
| 221 | C4⋊C4 | — | 3 | 4 | no | — |
| 222 | C22⋊C4 | — | 3 | 4 | no | — |
| 223 | C42 | — | 3 | 4 | no | — |
| 224 | C42 | D3×Dic5 | 1 | 1 | yes | characteristic |
| 225 | C42 | D3×Dic5 | 1 | 1 | yes | characteristic |
| 226 | C42 | C2×D3×D5 | 1 | 2 | yes | — |
| 227 | C3×C5 | #2960 | 1 | 1 | yes | characteristic, complemented, Hall |
| 228 | C22×C3 | — | 2 | 2 | no | — |
| 229 | C22×C3 | Q8×Dic5 | 1 | 2 | yes | — |
| 230 | C22×C3 | C23×Dic5 | 1 | 1 | yes | characteristic |
| 231 | C3×C4 | — | 2 | 2 | no | — |
| 232 | C3×C4 | Q8×Dic5 | 1 | 2 | yes | — |
| 233 | C3×C4 | #4094 | 1 | 4 | yes | — |
| 234 | C3×C4 | C2×Q8×D5 | 1 | 1 | yes | characteristic |
| 235 | C3×C4 | C2×#713 | 1 | 1 | yes | characteristic |
| 236 | Dic3 | — | 2 | 8 | no | — |
| 237 | C2×C5 | — | 2 | 2 | no | — |
| 238 | C2×C5 | C2×C4×Dic6 | 1 | 1 | yes | characteristic |
| 239 | C2×C5 | C2×#901 | 1 | 2 | yes | — |
| 240 | C23 | C2×D3×Dic5 | 1 | 1 | yes | characteristic |
| 241 | Q8 | — | 6 | 8 | no | — |
| 242 | C2×C4 | — | 2 | 4 | no | — |
| 243 | C2×C4 | — | 3 | 8 | no | — |
| 244 | C2×C4 | D5×Dic6 | 1 | 2 | yes | — |
| 245 | C2×C4 | #6773 | 1 | 2 | yes | — |
| 246 | C2×C4 | C2×D3×Dic5 | 1 | 2 | yes | — |
| 247 | C2×C4 | C2×D3×Dic5 | 1 | 2 | yes | — |
| 248 | C2×C4 | C2×D3×Dic5 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | C22×D3×D5 | 1 | 1 | yes | characteristic |
| 250 | C8 | — | 5 | 2 | no | — |
| 251 | C8 | — | 5 | 2 | no | — |
| 252 | C8 | — | 10 | 2 | no | — |
| 253 | C8 | — | 30 | 4 | no | — |
| 254 | C2×C3 | — | 2 | 2 | no | — |
| 255 | C2×C3 | C2×Q8×Dic5 | 1 | 1 | yes | characteristic |
| 256 | C2×C3 | C2×#4094 | 1 | 2 | yes | — |
| 257 | C5 | #78445 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 258 | C22 | — | 2 | 2 | no | — |
| 259 | C22 | Dic5×Dic6 | 1 | 2 | yes | — |
| 260 | C22 | C22×D3×Dic5 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 2 | 2 | no | — |
| 262 | C4 | — | 6 | 8 | no | — |
| 263 | C4 | #91286 | 1 | 4 | yes | — |
| 264 | C4 | Dic5×Dic6 | 1 | 2 | yes | — |
| 265 | C4 | C2×D5×Dic6 | 1 | 1 | yes | characteristic |
| 266 | C4 | C2×#6773 | 1 | 1 | yes | characteristic |
| 267 | C3 | #10615059 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 268 | C2 | — | 2 | 2 | no | — |
| 269 | C2 | C2×#91286 | 1 | 2 | yes | — |
| 270 | C2 | C2×Dic5×Dic6 | 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