← #14285596
Subgroups — #14285596
Order 1920.
360 conjugacy classes of subgroups.
The subgroup lattice has 268 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×C4×C5×M16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11751026 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C4⋊(C3⋊#168) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11746279 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11746278 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10600643 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C3×C5×M16 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×C3×C5×M16 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×C42×C5 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C3⋊#168 | — | 2 | 1 | no | — |
| 11 | C3×C5×#97 | C4 | 1 | 2 | yes | — |
| 12 | C3×C4×C5×C8 | C22 | 1 | 2 | yes | — |
| 13 | #91114 | — | 2 | 1 | no | — |
| 14 | C4⋊(C3⋊#168) | — | 2 | 1 | no | — |
| 15 | C4⋊(C3⋊#168) | C22 | 1 | 2 | yes | — |
| 16 | C4⋊(C3⋊#168) | C22 | 1 | 1 | yes | characteristic |
| 17 | C4⋊(C3⋊#168) | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×C3⋊#168 | — | 2 | 1 | no | — |
| 19 | #69850 | — | 5 | 1 | no | maximal, Hall |
| 20 | C4×C5×M16 | D3 | 1 | 1 | yes | characteristic |
| 21 | #65214 | — | 3 | 1 | no | — |
| 22 | C2×C4⋊#168 | — | 3 | 1 | no | — |
| 23 | #64678 | — | 3 | 2 | no | — |
| 24 | #64677 | — | 3 | 2 | no | — |
| 25 | C22×C3×C4×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C22×C3×C4×C5 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×C3×C4×C5 | Q8 | 1 | 1 | yes | characteristic |
| 28 | C3×C5×M16 | — | 2 | 2 | no | — |
| 29 | C3×C5×M16 | C8 | 1 | 4 | yes | complemented |
| 30 | C2×C3×C5×C8 | C2×C4 | 1 | 2 | yes | — |
| 31 | C2×C3×C5×C8 | C2×C4 | 1 | 2 | yes | — |
| 32 | C3×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C3×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 34 | C3×C42×C5 | Q8 | 1 | 1 | yes | characteristic |
| 35 | C3×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×C3⋊#168 | — | 2 | 2 | no | — |
| 37 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 38 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 39 | C2×C3⋊#168 | — | 4 | 2 | no | — |
| 40 | C3×C4×M16 | D5 | 1 | 1 | yes | characteristic |
| 41 | #4925 | — | 5 | 1 | no | — |
| 42 | C2×C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 43 | #4458 | — | 5 | 2 | no | — |
| 44 | #4457 | — | 5 | 2 | no | — |
| 45 | C2×C5×M16 | Dic3 | 1 | 1 | yes | characteristic |
| 46 | C2×C5×M16 | Dic3 | 1 | 1 | yes | characteristic |
| 47 | C2×C42×C5 | C2×D3 | 1 | 1 | yes | characteristic |
| 48 | C22×#168 | — | 6 | 1 | no | — |
| 49 | C5×#97 | Dic3 | 1 | 2 | yes | — |
| 50 | C4×C5×C8 | C2×D3 | 1 | 2 | yes | — |
| 51 | C22⋊#168 | — | 6 | 1 | no | — |
| 52 | C4⋊#168 | — | 3 | 1 | no | — |
| 53 | C4⋊#168 | — | 3 | 2 | no | — |
| 54 | C4⋊#168 | — | 3 | 1 | no | — |
| 55 | C4⋊#168 | — | 6 | 1 | no | — |
| 56 | C4×#168 | — | 6 | 1 | no | — |
| 57 | #1565 | — | 15 | 1 | no | Sylow |
| 58 | C23×C3×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 60 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 61 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 62 | C2×C3×C4×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C3×C4×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C2×C3×C4×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 65 | C2×C3×C4×C5 | C2×C8 | 1 | 2 | yes | — |
| 66 | C2×C3×C4×C5 | M16 | 1 | 2 | yes | — |
| 67 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C2×C3×C4×C5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 70 | C3×C5×C8 | — | 2 | 2 | no | — |
| 71 | C3×C5×C8 | C2×C8 | 1 | 4 | yes | complemented |
| 72 | C3⋊#168 | — | 4 | 1 | no | — |
| 73 | C3⋊#168 | — | 4 | 2 | no | — |
| 74 | C3⋊#168 | — | 4 | 1 | no | — |
| 75 | C2×C3×M16 | Dic5 | 1 | 1 | yes | characteristic |
| 76 | C2×C3×M16 | Dic5 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×C42 | C2×D5 | 1 | 1 | yes | characteristic |
| 78 | C22×C3⋊C8 | — | 10 | 1 | no | — |
| 79 | C3×#97 | Dic5 | 1 | 2 | yes | — |
| 80 | C3×C4×C8 | C2×D5 | 1 | 2 | yes | — |
| 81 | #830 | — | 10 | 1 | no | — |
| 82 | C4⋊(C3⋊C8) | — | 5 | 2 | no | — |
| 83 | C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 84 | C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 85 | C4⋊(C3⋊C8) | — | 10 | 1 | no | — |
| 86 | C4×C3⋊C8 | — | 10 | 1 | no | — |
| 87 | C22×C4×C5 | Dic6 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C5 | D12 | 1 | 1 | yes | characteristic |
| 89 | C22×C4×C5 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 90 | C5×M16 | — | 2 | 2 | no | — |
| 91 | C5×M16 | C3⋊C8 | 1 | 4 | yes | complemented |
| 92 | C2×C5×C8 | C2×Dic3 | 1 | 2 | yes | — |
| 93 | C2×C5×C8 | C2×Dic3 | 1 | 2 | yes | — |
| 94 | C42×C5 | Dic6 | 1 | 1 | yes | characteristic |
| 95 | C42×C5 | D12 | 1 | 1 | yes | characteristic |
| 96 | C42×C5 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 97 | C42×C5 | C22×D3 | 1 | 1 | yes | characteristic |
| 98 | C2×#168 | — | 6 | 1 | no | — |
| 99 | C2×#168 | — | 6 | 1 | no | — |
| 100 | C2×#168 | — | 6 | 2 | no | — |
| 101 | C2×#168 | — | 12 | 2 | no | — |
| 102 | #431 | — | 15 | 1 | no | — |
| 103 | C2×C4⋊C8 | — | 15 | 1 | no | — |
| 104 | C4×M16 | D15 | 1 | 1 | yes | characteristic |
| 105 | #335 | — | 15 | 2 | no | — |
| 106 | #334 | — | 15 | 2 | no | — |
| 107 | C22×C3×C5 | — | 2 | 1 | no | — |
| 108 | C22×C3×C5 | — | 2 | 1 | no | — |
| 109 | C22×C3×C5 | C4⋊C8 | 1 | 2 | yes | — |
| 110 | C22×C3×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 111 | C3×C4×C5 | — | 2 | 1 | no | — |
| 112 | C3×C4×C5 | — | 2 | 1 | no | — |
| 113 | C3×C4×C5 | — | 2 | 1 | no | — |
| 114 | C3×C4×C5 | C4⋊C8 | 1 | 2 | yes | — |
| 115 | C3×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 116 | C3×C4×C5 | C2×M16 | 1 | 1 | yes | characteristic |
| 117 | C3×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 118 | C3×C4×C5 | #137 | 1 | 1 | yes | characteristic |
| 119 | C22×C3×C4 | Dic10 | 1 | 1 | yes | characteristic |
| 120 | C22×C3×C4 | D20 | 1 | 1 | yes | characteristic |
| 121 | C22×C3×C4 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 122 | C3×M16 | — | 2 | 2 | no | — |
| 123 | C3×M16 | #168 | 1 | 4 | yes | complemented |
| 124 | C2×C3×C8 | C2×Dic5 | 1 | 2 | yes | — |
| 125 | C2×C3×C8 | C2×Dic5 | 1 | 2 | yes | — |
| 126 | C3×C42 | Dic10 | 1 | 1 | yes | characteristic |
| 127 | C3×C42 | D20 | 1 | 1 | yes | characteristic |
| 128 | C3×C42 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 129 | C3×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 130 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 131 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 132 | C2×C3⋊C8 | — | 10 | 2 | no | — |
| 133 | C2×C3⋊C8 | — | 20 | 2 | no | — |
| 134 | C23×C5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C5 | — | 2 | 1 | no | — |
| 136 | C2×C4×C5 | — | 2 | 1 | no | — |
| 137 | C2×C4×C5 | — | 4 | 1 | no | — |
| 138 | C2×C4×C5 | C2×C3⋊C8 | 1 | 2 | yes | — |
| 139 | C2×C4×C5 | #208 | 1 | 2 | yes | — |
| 140 | C2×C4×C5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 141 | C2×C4×C5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 142 | C2×C4×C5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 143 | C2×C4×C5 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 144 | C2×C4×C5 | C2×D12 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 146 | C5×C8 | — | 2 | 2 | no | — |
| 147 | C5×C8 | C2×C3⋊C8 | 1 | 4 | yes | complemented |
| 148 | #168 | — | 12 | 2 | no | — |
| 149 | #168 | — | 12 | 1 | no | — |
| 150 | #168 | — | 12 | 1 | no | — |
| 151 | C2×M16 | Dic15 | 1 | 1 | yes | characteristic |
| 152 | C2×M16 | Dic15 | 1 | 1 | yes | characteristic |
| 153 | C22×C8 | — | 30 | 1 | no | — |
| 154 | C2×C42 | C2×D15 | 1 | 1 | yes | characteristic |
| 155 | C4⋊C8 | — | 15 | 1 | no | — |
| 156 | C4⋊C8 | — | 15 | 2 | no | — |
| 157 | C4⋊C8 | — | 15 | 1 | no | — |
| 158 | C4⋊C8 | — | 30 | 1 | no | — |
| 159 | C22⋊C8 | — | 30 | 1 | no | — |
| 160 | #97 | Dic15 | 1 | 2 | yes | — |
| 161 | C4×C8 | — | 30 | 1 | no | — |
| 162 | C4×C8 | C2×D15 | 1 | 2 | yes | — |
| 163 | C2×C3×C5 | — | 2 | 2 | no | — |
| 164 | C2×C3×C5 | C2×C4⋊C8 | 1 | 1 | yes | characteristic |
| 165 | C2×C3×C5 | #428 | 1 | 1 | yes | characteristic |
| 166 | C2×C3×C5 | #430 | 1 | 1 | yes | characteristic |
| 167 | C23×C3 | C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 168 | C2×C3×C4 | — | 2 | 1 | no | — |
| 169 | C2×C3×C4 | — | 2 | 1 | no | — |
| 170 | C2×C3×C4 | — | 4 | 1 | no | — |
| 171 | C2×C3×C4 | C2×#168 | 1 | 2 | yes | — |
| 172 | C2×C3×C4 | #681 | 1 | 2 | yes | — |
| 173 | C2×C3×C4 | C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 174 | C2×C3×C4 | C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 176 | C2×C3×C4 | C2×Dic10 | 1 | 1 | yes | characteristic |
| 177 | C2×C3×C4 | C2×D20 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | C22×Dic5 | 1 | 1 | yes | characteristic |
| 179 | C3×C8 | — | 2 | 2 | no | — |
| 180 | C3×C8 | C2×#168 | 1 | 4 | yes | complemented |
| 181 | C3⋊C8 | — | 20 | 1 | no | — |
| 182 | C3⋊C8 | — | 20 | 2 | no | — |
| 183 | C3⋊C8 | — | 20 | 1 | no | — |
| 184 | C22×C5 | — | 2 | 1 | no | — |
| 185 | C22×C5 | — | 2 | 1 | no | — |
| 186 | C22×C5 | C4⋊(C3⋊C8) | 1 | 2 | yes | — |
| 187 | C22×C5 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 188 | C4×C5 | — | 2 | 1 | no | — |
| 189 | C4×C5 | — | 2 | 1 | no | — |
| 190 | C4×C5 | — | 2 | 1 | no | — |
| 191 | C4×C5 | C4⋊(C3⋊C8) | 1 | 2 | yes | — |
| 192 | C4×C5 | #908 | 1 | 1 | yes | characteristic |
| 193 | C4×C5 | #909 | 1 | 1 | yes | characteristic |
| 194 | C4×C5 | C22×C3⋊C8 | 1 | 1 | yes | characteristic |
| 195 | C4×C5 | C2×#208 | 1 | 1 | yes | characteristic |
| 196 | C22×C4 | Dic30 | 1 | 1 | yes | characteristic |
| 197 | C22×C4 | D60 | 1 | 1 | yes | characteristic |
| 198 | C22×C4 | C2×Dic15 | 1 | 1 | yes | characteristic |
| 199 | M16 | — | 2 | 2 | no | — |
| 200 | M16 | C3⋊#168 | 1 | 4 | yes | complemented |
| 201 | C2×C8 | — | 30 | 1 | no | — |
| 202 | C2×C8 | — | 30 | 1 | no | — |
| 203 | C2×C8 | — | 30 | 2 | no | — |
| 204 | C2×C8 | — | 60 | 2 | no | — |
| 205 | C2×C8 | C2×Dic15 | 1 | 2 | yes | — |
| 206 | C2×C8 | C2×Dic15 | 1 | 2 | yes | — |
| 207 | C42 | Dic30 | 1 | 1 | yes | characteristic |
| 208 | C42 | D60 | 1 | 1 | yes | characteristic |
| 209 | C42 | C2×Dic15 | 1 | 1 | yes | characteristic |
| 210 | C42 | C22×D15 | 1 | 1 | yes | characteristic |
| 211 | C3×C5 | #1565 | 1 | 1 | yes | characteristic, complemented, Hall |
| 212 | C22×C3 | — | 2 | 1 | no | — |
| 213 | C22×C3 | — | 2 | 1 | no | — |
| 214 | C22×C3 | C4⋊#168 | 1 | 2 | yes | — |
| 215 | C22×C3 | C2×C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 216 | C3×C4 | — | 2 | 1 | no | — |
| 217 | C3×C4 | — | 2 | 1 | no | — |
| 218 | C3×C4 | — | 2 | 1 | no | — |
| 219 | C3×C4 | C4⋊#168 | 1 | 2 | yes | — |
| 220 | C3×C4 | #4054 | 1 | 1 | yes | characteristic |
| 221 | C3×C4 | #4055 | 1 | 1 | yes | characteristic |
| 222 | C3×C4 | C22×#168 | 1 | 1 | yes | characteristic |
| 223 | C3×C4 | C2×#681 | 1 | 1 | yes | characteristic |
| 224 | C2×C5 | — | 2 | 2 | no | — |
| 225 | C2×C5 | C2×C4⋊(C3⋊C8) | 1 | 1 | yes | characteristic |
| 226 | C2×C5 | #5121 | 1 | 1 | yes | characteristic |
| 227 | C2×C5 | #5125 | 1 | 1 | yes | characteristic |
| 228 | C23 | C4⋊Dic15 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | — | 2 | 1 | no | — |
| 230 | C2×C4 | — | 2 | 1 | no | — |
| 231 | C2×C4 | — | 4 | 1 | no | — |
| 232 | C2×C4 | C2×C3⋊#168 | 1 | 2 | yes | — |
| 233 | C2×C4 | #6710 | 1 | 2 | yes | — |
| 234 | C2×C4 | C4⋊Dic15 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | C4⋊Dic15 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | C4⋊Dic15 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | C2×Dic30 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | C2×D60 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | C22×Dic15 | 1 | 1 | yes | characteristic |
| 240 | C8 | — | 2 | 2 | no | — |
| 241 | C8 | — | 60 | 2 | no | — |
| 242 | C8 | — | 60 | 1 | no | — |
| 243 | C8 | — | 60 | 1 | no | — |
| 244 | C8 | C2×C3⋊#168 | 1 | 4 | yes | complemented |
| 245 | C2×C3 | — | 2 | 2 | no | — |
| 246 | C2×C3 | C2×C4⋊#168 | 1 | 1 | yes | characteristic |
| 247 | C2×C3 | #65410 | 1 | 1 | yes | characteristic |
| 248 | C2×C3 | #65414 | 1 | 1 | yes | characteristic |
| 249 | C5 | #69850 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 250 | C22 | — | 2 | 1 | no | — |
| 251 | C22 | — | 2 | 1 | no | — |
| 252 | C22 | C4⋊(C3⋊#168) | 1 | 2 | yes | — |
| 253 | C22 | C2×C4⋊Dic15 | 1 | 1 | yes | characteristic |
| 254 | C4 | — | 2 | 1 | no | — |
| 255 | C4 | — | 2 | 1 | no | — |
| 256 | C4 | — | 2 | 1 | no | — |
| 257 | C4 | C4⋊(C3⋊#168) | 1 | 2 | yes | — |
| 258 | C4 | #91797 | 1 | 1 | yes | characteristic |
| 259 | C4 | #91798 | 1 | 1 | yes | characteristic |
| 260 | C4 | C22×C3⋊#168 | 1 | 1 | yes | characteristic |
| 261 | C4 | C2×#6710 | 1 | 1 | yes | characteristic |
| 262 | C3 | #10600643 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 263 | C2 | — | 2 | 2 | no | — |
| 264 | C2 | C2×C4⋊(C3⋊#168) | 1 | 1 | yes | characteristic |
| 265 | C2 | #11751222 | 1 | 1 | yes | characteristic |
| 266 | C2 | #11751226 | 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