← #10778261
Subgroups — #10778261
Order 768.
714 conjugacy classes of subgroups.
The subgroup lattice has 338 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×#494 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #83215 | C2 | 1 | 2 | yes | maximal |
| 3 | #78458 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #71773 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #71306 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #71305 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #69729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #19188 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×C3×Q16 | — | 2 | 1 | no | — |
| 10 | C22×C3×Q16 | — | 2 | 1 | no | — |
| 11 | C2×C3×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 12 | C2×#901 | — | 2 | 1 | no | — |
| 13 | #5477 | C22 | 1 | 1 | yes | characteristic |
| 14 | C3×#494 | — | 2 | 1 | no | — |
| 15 | C3×#494 | — | 2 | 1 | no | — |
| 16 | C3×#494 | C22 | 1 | 2 | yes | — |
| 17 | C3×#494 | C22 | 1 | 2 | yes | — |
| 18 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 19 | #5110 | — | 2 | 1 | no | — |
| 20 | #5088 | — | 2 | 2 | no | — |
| 21 | #5087 | C22 | 1 | 8 | yes | — |
| 22 | #5078 | — | 2 | 2 | no | — |
| 23 | #4967 | — | 2 | 4 | no | — |
| 24 | #4925 | C22 | 1 | 2 | yes | — |
| 25 | #4924 | C22 | 1 | 1 | yes | characteristic |
| 26 | #4688 | — | 2 | 1 | no | — |
| 27 | #4688 | C22 | 1 | 2 | yes | — |
| 28 | C8×D12 | — | 2 | 1 | no | — |
| 29 | C8×D12 | C22 | 1 | 2 | yes | — |
| 30 | #4547 | C22 | 1 | 8 | yes | — |
| 31 | #4458 | C22 | 1 | 2 | yes | — |
| 32 | #4457 | C22 | 1 | 2 | yes | — |
| 33 | #3226 | — | 3 | 2 | no | — |
| 34 | C2×#494 | D3 | 1 | 1 | yes | characteristic, complemented |
| 35 | #2931 | — | 3 | 1 | no | — |
| 36 | #1689 | — | 3 | 4 | no | — |
| 37 | #1669 | — | 3 | 4 | no | — |
| 38 | #1564 | — | 3 | 1 | no | — |
| 39 | #1546 | — | 3 | 2 | no | — |
| 40 | C22×C3×Q8 | — | 2 | 2 | no | — |
| 41 | C2×#235 | — | 2 | 1 | no | — |
| 42 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 43 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 44 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 45 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 46 | C2×C3×Q16 | — | 4 | 4 | no | — |
| 47 | C2×C3×Q16 | — | 4 | 4 | no | — |
| 48 | C22×C3×C8 | D4 | 1 | 1 | yes | characteristic |
| 49 | C22×C3×C8 | D4 | 1 | 1 | yes | characteristic |
| 50 | C3×C4⋊Q8 | — | 2 | 2 | no | — |
| 51 | C3×C4⋊Q8 | — | 2 | 2 | no | — |
| 52 | C3×C4⋊Q8 | — | 2 | 2 | no | — |
| 53 | C3×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 54 | C2×C3×C4⋊C4 | — | 2 | 2 | no | — |
| 55 | C2×C3×C4⋊C4 | — | 2 | 2 | no | — |
| 56 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 57 | #930 | — | 2 | 1 | no | — |
| 58 | #930 | D4 | 1 | 2 | yes | — |
| 59 | C2×#208 | — | 2 | 1 | no | — |
| 60 | #901 | — | 4 | 4 | no | — |
| 61 | C2×ΓL(1,𝔽25) | — | 2 | 2 | no | — |
| 62 | C2×D3×C8 | — | 2 | 2 | no | — |
| 63 | #875 | — | 2 | 1 | no | — |
| 64 | #874 | D4 | 1 | 1 | yes | characteristic |
| 65 | C4×D12 | — | 2 | 1 | no | — |
| 66 | C4×D12 | C23 | 1 | 2 | yes | — |
| 67 | #869 | D4 | 1 | 1 | yes | characteristic |
| 68 | C3×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 69 | C3×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 70 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 71 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 72 | #830 | — | 2 | 2 | no | — |
| 73 | (C2×D3)⋊C8 | — | 2 | 2 | no | — |
| 74 | (C2×D3)⋊C8 | — | 2 | 2 | no | — |
| 75 | #809 | — | 2 | 8 | no | — |
| 76 | #807 | — | 2 | 8 | no | — |
| 77 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 78 | C4⋊(C3⋊C8) | C23 | 1 | 2 | yes | — |
| 79 | C4⋊(C3⋊C8) | C23 | 1 | 4 | yes | — |
| 80 | C4×C3⋊C8 | — | 2 | 2 | no | — |
| 81 | C22×Q16 | — | 2 | 1 | no | — |
| 82 | C22×Q16 | — | 2 | 1 | no | — |
| 83 | C2×#131 | — | 6 | 1 | no | — |
| 84 | #532 | — | 3 | 1 | no | — |
| 85 | C2×C4⋊Q8 | C2×D3 | 1 | 2 | yes | — |
| 86 | #500 | — | 6 | 2 | no | — |
| 87 | #494 | — | 2 | 1 | no | — |
| 88 | #494 | — | 2 | 1 | no | — |
| 89 | #494 | C2×D3 | 1 | 2 | yes | — |
| 90 | #494 | C2×D3 | 1 | 2 | yes | — |
| 91 | #486 | — | 6 | 2 | no | — |
| 92 | #480 | — | 6 | 4 | no | — |
| 93 | #474 | — | 3 | 8 | no | — |
| 94 | #436 | — | 3 | 2 | no | — |
| 95 | #436 | — | 6 | 1 | no | — |
| 96 | C8×D4 | — | 3 | 2 | no | — |
| 97 | C8×D4 | — | 6 | 1 | no | — |
| 98 | #431 | — | 3 | 2 | no | — |
| 99 | #423 | — | 3 | 1 | no | — |
| 100 | #408 | — | 6 | 1 | no | — |
| 101 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 102 | #335 | — | 3 | 2 | no | — |
| 103 | #334 | — | 3 | 2 | no | — |
| 104 | #332 | — | 3 | 8 | no | — |
| 105 | C2×C3×Q8 | — | 2 | 4 | no | — |
| 106 | C2×C3×Q8 | — | 4 | 2 | no | — |
| 107 | C2×C3×Q8 | — | 4 | 4 | no | — |
| 108 | C2×C3×Q8 | — | 4 | 4 | no | — |
| 109 | C22×C3×C4 | — | 4 | 2 | no | — |
| 110 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 113 | C2×#67 | — | 2 | 2 | no | — |
| 114 | #235 | — | 4 | 2 | no | — |
| 115 | #235 | — | 4 | 2 | no | — |
| 116 | C2×D12 | — | 2 | 1 | no | — |
| 117 | C2×D12 | — | 2 | 1 | no | — |
| 118 | C2×D12 | C2×D4 | 1 | 2 | yes | — |
| 119 | C2×C4×D3 | — | 2 | 2 | no | — |
| 120 | C2×Dic6 | — | 2 | 1 | no | — |
| 121 | C3×Q16 | — | 4 | 4 | no | — |
| 122 | C3×Q16 | — | 4 | 4 | no | — |
| 123 | C3×Q16 | — | 4 | 4 | no | — |
| 124 | C3×Q16 | — | 4 | 4 | no | — |
| 125 | C2×C3×C8 | — | 2 | 2 | no | — |
| 126 | C2×C3×C8 | D8 | 1 | 4 | yes | — |
| 127 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 128 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 129 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 131 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 132 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 133 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 134 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 135 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 136 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 137 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 138 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 139 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 140 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 141 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 142 | C4⋊Dic3 | C2×D4 | 1 | 2 | yes | — |
| 143 | #208 | — | 4 | 2 | no | — |
| 144 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 145 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 146 | ΓL(1,𝔽25) | — | 4 | 4 | no | — |
| 147 | D3×C8 | — | 4 | 4 | no | — |
| 148 | C2×#41 | — | 6 | 1 | no | — |
| 149 | C22×Q8 | — | 2 | 2 | no | — |
| 150 | C2×Q16 | — | 2 | 2 | no | — |
| 151 | C2×Q16 | — | 2 | 2 | no | — |
| 152 | C2×Q16 | — | 2 | 2 | no | — |
| 153 | C2×Q16 | — | 2 | 2 | no | — |
| 154 | C2×Q16 | — | 4 | 4 | no | — |
| 155 | C2×Q16 | — | 4 | 4 | no | — |
| 156 | #131 | — | 12 | 4 | no | — |
| 157 | C2×M16 | — | 6 | 1 | no | — |
| 158 | C2×M16 | — | 6 | 2 | no | — |
| 159 | C22×C8 | — | 6 | 2 | no | — |
| 160 | C22×C8 | #67 | 1 | 1 | yes | characteristic |
| 161 | C22×C8 | #67 | 1 | 1 | yes | characteristic |
| 162 | C4⋊Q8 | — | 2 | 2 | no | — |
| 163 | C4⋊Q8 | — | 2 | 2 | no | — |
| 164 | C4⋊Q8 | — | 2 | 2 | no | — |
| 165 | C4⋊Q8 | — | 3 | 1 | no | — |
| 166 | C4⋊Q8 | C22×D3 | 1 | 4 | yes | — |
| 167 | #127 | — | 3 | 1 | no | — |
| 168 | #124 | — | 6 | 1 | no | — |
| 169 | #121 | — | 3 | 2 | no | — |
| 170 | #121 | — | 6 | 1 | no | — |
| 171 | C4×D4 | — | 3 | 2 | no | — |
| 172 | C4×D4 | — | 6 | 1 | no | — |
| 173 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 174 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 175 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 176 | #107 | — | 6 | 8 | no | — |
| 177 | C4⋊C8 | — | 3 | 2 | no | — |
| 178 | C4⋊C8 | — | 3 | 4 | no | — |
| 179 | C4⋊C8 | — | 6 | 1 | no | — |
| 180 | #102 | — | 6 | 8 | no | — |
| 181 | C22⋊C8 | — | 6 | 2 | no | — |
| 182 | C22⋊C8 | — | 6 | 2 | no | — |
| 183 | C22⋊C8 | — | 6 | 2 | no | — |
| 184 | C4×C8 | — | 6 | 2 | no | — |
| 185 | C4×C8 | #67 | 1 | 1 | yes | characteristic |
| 186 | C4×C8 | #67 | 1 | 1 | yes | characteristic |
| 187 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 188 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 189 | C23×C3 | #121 | 1 | 1 | yes | characteristic |
| 190 | C22×D3 | — | 2 | 2 | no | — |
| 191 | C3×Q8 | — | 4 | 4 | no | — |
| 192 | C3×Q8 | — | 4 | 4 | no | — |
| 193 | C3×Q8 | — | 8 | 4 | no | — |
| 194 | C2×C3×C4 | — | 2 | 2 | no | — |
| 195 | C2×C3×C4 | — | 2 | 2 | no | — |
| 196 | C2×C3×C4 | — | 4 | 4 | no | — |
| 197 | C2×C3×C4 | — | 8 | 4 | no | — |
| 198 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 199 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 200 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 201 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 202 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 203 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 204 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 205 | C2×C3×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 206 | #67 | — | 4 | 4 | no | — |
| 207 | C2×Dic3 | — | 2 | 2 | no | — |
| 208 | D12 | — | 2 | 4 | no | — |
| 209 | D12 | — | 4 | 2 | no | — |
| 210 | D12 | — | 4 | 2 | no | — |
| 211 | C4×D3 | — | 4 | 2 | no | — |
| 212 | C4×D3 | — | 4 | 2 | no | — |
| 213 | Dic6 | — | 4 | 2 | no | — |
| 214 | C3×C8 | — | 2 | 1 | no | — |
| 215 | C3×C8 | — | 2 | 1 | no | — |
| 216 | C3×C8 | C2×D8 | 1 | 2 | yes | — |
| 217 | C3×C8 | C2×D8 | 1 | 2 | yes | — |
| 218 | C3⋊C8 | — | 4 | 2 | no | — |
| 219 | C3⋊C8 | — | 4 | 4 | no | — |
| 220 | #41 | — | 12 | 2 | no | — |
| 221 | #41 | — | 12 | 2 | no | — |
| 222 | C2×Q8 | — | 2 | 4 | no | — |
| 223 | C2×Q8 | — | 4 | 2 | no | — |
| 224 | C2×Q8 | — | 4 | 4 | no | — |
| 225 | C2×Q8 | — | 4 | 4 | no | — |
| 226 | C2×Q8 | — | 6 | 1 | no | — |
| 227 | C2×D4 | — | 3 | 2 | no | — |
| 228 | C2×D4 | — | 6 | 2 | no | — |
| 229 | C2×D4 | — | 6 | 1 | no | — |
| 230 | C2×D4 | — | 6 | 1 | no | — |
| 231 | C22×C4 | — | 4 | 2 | no | — |
| 232 | C22×C4 | — | 6 | 2 | no | — |
| 233 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 234 | C22×C4 | #237 | 1 | 1 | yes | characteristic |
| 235 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 236 | Q16 | — | 4 | 4 | no | — |
| 237 | Q16 | — | 4 | 4 | no | — |
| 238 | Q16 | — | 4 | 4 | no | — |
| 239 | Q16 | — | 4 | 4 | no | — |
| 240 | M16 | — | 12 | 2 | no | — |
| 241 | M16 | — | 12 | 4 | no | — |
| 242 | C2×C8 | — | 2 | 2 | no | — |
| 243 | C2×C8 | — | 6 | 4 | no | — |
| 244 | C2×C8 | — | 6 | 2 | no | — |
| 245 | C2×C8 | — | 12 | 4 | no | — |
| 246 | C2×C8 | #213 | 1 | 4 | yes | — |
| 247 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 251 | C4⋊C4 | — | 2 | 4 | no | — |
| 252 | C4⋊C4 | — | 2 | 4 | no | — |
| 253 | C4⋊C4 | — | 3 | 2 | no | — |
| 254 | C4⋊C4 | — | 4 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 2 | no | — |
| 256 | C4⋊C4 | — | 6 | 1 | no | — |
| 257 | C22⋊C4 | — | 6 | 2 | no | — |
| 258 | C22⋊C4 | — | 6 | 2 | no | — |
| 259 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 260 | C42 | #237 | 1 | 1 | yes | characteristic |
| 261 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 262 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 263 | C22×C3 | — | 2 | 1 | no | — |
| 264 | C22×C3 | — | 2 | 1 | no | — |
| 265 | C22×C3 | #459 | 1 | 2 | yes | — |
| 266 | C22×C3 | C2×#121 | 1 | 1 | yes | characteristic |
| 267 | C2×D3 | — | 4 | 2 | no | — |
| 268 | C2×D3 | — | 4 | 2 | no | — |
| 269 | C2×D3 | — | 4 | 2 | no | — |
| 270 | C3×C4 | — | 2 | 1 | no | — |
| 271 | C3×C4 | — | 2 | 1 | no | — |
| 272 | C3×C4 | — | 8 | 4 | no | — |
| 273 | C3×C4 | #459 | 1 | 2 | yes | — |
| 274 | C3×C4 | #473 | 1 | 2 | yes | — |
| 275 | C3×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 276 | C3×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 277 | C3×C4 | #578 | 1 | 2 | yes | — |
| 278 | Dic3 | — | 4 | 2 | no | — |
| 279 | C23 | — | 6 | 2 | no | — |
| 280 | C23 | #938 | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 4 | 4 | no | — |
| 282 | Q8 | — | 4 | 4 | no | — |
| 283 | Q8 | — | 8 | 4 | no | — |
| 284 | Q8 | — | 12 | 2 | no | — |
| 285 | D4 | — | 6 | 4 | no | — |
| 286 | D4 | — | 12 | 4 | no | — |
| 287 | D4 | — | 12 | 2 | no | — |
| 288 | D4 | — | 12 | 2 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 2 | no | — |
| 291 | C2×C4 | — | 4 | 4 | no | — |
| 292 | C2×C4 | — | 6 | 2 | no | — |
| 293 | C2×C4 | — | 8 | 4 | no | — |
| 294 | C2×C4 | — | 12 | 2 | no | — |
| 295 | C2×C4 | — | 12 | 2 | no | — |
| 296 | C2×C4 | C2×#213 | 1 | 2 | yes | — |
| 297 | C2×C4 | #932 | 1 | 2 | yes | — |
| 298 | C2×C4 | #938 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | #938 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #938 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | C2×#237 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C22×#67 | 1 | 1 | yes | characteristic |
| 304 | C8 | — | 2 | 1 | no | — |
| 305 | C8 | — | 2 | 1 | no | — |
| 306 | C8 | — | 12 | 4 | no | — |
| 307 | C8 | — | 12 | 2 | no | — |
| 308 | C8 | C2×#213 | 1 | 2 | yes | — |
| 309 | C8 | C2×#213 | 1 | 2 | yes | — |
| 310 | C2×C3 | — | 2 | 2 | no | — |
| 311 | C2×C3 | C2×#459 | 1 | 1 | yes | characteristic |
| 312 | C2×C3 | #3059 | 1 | 1 | yes | characteristic |
| 313 | C2×C3 | C2×#473 | 1 | 1 | yes | characteristic |
| 314 | D3 | — | 8 | 2 | no | — |
| 315 | C22 | — | 2 | 1 | no | — |
| 316 | C22 | — | 2 | 1 | no | — |
| 317 | C22 | — | 12 | 2 | no | — |
| 318 | C22 | — | 12 | 2 | no | — |
| 319 | C22 | — | 12 | 2 | no | — |
| 320 | C22 | #5072 | 1 | 2 | yes | — |
| 321 | C22 | C2×#938 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 8 | 4 | no | — |
| 325 | C4 | — | 12 | 2 | no | — |
| 326 | C4 | #5072 | 1 | 2 | yes | — |
| 327 | C4 | #5192 | 1 | 2 | yes | — |
| 328 | C4 | #5766 | 1 | 2 | yes | — |
| 329 | C4 | C22×#213 | 1 | 1 | yes | characteristic |
| 330 | C4 | C2×#932 | 1 | 1 | yes | characteristic |
| 331 | C3 | #19188 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 332 | C2 | — | 2 | 2 | no | — |
| 333 | C2 | — | 24 | 2 | no | — |
| 334 | C2 | C2×#5072 | 1 | 1 | yes | characteristic |
| 335 | C2 | #83887 | 1 | 1 | yes | characteristic |
| 336 | C2 | C2×#5192 | 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