← #10765752
Subgroups — #10765752
Order 768.
734 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 | #79041 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #78470 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×#1566 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #70498 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#4455 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #68525 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #68518 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #17962 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×#907 | — | 2 | 2 | no | — |
| 10 | #5477 | C22 | 1 | 2 | yes | — |
| 11 | C3×#423 | C22 | 1 | 2 | yes | — |
| 12 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5129 | — | 2 | 2 | no | — |
| 14 | #5115 | C4 | 1 | 4 | yes | — |
| 15 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 17 | #4838 | — | 2 | 2 | no | — |
| 18 | #4838 | C22 | 1 | 4 | yes | — |
| 19 | #4834 | — | 2 | 2 | no | — |
| 20 | #4834 | C22 | 1 | 4 | yes | — |
| 21 | #4695 | C4 | 1 | 1 | yes | characteristic |
| 22 | #4694 | C4 | 1 | 2 | yes | — |
| 23 | C4×D24 | C22 | 1 | 4 | yes | — |
| 24 | #4682 | C4 | 1 | 1 | yes | characteristic |
| 25 | C3×#334 | C22 | 1 | 4 | yes | — |
| 26 | #4483 | C22 | 1 | 8 | yes | — |
| 27 | #4455 | C22 | 1 | 2 | yes | — |
| 28 | #4455 | C22 | 1 | 2 | yes | — |
| 29 | #3169 | — | 3 | 1 | no | — |
| 30 | #2931 | — | 3 | 2 | no | — |
| 31 | #1594 | — | 3 | 4 | no | — |
| 32 | #1580 | — | 3 | 4 | no | — |
| 33 | #1566 | D3 | 1 | 1 | yes | characteristic, complemented |
| 34 | #1492 | — | 3 | 2 | no | — |
| 35 | C2×#322 | — | 3 | 1 | no | — |
| 36 | C2×#235 | — | 2 | 2 | no | — |
| 37 | C2×C3×M16 | — | 2 | 2 | no | — |
| 38 | C22×C3×C8 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 40 | #930 | — | 2 | 2 | no | — |
| 41 | #930 | C2×C4 | 1 | 4 | yes | — |
| 42 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 43 | C22×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 44 | #907 | — | 4 | 8 | no | — |
| 45 | C2×D24 | C2×C4 | 1 | 4 | yes | — |
| 46 | C2×ΓL(1,𝔽25) | — | 2 | 4 | no | — |
| 47 | C2×D3×C8 | — | 2 | 4 | no | — |
| 48 | #875 | — | 2 | 2 | no | — |
| 49 | #874 | C2×C4 | 1 | 2 | yes | — |
| 50 | C4×D12 | — | 2 | 2 | no | — |
| 51 | C4×D12 | C23 | 1 | 4 | yes | — |
| 52 | #869 | C2×C4 | 1 | 2 | yes | — |
| 53 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 54 | C3×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 55 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 57 | C3×C4×C8 | C23 | 1 | 2 | yes | — |
| 58 | #821 | C2×C4 | 1 | 8 | yes | — |
| 59 | (C2×D3)⋊C8 | — | 2 | 4 | no | — |
| 60 | (C2×D3)⋊C8 | — | 2 | 4 | no | — |
| 61 | #818 | C2×C4 | 1 | 4 | yes | — |
| 62 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 63 | C4×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 64 | C4×C3⋊C8 | C23 | 1 | 2 | yes | — |
| 65 | C4×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 67 | C2×#131 | — | 6 | 2 | no | — |
| 68 | #532 | — | 3 | 2 | no | — |
| 69 | #500 | — | 3 | 1 | no | — |
| 70 | #493 | — | 3 | 1 | no | — |
| 71 | #486 | — | 3 | 2 | no | — |
| 72 | #466 | — | 3 | 4 | no | — |
| 73 | C4×D8 | — | 3 | 4 | no | — |
| 74 | #436 | — | 3 | 4 | no | — |
| 75 | #436 | — | 6 | 2 | no | — |
| 76 | C8×D4 | — | 3 | 4 | no | — |
| 77 | C8×D4 | — | 6 | 2 | no | — |
| 78 | #423 | C2×D3 | 1 | 2 | yes | — |
| 79 | #408 | — | 6 | 2 | no | — |
| 80 | C2×C4×C8 | — | 3 | 1 | no | — |
| 81 | C2×C4×C8 | — | 3 | 1 | no | — |
| 82 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 83 | #334 | C2×D3 | 1 | 4 | yes | — |
| 84 | D4⋊C8 | — | 3 | 8 | no | — |
| 85 | #322 | — | 3 | 2 | no | — |
| 86 | #322 | — | 3 | 2 | no | — |
| 87 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 90 | C2×#67 | — | 2 | 4 | no | — |
| 91 | #235 | — | 4 | 4 | no | — |
| 92 | #235 | — | 4 | 4 | no | — |
| 93 | C2×D12 | — | 2 | 2 | no | — |
| 94 | C2×D12 | — | 2 | 2 | no | — |
| 95 | C2×D12 | C22×C4 | 1 | 4 | yes | — |
| 96 | C2×C4×D3 | — | 2 | 4 | no | — |
| 97 | C2×Dic6 | — | 2 | 2 | no | — |
| 98 | C3×M16 | — | 4 | 4 | no | — |
| 99 | C2×C3×C8 | — | 2 | 4 | no | — |
| 100 | C2×C3×C8 | — | 2 | 4 | no | — |
| 101 | C2×C3×C8 | C22×C4 | 1 | 2 | yes | — |
| 102 | C2×C3×C8 | C22×C4 | 1 | 2 | yes | — |
| 103 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 104 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 106 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 107 | (C2×D3)⋊C4 | — | 2 | 4 | no | — |
| 108 | (C2×D3)⋊C4 | — | 2 | 4 | no | — |
| 109 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 110 | C4⋊Dic3 | C22×C4 | 1 | 4 | yes | — |
| 111 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 112 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 113 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 114 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 115 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 116 | D24 | — | 2 | 8 | no | — |
| 117 | ΓL(1,𝔽25) | — | 4 | 8 | no | — |
| 118 | D3×C8 | — | 4 | 8 | no | — |
| 119 | C2×#41 | — | 6 | 2 | no | — |
| 120 | C2×D8 | — | 3 | 4 | no | — |
| 121 | #131 | — | 12 | 8 | no | — |
| 122 | C2×M16 | — | 2 | 2 | no | — |
| 123 | C2×M16 | — | 6 | 4 | no | — |
| 124 | C22×C8 | — | 3 | 2 | no | — |
| 125 | C22×C8 | — | 6 | 4 | no | — |
| 126 | C22×C8 | — | 6 | 1 | no | — |
| 127 | C22×C8 | C4×D3 | 1 | 2 | yes | — |
| 128 | C4⋊Q8 | — | 3 | 2 | no | — |
| 129 | #127 | — | 3 | 2 | no | — |
| 130 | #124 | — | 6 | 2 | no | — |
| 131 | #121 | — | 3 | 4 | no | — |
| 132 | #121 | — | 6 | 2 | no | — |
| 133 | C4×D4 | — | 3 | 4 | no | — |
| 134 | C4×D4 | — | 6 | 2 | no | — |
| 135 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 136 | #107 | — | 3 | 4 | no | — |
| 137 | C4⋊C8 | — | 2 | 2 | no | — |
| 138 | C4⋊C8 | C22×D3 | 1 | 4 | yes | — |
| 139 | #102 | — | 3 | 8 | no | — |
| 140 | C22⋊C8 | — | 6 | 4 | no | — |
| 141 | C22⋊C8 | — | 6 | 4 | no | — |
| 142 | C4×C8 | — | 3 | 2 | no | — |
| 143 | C4×C8 | — | 3 | 1 | no | — |
| 144 | C4×C8 | — | 3 | 1 | no | — |
| 145 | C4×C8 | — | 3 | 2 | no | — |
| 146 | C4×C8 | — | 6 | 1 | no | — |
| 147 | C4×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 148 | C4×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 149 | C4×C8 | C22×D3 | 1 | 2 | yes | — |
| 150 | C23×C3 | C4×D4 | 1 | 1 | yes | characteristic |
| 151 | C22×D3 | — | 2 | 4 | no | — |
| 152 | C2×C3×C4 | — | 2 | 2 | no | — |
| 153 | C2×C3×C4 | — | 2 | 2 | no | — |
| 154 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 155 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 156 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 157 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 158 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 159 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 160 | C2×C3×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 161 | #67 | — | 4 | 8 | no | — |
| 162 | C2×Dic3 | — | 2 | 4 | no | — |
| 163 | D12 | — | 2 | 8 | no | — |
| 164 | D12 | — | 4 | 4 | no | — |
| 165 | D12 | — | 4 | 4 | no | — |
| 166 | C4×D3 | — | 4 | 4 | no | — |
| 167 | C4×D3 | — | 4 | 4 | no | — |
| 168 | Dic6 | — | 4 | 4 | no | — |
| 169 | C3×C8 | — | 2 | 2 | no | — |
| 170 | C3×C8 | — | 4 | 4 | no | — |
| 171 | C3×C8 | #131 | 1 | 4 | yes | complemented |
| 172 | C3⋊C8 | — | 2 | 2 | no | — |
| 173 | C3⋊C8 | — | 2 | 2 | no | — |
| 174 | C3⋊C8 | — | 4 | 2 | no | — |
| 175 | #41 | — | 12 | 4 | no | — |
| 176 | #41 | — | 12 | 4 | no | — |
| 177 | C2×Q8 | — | 6 | 2 | no | — |
| 178 | C2×D4 | — | 3 | 4 | no | — |
| 179 | C2×D4 | — | 6 | 4 | no | — |
| 180 | C2×D4 | — | 6 | 2 | no | — |
| 181 | C2×D4 | — | 6 | 2 | no | — |
| 182 | C22×C4 | — | 6 | 4 | no | — |
| 183 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 184 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 185 | C22×C4 | #239 | 1 | 1 | yes | characteristic |
| 186 | D8 | — | 6 | 8 | no | — |
| 187 | M16 | — | 4 | 4 | no | — |
| 188 | M16 | — | 12 | 8 | no | — |
| 189 | C2×C8 | — | 2 | 4 | no | — |
| 190 | C2×C8 | — | 2 | 4 | no | — |
| 191 | C2×C8 | — | 3 | 2 | no | — |
| 192 | C2×C8 | — | 3 | 2 | no | — |
| 193 | C2×C8 | — | 6 | 4 | no | — |
| 194 | C2×C8 | — | 6 | 2 | no | — |
| 195 | C2×C8 | — | 12 | 8 | no | — |
| 196 | C2×C8 | — | 12 | 2 | no | — |
| 197 | C2×C8 | C2×C4×D3 | 1 | 2 | yes | — |
| 198 | C2×C8 | C2×C4×D3 | 1 | 2 | yes | — |
| 199 | C4⋊C4 | — | 3 | 4 | no | — |
| 200 | C4⋊C4 | — | 6 | 2 | no | — |
| 201 | C22⋊C4 | — | 6 | 4 | no | — |
| 202 | C22⋊C4 | — | 6 | 4 | no | — |
| 203 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 204 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 205 | C42 | #239 | 1 | 1 | yes | characteristic |
| 206 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 207 | C22×C3 | — | 2 | 1 | no | — |
| 208 | C22×C3 | — | 2 | 1 | no | — |
| 209 | C22×C3 | #442 | 1 | 2 | yes | — |
| 210 | C22×C3 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 211 | C2×D3 | — | 4 | 4 | no | — |
| 212 | C2×D3 | — | 4 | 4 | no | — |
| 213 | C2×D3 | — | 4 | 4 | no | — |
| 214 | C3×C4 | — | 2 | 1 | no | — |
| 215 | C3×C4 | — | 2 | 1 | no | — |
| 216 | C3×C4 | #442 | 1 | 2 | yes | — |
| 217 | C3×C4 | #444 | 1 | 2 | yes | — |
| 218 | C3×C4 | C2×#131 | 1 | 2 | yes | — |
| 219 | C3×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 220 | Dic3 | — | 4 | 4 | no | — |
| 221 | C23 | — | 6 | 4 | no | — |
| 222 | C23 | #893 | 1 | 1 | yes | characteristic |
| 223 | Q8 | — | 12 | 4 | no | — |
| 224 | D4 | — | 6 | 8 | no | — |
| 225 | D4 | — | 12 | 8 | no | — |
| 226 | D4 | — | 12 | 4 | no | — |
| 227 | D4 | — | 12 | 4 | no | — |
| 228 | C2×C4 | — | 2 | 2 | no | — |
| 229 | C2×C4 | — | 2 | 2 | no | — |
| 230 | C2×C4 | — | 6 | 4 | no | — |
| 231 | C2×C4 | — | 12 | 4 | no | — |
| 232 | C2×C4 | — | 12 | 4 | no | — |
| 233 | C2×C4 | #893 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | #893 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | #893 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | #914 | 1 | 4 | yes | — |
| 237 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | C2×#239 | 1 | 1 | yes | characteristic |
| 240 | C8 | — | 2 | 2 | no | — |
| 241 | C8 | — | 4 | 4 | no | — |
| 242 | C8 | — | 6 | 2 | no | — |
| 243 | C8 | — | 6 | 2 | no | — |
| 244 | C8 | — | 12 | 2 | no | — |
| 245 | C8 | #907 | 1 | 4 | yes | complemented |
| 246 | C2×C3 | — | 2 | 2 | no | — |
| 247 | C2×C3 | #2931 | 1 | 1 | yes | characteristic |
| 248 | C2×C3 | C2×#442 | 1 | 1 | yes | characteristic |
| 249 | C2×C3 | C2×#444 | 1 | 1 | yes | characteristic |
| 250 | D3 | — | 8 | 4 | no | — |
| 251 | C22 | — | 2 | 1 | no | — |
| 252 | C22 | — | 2 | 1 | no | — |
| 253 | C22 | — | 12 | 4 | no | — |
| 254 | C22 | — | 12 | 4 | no | — |
| 255 | C22 | — | 12 | 4 | no | — |
| 256 | C22 | #4869 | 1 | 2 | yes | — |
| 257 | C22 | C2×#893 | 1 | 1 | yes | characteristic |
| 258 | C4 | — | 2 | 1 | no | — |
| 259 | C4 | — | 2 | 1 | no | — |
| 260 | C4 | — | 12 | 4 | no | — |
| 261 | C4 | #4869 | 1 | 2 | yes | — |
| 262 | C4 | #4894 | 1 | 2 | yes | — |
| 263 | C4 | C2×#907 | 1 | 2 | yes | — |
| 264 | C4 | C2×#914 | 1 | 2 | yes | — |
| 265 | C3 | #17962 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 266 | C2 | — | 2 | 2 | no | — |
| 267 | C2 | — | 24 | 4 | no | — |
| 268 | C2 | #79041 | 1 | 1 | yes | characteristic |
| 269 | C2 | C2×#4869 | 1 | 1 | yes | characteristic |
| 270 | C2 | C2×#4894 | 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