← #10710560
Subgroups — #10710560
Order 768.
296 conjugacy classes of subgroups.
The subgroup lattice has 256 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #71425 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#4511 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | Dic3×M32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #69889 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #69884 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C3×#1378 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67813 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #12667 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#405 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#822 | C4 | 1 | 1 | yes | characteristic |
| 13 | #5101 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 17 | #4998 | C4 | 1 | 1 | yes | characteristic |
| 18 | #4555 | C4 | 1 | 1 | yes | characteristic |
| 19 | #4532 | C4 | 1 | 1 | yes | characteristic |
| 20 | #4512 | — | 2 | 2 | no | — |
| 21 | #4511 | C4 | 1 | 2 | yes | complemented |
| 22 | #4511 | C4 | 1 | 2 | yes | complemented |
| 23 | #4502 | C4 | 1 | 2 | yes | complemented |
| 24 | Dic3×C16 | C4 | 1 | 2 | yes | complemented |
| 25 | #4466 | — | 2 | 2 | no | — |
| 26 | C2×#364 | — | 3 | 1 | no | — |
| 27 | #2110 | — | 3 | 1 | no | — |
| 28 | #2110 | — | 3 | 1 | no | — |
| 29 | C4×M32 | — | 3 | 1 | no | — |
| 30 | #1772 | — | 3 | 1 | no | — |
| 31 | #1378 | — | 3 | 1 | no | — |
| 32 | #1378 | D3 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 35 | C3×#117 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | #926 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C2×C4×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C2×#208 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C22×C3⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C3×M32 | — | 2 | 1 | no | — |
| 41 | C3×M32 | — | 2 | 1 | no | — |
| 42 | C3×M32 | — | 2 | 1 | no | — |
| 43 | C3×M32 | C2×C4 | 1 | 2 | yes | — |
| 44 | C2×C3×C16 | — | 2 | 1 | no | — |
| 45 | C2×C3×C16 | C2×C4 | 1 | 2 | yes | — |
| 46 | C3×#97 | — | 2 | 1 | no | — |
| 47 | C3×C4×C8 | — | 2 | 1 | no | — |
| 48 | #822 | — | 2 | 1 | no | — |
| 49 | #822 | — | 2 | 1 | no | — |
| 50 | #815 | — | 2 | 1 | no | — |
| 51 | C8×Dic3 | — | 2 | 1 | no | — |
| 52 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | #812 | — | 2 | 1 | no | — |
| 57 | #812 | C2×C4 | 1 | 2 | yes | — |
| 58 | #812 | D4 | 1 | 2 | yes | — |
| 59 | #812 | Q8 | 1 | 2 | yes | — |
| 60 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 61 | C2×C3⋊C16 | C2×C4 | 1 | 2 | yes | — |
| 62 | #808 | — | 2 | 1 | no | — |
| 63 | #807 | — | 2 | 1 | no | — |
| 64 | C2×M32 | — | 3 | 1 | no | — |
| 65 | C2×M32 | — | 3 | 1 | no | — |
| 66 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 67 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 68 | C2×#108 | — | 3 | 1 | no | — |
| 69 | #427 | — | 3 | 1 | no | — |
| 70 | #405 | — | 3 | 1 | no | — |
| 71 | #405 | C2×D3 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×C8 | — | 3 | 1 | no | — |
| 73 | #364 | — | 3 | 2 | no | — |
| 74 | #364 | — | 3 | 2 | no | — |
| 75 | #347 | — | 6 | 2 | no | — |
| 76 | #347 | — | 6 | 2 | no | — |
| 77 | #346 | — | 3 | 2 | no | — |
| 78 | C4×C16 | — | 3 | 2 | no | — |
| 79 | #339 | — | 3 | 1 | no | — |
| 80 | #339 | — | 3 | 1 | no | — |
| 81 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C22×Dic3 | — | 2 | 1 | no | — |
| 83 | C3×M16 | — | 2 | 1 | no | — |
| 84 | C3×M16 | — | 2 | 1 | no | — |
| 85 | C2×C3×C8 | — | 2 | 1 | no | — |
| 86 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 87 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 90 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 91 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 92 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 93 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 94 | C3×C42 | — | 2 | 1 | no | — |
| 95 | #217 | — | 2 | 1 | no | — |
| 96 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 97 | C4×Dic3 | — | 2 | 1 | no | — |
| 98 | C4×Dic3 | — | 2 | 1 | no | — |
| 99 | C4×Dic3 | C42 | 1 | 1 | yes | characteristic |
| 100 | C4×Dic3 | C42 | 1 | 1 | yes | characteristic |
| 101 | #208 | — | 2 | 1 | no | — |
| 102 | #208 | — | 2 | 1 | no | — |
| 103 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 104 | C2×C3⋊C8 | C42 | 1 | 1 | yes | characteristic |
| 105 | C2×C3⋊C8 | C42 | 1 | 1 | yes | characteristic |
| 106 | C2×C3⋊C8 | M16 | 1 | 1 | yes | characteristic |
| 107 | C2×C3⋊C8 | M16 | 1 | 1 | yes | characteristic |
| 108 | C2×C3⋊C8 | M16 | 1 | 1 | yes | characteristic |
| 109 | C2×C3⋊C8 | M16 | 1 | 1 | yes | characteristic |
| 110 | C3×C16 | — | 2 | 2 | no | — |
| 111 | C3×C16 | — | 2 | 2 | no | — |
| 112 | C3⋊C16 | — | 2 | 2 | no | — |
| 113 | C3⋊C16 | C4⋊C4 | 1 | 4 | yes | — |
| 114 | C2×M16 | — | 3 | 1 | no | — |
| 115 | C2×M16 | C4×D3 | 1 | 1 | yes | characteristic |
| 116 | C22×C8 | — | 3 | 1 | no | — |
| 117 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 118 | #117 | — | 3 | 1 | no | — |
| 119 | #117 | C4×D3 | 1 | 1 | yes | characteristic |
| 120 | C2×C42 | — | 3 | 1 | no | — |
| 121 | M32 | — | 2 | 1 | no | — |
| 122 | M32 | — | 2 | 1 | no | — |
| 123 | M32 | — | 2 | 1 | no | — |
| 124 | M32 | — | 3 | 2 | no | — |
| 125 | M32 | — | 3 | 2 | no | — |
| 126 | M32 | — | 3 | 2 | no | — |
| 127 | M32 | — | 6 | 1 | no | — |
| 128 | M32 | C4×D3 | 1 | 2 | yes | — |
| 129 | C2×C16 | — | 2 | 1 | no | — |
| 130 | C2×C16 | — | 3 | 2 | no | — |
| 131 | C2×C16 | — | 6 | 1 | no | — |
| 132 | C2×C16 | C4×D3 | 1 | 2 | yes | — |
| 133 | #108 | — | 6 | 1 | no | — |
| 134 | #108 | — | 6 | 1 | no | — |
| 135 | #107 | — | 6 | 1 | no | — |
| 136 | #106 | — | 6 | 1 | no | — |
| 137 | #97 | — | 2 | 1 | no | — |
| 138 | #97 | — | 6 | 1 | no | — |
| 139 | C4×C8 | — | 2 | 1 | no | — |
| 140 | C4×C8 | — | 3 | 1 | no | — |
| 141 | C4×C8 | — | 3 | 1 | no | — |
| 142 | C4×C8 | — | 3 | 1 | no | — |
| 143 | C4×C8 | — | 3 | 1 | no | — |
| 144 | C4×C8 | — | 6 | 1 | no | — |
| 145 | C23×C3 | #131 | 1 | 1 | yes | characteristic |
| 146 | C2×C3×C4 | — | 4 | 1 | no | — |
| 147 | C2×C3×C4 | C2×C42 | 1 | 1 | yes | characteristic |
| 148 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 149 | C2×C3×C4 | C4×Q8 | 1 | 1 | yes | characteristic |
| 150 | C2×C3×C4 | C2×M16 | 1 | 1 | yes | characteristic |
| 151 | C2×C3×C4 | C2×M16 | 1 | 1 | yes | characteristic |
| 152 | C2×C3×C4 | #131 | 1 | 1 | yes | characteristic |
| 153 | C2×Dic3 | — | 2 | 1 | no | — |
| 154 | C2×Dic3 | — | 2 | 1 | no | — |
| 155 | C2×Dic3 | — | 4 | 2 | no | — |
| 156 | C2×Dic3 | — | 4 | 1 | no | — |
| 157 | C3×C8 | — | 4 | 1 | no | — |
| 158 | C3×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 159 | C3×C8 | #117 | 1 | 1 | yes | characteristic |
| 160 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 161 | C3×C8 | C4×Q8 | 1 | 1 | yes | characteristic |
| 162 | C3⋊C8 | — | 4 | 1 | no | — |
| 163 | C3⋊C8 | #97 | 1 | 1 | yes | characteristic |
| 164 | C3⋊C8 | #97 | 1 | 1 | yes | characteristic |
| 165 | C3⋊C8 | #97 | 1 | 1 | yes | characteristic |
| 166 | C3⋊C8 | #97 | 1 | 1 | yes | characteristic |
| 167 | C22×C4 | — | 6 | 1 | no | — |
| 168 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 169 | M16 | — | 2 | 1 | no | — |
| 170 | M16 | — | 2 | 1 | no | — |
| 171 | M16 | — | 6 | 1 | no | — |
| 172 | M16 | — | 6 | 1 | no | — |
| 173 | C2×C8 | — | 2 | 1 | no | — |
| 174 | C2×C8 | — | 3 | 1 | no | — |
| 175 | C2×C8 | — | 3 | 1 | no | — |
| 176 | C2×C8 | — | 3 | 1 | no | — |
| 177 | C2×C8 | — | 3 | 1 | no | — |
| 178 | C2×C8 | — | 3 | 1 | no | — |
| 179 | C2×C8 | — | 3 | 1 | no | — |
| 180 | C2×C8 | — | 6 | 1 | no | — |
| 181 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 182 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | D3×Q8 | 1 | 1 | yes | characteristic |
| 187 | C4⋊C4 | — | 2 | 1 | no | — |
| 188 | C4⋊C4 | — | 6 | 1 | no | — |
| 189 | C22⋊C4 | — | 2 | 1 | no | — |
| 190 | C22⋊C4 | — | 6 | 1 | no | — |
| 191 | C42 | — | 2 | 1 | no | — |
| 192 | C42 | — | 3 | 1 | no | — |
| 193 | C42 | — | 3 | 1 | no | — |
| 194 | C42 | — | 6 | 1 | no | — |
| 195 | C42 | — | 6 | 1 | no | — |
| 196 | C16 | — | 2 | 2 | no | — |
| 197 | C16 | — | 2 | 2 | no | — |
| 198 | C16 | — | 3 | 4 | no | — |
| 199 | C16 | — | 6 | 2 | no | — |
| 200 | C22×C3 | — | 4 | 1 | no | — |
| 201 | C22×C3 | #405 | 1 | 1 | yes | characteristic |
| 202 | C22×C3 | #436 | 1 | 1 | yes | characteristic |
| 203 | C22×C3 | #446 | 1 | 1 | yes | characteristic |
| 204 | C3×C4 | — | 8 | 1 | no | — |
| 205 | C3×C4 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 206 | C3×C4 | C2×#97 | 1 | 1 | yes | characteristic |
| 207 | C3×C4 | #436 | 1 | 1 | yes | characteristic |
| 208 | C3×C4 | #446 | 1 | 1 | yes | characteristic |
| 209 | Dic3 | — | 2 | 2 | no | — |
| 210 | Dic3 | — | 4 | 1 | no | — |
| 211 | Dic3 | — | 8 | 1 | no | — |
| 212 | C23 | #901 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | — | 4 | 1 | no | — |
| 214 | C2×C4 | — | 6 | 1 | no | — |
| 215 | C2×C4 | — | 6 | 1 | no | — |
| 216 | C2×C4 | — | 12 | 2 | no | — |
| 217 | C2×C4 | — | 12 | 1 | no | — |
| 218 | C2×C4 | C42×D3 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | #887 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #901 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | D3×M16 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | D3×M16 | 1 | 1 | yes | characteristic |
| 224 | C8 | — | 3 | 1 | no | — |
| 225 | C8 | — | 3 | 1 | no | — |
| 226 | C8 | — | 3 | 1 | no | — |
| 227 | C8 | — | 3 | 1 | no | — |
| 228 | C8 | — | 4 | 1 | no | — |
| 229 | C8 | — | 12 | 1 | no | — |
| 230 | C8 | #877 | 1 | 1 | yes | characteristic |
| 231 | C8 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 232 | C8 | #887 | 1 | 1 | yes | characteristic |
| 233 | C8 | D3×C4⋊C4 | 1 | 1 | yes | characteristic |
| 234 | C2×C3 | — | 2 | 1 | no | — |
| 235 | C2×C3 | — | 2 | 1 | no | — |
| 236 | C2×C3 | — | 2 | 1 | no | — |
| 237 | C2×C3 | #1770 | 1 | 1 | yes | characteristic |
| 238 | C22 | — | 4 | 1 | no | — |
| 239 | C22 | #4689 | 1 | 1 | yes | characteristic |
| 240 | C22 | #4729 | 1 | 1 | yes | characteristic |
| 241 | C22 | #4828 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 6 | 2 | no | — |
| 243 | C4 | — | 8 | 1 | no | — |
| 244 | C4 | — | 12 | 1 | no | — |
| 245 | C4 | — | 24 | 1 | no | — |
| 246 | C4 | #4649 | 1 | 1 | yes | characteristic |
| 247 | C4 | D3×#97 | 1 | 1 | yes | characteristic |
| 248 | C4 | #4729 | 1 | 1 | yes | characteristic |
| 249 | C4 | #4828 | 1 | 1 | yes | characteristic |
| 250 | C3 | #12667 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 251 | C2 | — | 2 | 1 | no | — |
| 252 | C2 | — | 2 | 1 | no | — |
| 253 | C2 | — | 2 | 1 | no | — |
| 254 | C2 | #68652 | 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