← #10711770
Subgroups — #10711770
Order 768.
394 conjugacy classes of subgroups.
The subgroup lattice has 306 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #83784 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#1814 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72006 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #70685 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#4494 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #67726 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #67725 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #12934 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×#131 | C22 | 1 | 1 | yes | characteristic |
| 10 | #5827 | — | 2 | 1 | no | — |
| 11 | C2×C3×#108 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×#427 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×#421 | — | 2 | 1 | no | — |
| 14 | #5121 | — | 2 | 1 | no | — |
| 15 | #5121 | — | 2 | 2 | no | — |
| 16 | #5106 | — | 2 | 1 | no | — |
| 17 | #5106 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#817 | — | 2 | 1 | no | — |
| 19 | C2×#817 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 22 | C3×#344 | — | 2 | 1 | no | — |
| 23 | #4561 | C4 | 1 | 4 | yes | complemented |
| 24 | #4494 | — | 2 | 2 | no | — |
| 25 | #4494 | — | 2 | 1 | no | — |
| 26 | #4494 | — | 2 | 1 | no | — |
| 27 | #2912 | — | 3 | 1 | no | — |
| 28 | #2155 | — | 3 | 1 | no | — |
| 29 | C2×#365 | — | 3 | 1 | no | — |
| 30 | #2117 | — | 3 | 1 | no | — |
| 31 | #1814 | D3 | 1 | 1 | yes | characteristic |
| 32 | #1386 | — | 3 | 1 | no | — |
| 33 | #1384 | — | 3 | 1 | no | — |
| 34 | C2×C3×#41 | D4 | 1 | 1 | yes | characteristic |
| 35 | C3×#131 | — | 2 | 1 | no | — |
| 36 | C3×#131 | — | 2 | 1 | no | — |
| 37 | C3×#131 | C2×C4 | 1 | 2 | yes | — |
| 38 | C3×#131 | Q8 | 1 | 2 | yes | — |
| 39 | C2×C3×M16 | — | 2 | 1 | no | — |
| 40 | C2×C3×M16 | — | 2 | 1 | no | — |
| 41 | C2×C3×M16 | D4 | 1 | 1 | yes | characteristic |
| 42 | C22×C3×C8 | — | 2 | 1 | no | — |
| 43 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C3×#117 | — | 2 | 1 | no | — |
| 45 | Q8×Dic3 | — | 4 | 1 | no | — |
| 46 | D4×Dic3 | — | 4 | 2 | no | — |
| 47 | D4×Dic3 | — | 4 | 1 | no | — |
| 48 | #926 | — | 2 | 2 | no | — |
| 49 | #926 | — | 2 | 1 | no | — |
| 50 | C2×C4⋊Dic3 | — | 2 | 2 | no | — |
| 51 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 52 | C3×#108 | — | 2 | 1 | no | — |
| 53 | C3×#108 | — | 2 | 1 | no | — |
| 54 | C3×#107 | — | 2 | 1 | no | — |
| 55 | C3×#106 | — | 2 | 1 | no | — |
| 56 | C3×#104 | — | 4 | 2 | no | — |
| 57 | #818 | — | 2 | 2 | no | — |
| 58 | #818 | — | 2 | 1 | no | — |
| 59 | #817 | — | 2 | 2 | no | — |
| 60 | #817 | — | 2 | 1 | no | — |
| 61 | #817 | D4 | 1 | 2 | yes | — |
| 62 | #817 | D4 | 1 | 1 | yes | characteristic |
| 63 | #817 | D4 | 1 | 1 | yes | characteristic |
| 64 | #812 | — | 2 | 1 | no | — |
| 65 | #812 | — | 2 | 1 | no | — |
| 66 | #812 | — | 2 | 1 | no | — |
| 67 | #812 | C2×C4 | 1 | 2 | yes | complemented |
| 68 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 69 | C2×C3⋊C16 | C2×C4 | 1 | 2 | yes | complemented |
| 70 | C2×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 71 | #520 | — | 6 | 1 | no | — |
| 72 | C2×M32 | — | 3 | 1 | no | — |
| 73 | C2×M32 | — | 3 | 1 | no | — |
| 74 | C2×#108 | C2×D3 | 1 | 1 | yes | characteristic |
| 75 | #428 | — | 6 | 2 | no | — |
| 76 | #428 | — | 6 | 1 | no | — |
| 77 | #427 | — | 3 | 1 | no | — |
| 78 | #427 | — | 6 | 1 | no | — |
| 79 | #427 | C2×D3 | 1 | 1 | yes | characteristic |
| 80 | C2×#106 | — | 3 | 1 | no | — |
| 81 | C2×#106 | — | 6 | 1 | no | — |
| 82 | #421 | — | 2 | 1 | no | — |
| 83 | #365 | — | 6 | 2 | no | — |
| 84 | #365 | — | 6 | 1 | no | — |
| 85 | #365 | — | 6 | 1 | no | — |
| 86 | #350 | — | 3 | 4 | no | — |
| 87 | #344 | — | 2 | 1 | no | — |
| 88 | C3×#41 | — | 2 | 1 | no | — |
| 89 | C3×#41 | — | 2 | 1 | no | — |
| 90 | C3×#41 | C4⋊C4 | 1 | 2 | yes | — |
| 91 | C3×#41 | SD16 | 1 | 2 | yes | — |
| 92 | C2×C3×Q8 | SD16 | 1 | 1 | yes | characteristic |
| 93 | C2×C3×D4 | — | 2 | 1 | no | — |
| 94 | C2×C3×D4 | SD16 | 1 | 1 | yes | characteristic |
| 95 | C22×C3×C4 | — | 2 | 1 | no | — |
| 96 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C22×Dic3 | — | 4 | 2 | no | — |
| 98 | C22×Dic3 | — | 4 | 1 | no | — |
| 99 | C3×M16 | — | 2 | 1 | no | — |
| 100 | C3×M16 | — | 2 | 2 | no | — |
| 101 | C3×M16 | — | 2 | 2 | no | — |
| 102 | C3×M16 | — | 4 | 1 | no | — |
| 103 | C3×M16 | — | 4 | 1 | no | — |
| 104 | C3×M16 | C4⋊C4 | 1 | 2 | yes | — |
| 105 | C2×C3×C8 | — | 2 | 1 | no | — |
| 106 | C2×C3×C8 | — | 2 | 2 | no | — |
| 107 | C2×C3×C8 | — | 2 | 2 | no | — |
| 108 | C2×C3×C8 | — | 4 | 1 | no | — |
| 109 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 113 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 114 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 115 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 116 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 117 | C3×C42 | — | 4 | 1 | no | — |
| 118 | #217 | — | 4 | 2 | no | — |
| 119 | #217 | — | 4 | 1 | no | — |
| 120 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 121 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 122 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 123 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 124 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 125 | C4×Dic3 | — | 4 | 2 | no | — |
| 126 | C4×Dic3 | — | 4 | 1 | no | — |
| 127 | C3⋊C16 | — | 2 | 2 | no | — |
| 128 | C3⋊C16 | — | 2 | 2 | no | — |
| 129 | C2×#41 | #67 | 1 | 1 | yes | characteristic |
| 130 | #131 | — | 2 | 1 | no | — |
| 131 | #131 | — | 2 | 1 | no | — |
| 132 | #131 | Dic6 | 1 | 2 | yes | — |
| 133 | #131 | C4×D3 | 1 | 2 | yes | — |
| 134 | C2×M16 | — | 2 | 1 | no | — |
| 135 | C2×M16 | — | 2 | 1 | no | — |
| 136 | C2×M16 | #67 | 1 | 1 | yes | characteristic |
| 137 | C22×C8 | — | 2 | 1 | no | — |
| 138 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 139 | C4×Q8 | — | 12 | 1 | no | — |
| 140 | C4×D4 | — | 12 | 2 | no | — |
| 141 | C4×D4 | — | 12 | 1 | no | — |
| 142 | #117 | — | 2 | 1 | no | — |
| 143 | #117 | — | 6 | 2 | no | — |
| 144 | #117 | — | 6 | 1 | no | — |
| 145 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 146 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 147 | M32 | — | 3 | 2 | no | — |
| 148 | M32 | — | 6 | 1 | no | — |
| 149 | M32 | — | 6 | 1 | no | — |
| 150 | M32 | — | 6 | 1 | no | — |
| 151 | C2×C16 | — | 3 | 2 | no | — |
| 152 | C2×C16 | — | 6 | 1 | no | — |
| 153 | #108 | — | 2 | 1 | no | — |
| 154 | #108 | — | 2 | 1 | no | — |
| 155 | #107 | — | 2 | 1 | no | — |
| 156 | #107 | — | 6 | 2 | no | — |
| 157 | #107 | — | 6 | 1 | no | — |
| 158 | #106 | — | 2 | 1 | no | — |
| 159 | #106 | — | 3 | 2 | no | — |
| 160 | #106 | — | 3 | 1 | no | — |
| 161 | #106 | — | 3 | 1 | no | — |
| 162 | #106 | — | 6 | 2 | no | — |
| 163 | #106 | — | 6 | 1 | no | — |
| 164 | #104 | — | 4 | 2 | no | — |
| 165 | C23×C3 | — | 2 | 1 | no | — |
| 166 | C23×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 167 | C3×Q8 | — | 2 | 1 | no | — |
| 168 | C3×Q8 | #106 | 1 | 2 | yes | — |
| 169 | C3×D4 | — | 2 | 2 | no | — |
| 170 | C3×D4 | — | 2 | 2 | no | — |
| 171 | C3×D4 | — | 2 | 1 | no | — |
| 172 | C3×D4 | #106 | 1 | 2 | yes | — |
| 173 | C2×C3×C4 | — | 2 | 1 | no | — |
| 174 | C2×C3×C4 | — | 2 | 2 | no | — |
| 175 | C2×C3×C4 | — | 2 | 2 | no | — |
| 176 | C2×C3×C4 | — | 8 | 1 | no | — |
| 177 | C2×C3×C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 179 | C2×C3×C4 | #123 | 1 | 1 | yes | characteristic |
| 180 | C2×C3×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 181 | C2×C3×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 182 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 183 | C2×Dic3 | — | 4 | 1 | no | — |
| 184 | C2×Dic3 | — | 4 | 2 | no | — |
| 185 | C2×Dic3 | — | 4 | 1 | no | — |
| 186 | C2×Dic3 | — | 16 | 1 | no | — |
| 187 | C2×Dic3 | — | 16 | 2 | no | — |
| 188 | C3×C8 | — | 2 | 2 | no | — |
| 189 | C3×C8 | — | 8 | 1 | no | — |
| 190 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 191 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 192 | C3×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 193 | C3×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 194 | #41 | — | 2 | 1 | no | — |
| 195 | #41 | — | 2 | 1 | no | — |
| 196 | #41 | Dic3⋊C4 | 1 | 2 | yes | — |
| 197 | #41 | #214 | 1 | 2 | yes | — |
| 198 | C2×Q8 | #215 | 1 | 1 | yes | characteristic |
| 199 | C2×D4 | — | 2 | 1 | no | — |
| 200 | C2×D4 | #215 | 1 | 1 | yes | characteristic |
| 201 | C22×C4 | — | 2 | 1 | no | — |
| 202 | C22×C4 | — | 12 | 2 | no | — |
| 203 | C22×C4 | — | 12 | 1 | no | — |
| 204 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 205 | M16 | — | 2 | 1 | no | — |
| 206 | M16 | — | 2 | 2 | no | — |
| 207 | M16 | — | 2 | 2 | no | — |
| 208 | M16 | — | 4 | 1 | no | — |
| 209 | M16 | — | 4 | 1 | no | — |
| 210 | M16 | Dic3⋊C4 | 1 | 2 | yes | — |
| 211 | C2×C8 | — | 2 | 1 | no | — |
| 212 | C2×C8 | — | 2 | 2 | no | — |
| 213 | C2×C8 | — | 2 | 2 | no | — |
| 214 | C2×C8 | — | 4 | 1 | no | — |
| 215 | C2×C8 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 216 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 217 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 218 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 219 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 220 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 221 | C4⋊C4 | — | 4 | 1 | no | — |
| 222 | C4⋊C4 | — | 6 | 2 | no | — |
| 223 | C4⋊C4 | — | 6 | 1 | no | — |
| 224 | C4⋊C4 | — | 6 | 1 | no | — |
| 225 | C4⋊C4 | — | 12 | 2 | no | — |
| 226 | C4⋊C4 | — | 12 | 1 | no | — |
| 227 | C22⋊C4 | — | 4 | 1 | no | — |
| 228 | C22⋊C4 | — | 12 | 2 | no | — |
| 229 | C22⋊C4 | — | 12 | 1 | no | — |
| 230 | C42 | — | 4 | 1 | no | — |
| 231 | C42 | — | 12 | 2 | no | — |
| 232 | C42 | — | 12 | 1 | no | — |
| 233 | C16 | — | 6 | 2 | no | — |
| 234 | C16 | — | 6 | 2 | no | — |
| 235 | C22×C3 | — | 2 | 2 | no | — |
| 236 | C22×C3 | — | 4 | 1 | no | — |
| 237 | C22×C3 | — | 4 | 2 | no | — |
| 238 | C22×C3 | #428 | 1 | 1 | yes | characteristic |
| 239 | C22×C3 | #450 | 1 | 1 | yes | characteristic |
| 240 | C22×C3 | #481 | 1 | 1 | yes | characteristic |
| 241 | C3×C4 | — | 2 | 2 | no | — |
| 242 | C3×C4 | — | 16 | 1 | no | — |
| 243 | C3×C4 | #385 | 1 | 1 | yes | characteristic |
| 244 | C3×C4 | C2×#106 | 1 | 1 | yes | characteristic |
| 245 | C3×C4 | #448 | 1 | 1 | yes | characteristic |
| 246 | C3×C4 | #483 | 1 | 1 | yes | characteristic |
| 247 | Dic3 | — | 8 | 1 | no | — |
| 248 | Dic3 | — | 8 | 2 | no | — |
| 249 | Dic3 | — | 8 | 1 | no | — |
| 250 | C23 | — | 2 | 1 | no | — |
| 251 | C23 | #949 | 1 | 1 | yes | characteristic |
| 252 | Q8 | — | 2 | 1 | no | — |
| 253 | Q8 | #808 | 1 | 2 | yes | — |
| 254 | D4 | — | 2 | 2 | no | — |
| 255 | D4 | — | 2 | 2 | no | — |
| 256 | D4 | — | 2 | 1 | no | — |
| 257 | D4 | #808 | 1 | 2 | yes | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 2 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 2 | no | — |
| 261 | C2×C4 | — | 8 | 1 | no | — |
| 262 | C2×C4 | — | 12 | 1 | no | — |
| 263 | C2×C4 | — | 12 | 2 | no | — |
| 264 | C2×C4 | — | 12 | 1 | no | — |
| 265 | C2×C4 | — | 48 | 1 | no | — |
| 266 | C2×C4 | — | 48 | 2 | no | — |
| 267 | C2×C4 | C2×Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 268 | C2×C4 | C2×#214 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | #935 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #937 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | C2×#215 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | #951 | 1 | 1 | yes | characteristic |
| 273 | C8 | — | 2 | 2 | no | — |
| 274 | C8 | — | 8 | 1 | no | — |
| 275 | C8 | #878 | 1 | 1 | yes | characteristic |
| 276 | C8 | #878 | 1 | 1 | yes | characteristic |
| 277 | C8 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 278 | C8 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 279 | C2×C3 | — | 2 | 1 | no | — |
| 280 | C2×C3 | — | 2 | 1 | no | — |
| 281 | C2×C3 | — | 2 | 1 | no | — |
| 282 | C2×C3 | — | 4 | 2 | no | — |
| 283 | C2×C3 | #1853 | 1 | 1 | yes | characteristic |
| 284 | C22 | — | 2 | 2 | no | — |
| 285 | C22 | — | 4 | 1 | no | — |
| 286 | C22 | — | 4 | 2 | no | — |
| 287 | C22 | #4995 | 1 | 1 | yes | characteristic |
| 288 | C22 | #5034 | 1 | 1 | yes | characteristic |
| 289 | C22 | #5046 | 1 | 1 | yes | characteristic |
| 290 | C4 | — | 2 | 2 | no | — |
| 291 | C4 | — | 16 | 1 | no | — |
| 292 | C4 | — | 24 | 2 | no | — |
| 293 | C4 | — | 24 | 1 | no | — |
| 294 | C4 | — | 24 | 1 | no | — |
| 295 | C4 | #4946 | 1 | 1 | yes | characteristic |
| 296 | C4 | C2×#808 | 1 | 1 | yes | characteristic |
| 297 | C4 | #5040 | 1 | 1 | yes | characteristic |
| 298 | C4 | #5043 | 1 | 1 | yes | characteristic |
| 299 | C3 | #12934 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 300 | C2 | — | 2 | 1 | no | — |
| 301 | C2 | — | 2 | 1 | no | — |
| 302 | C2 | — | 2 | 1 | no | — |
| 303 | C2 | — | 4 | 2 | no | — |
| 304 | C2 | #69505 | 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