← C3×#13456 #10632063
Subgroups — C3×#13456 #10632063
Order 768.
318 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×#3164 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#2159 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C3×#367 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C3×#1952 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C3×#1387 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C3×#1382 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C3×#1379 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #13456 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 9 | C3×#544 | — | 2 | 1 | no | — |
| 10 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×#501 | — | 2 | 2 | no | — |
| 13 | C3×#500 | — | 2 | 1 | no | — |
| 14 | C3×#499 | — | 2 | 1 | no | — |
| 15 | C2×C3×#108 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×#428 | — | 2 | 1 | no | — |
| 17 | C3×#428 | — | 2 | 1 | no | — |
| 18 | C3×#427 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C3×#107 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C3×#107 | C22 | 1 | 1 | yes | characteristic |
| 21 | C3×#405 | C22 | 1 | 1 | yes | characteristic |
| 22 | C3×#367 | — | 2 | 2 | no | — |
| 23 | C3×#367 | — | 2 | 1 | no | — |
| 24 | C3×#366 | — | 2 | 1 | no | — |
| 25 | C3×#343 | — | 2 | 1 | no | — |
| 26 | #3164 | C2×C3 | 1 | 1 | yes | characteristic |
| 27 | #2159 | C2×C3 | 1 | 1 | yes | characteristic |
| 28 | C2×#367 | C2×C3 | 1 | 1 | yes | characteristic |
| 29 | #1952 | C2×C3 | 1 | 1 | yes | characteristic |
| 30 | #1387 | C2×C3 | 1 | 1 | yes | characteristic |
| 31 | #1382 | C2×C3 | 1 | 1 | yes | characteristic |
| 32 | #1379 | C2×C3 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×M16 | — | 2 | 1 | no | — |
| 34 | C2×C3×M16 | D4 | 1 | 1 | yes | characteristic |
| 35 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 36 | C3×C4⋊Q8 | — | 2 | 2 | no | — |
| 37 | C3×C4⋊Q8 | — | 4 | 1 | no | — |
| 38 | C3×#125 | — | 2 | 2 | no | — |
| 39 | C3×#125 | — | 4 | 1 | no | — |
| 40 | C3×C22⋊Q8 | — | 4 | 2 | no | — |
| 41 | C3×#117 | — | 2 | 1 | no | — |
| 42 | C3×#117 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 44 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 45 | C3×M32 | — | 2 | 1 | no | — |
| 46 | C3×M32 | Q8 | 1 | 2 | yes | complemented |
| 47 | C2×C3×C16 | — | 2 | 1 | no | — |
| 48 | C2×C3×C16 | — | 2 | 1 | no | — |
| 49 | C2×C3×C16 | — | 2 | 1 | no | — |
| 50 | C2×C3×C16 | Q8 | 1 | 2 | yes | complemented |
| 51 | C3×#108 | — | 2 | 1 | no | — |
| 52 | C3×#108 | Q8 | 1 | 2 | yes | complemented |
| 53 | C3×#107 | — | 2 | 2 | no | — |
| 54 | C3×#107 | — | 2 | 1 | no | — |
| 55 | C3×#107 | — | 2 | 1 | no | — |
| 56 | C3×#107 | — | 2 | 1 | no | — |
| 57 | C3×#107 | — | 4 | 1 | no | — |
| 58 | C3×#107 | D4 | 1 | 1 | yes | characteristic |
| 59 | C3×#107 | D4 | 1 | 2 | yes | — |
| 60 | C3×#107 | D4 | 1 | 1 | yes | characteristic |
| 61 | C3×#106 | — | 2 | 2 | no | — |
| 62 | C3×#106 | — | 2 | 1 | no | — |
| 63 | C3×#106 | — | 4 | 1 | no | — |
| 64 | C3×#97 | — | 2 | 1 | no | — |
| 65 | C3×C4×C8 | — | 2 | 1 | no | — |
| 66 | #544 | — | 2 | 1 | no | — |
| 67 | C2×M32 | C22×C3 | 1 | 1 | yes | characteristic |
| 68 | C22×C16 | C22×C3 | 1 | 1 | yes | characteristic |
| 69 | #501 | — | 2 | 2 | no | — |
| 70 | #500 | — | 2 | 1 | no | — |
| 71 | #499 | — | 2 | 1 | no | — |
| 72 | C2×#108 | C22×C3 | 1 | 1 | yes | characteristic |
| 73 | #428 | — | 2 | 1 | no | — |
| 74 | #428 | — | 2 | 1 | no | — |
| 75 | #427 | C22×C3 | 1 | 1 | yes | characteristic |
| 76 | C2×#107 | C22×C3 | 1 | 1 | yes | characteristic |
| 77 | C2×#107 | C22×C3 | 1 | 1 | yes | characteristic |
| 78 | #405 | C22×C3 | 1 | 1 | yes | characteristic |
| 79 | #367 | — | 2 | 2 | no | — |
| 80 | #367 | — | 2 | 1 | no | — |
| 81 | #366 | — | 2 | 1 | no | — |
| 82 | #343 | — | 2 | 1 | no | — |
| 83 | C2×C3×Q8 | — | 4 | 2 | no | — |
| 84 | C22×C3×C4 | — | 4 | 1 | no | — |
| 85 | C22×C3×C4 | — | 4 | 1 | no | — |
| 86 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C3×M16 | — | 2 | 1 | no | — |
| 88 | C3×M16 | — | 2 | 2 | no | — |
| 89 | C3×M16 | — | 4 | 1 | no | — |
| 90 | C3×M16 | Q16 | 1 | 2 | yes | — |
| 91 | C2×C3×C8 | — | 2 | 1 | no | — |
| 92 | C2×C3×C8 | — | 4 | 1 | no | — |
| 93 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 96 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 97 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 98 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 99 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 100 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 101 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 102 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 103 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 104 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 105 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 106 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 107 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 108 | C3×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 109 | C3×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 110 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 111 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 112 | C3×C42 | — | 2 | 1 | no | — |
| 113 | C3×C42 | — | 4 | 1 | no | — |
| 114 | C3×C16 | — | 2 | 2 | no | — |
| 115 | C3×C16 | — | 2 | 2 | no | — |
| 116 | C2×M16 | — | 2 | 1 | no | — |
| 117 | C2×M16 | C3×D4 | 1 | 1 | yes | characteristic |
| 118 | C22×C8 | C23×C3 | 1 | 1 | yes | characteristic |
| 119 | C4⋊Q8 | — | 2 | 2 | no | — |
| 120 | C4⋊Q8 | — | 4 | 1 | no | — |
| 121 | #125 | — | 2 | 2 | no | — |
| 122 | #125 | — | 4 | 1 | no | — |
| 123 | C22⋊Q8 | — | 4 | 2 | no | — |
| 124 | #117 | — | 2 | 1 | no | — |
| 125 | #117 | C3×D4 | 1 | 1 | yes | characteristic |
| 126 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 127 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 128 | M32 | — | 2 | 1 | no | — |
| 129 | M32 | C3×Q8 | 1 | 2 | yes | complemented |
| 130 | C2×C16 | — | 2 | 1 | no | — |
| 131 | C2×C16 | — | 2 | 1 | no | — |
| 132 | C2×C16 | — | 2 | 1 | no | — |
| 133 | C2×C16 | C3×Q8 | 1 | 2 | yes | complemented |
| 134 | #108 | — | 2 | 1 | no | — |
| 135 | #108 | C3×Q8 | 1 | 2 | yes | complemented |
| 136 | #107 | — | 2 | 2 | no | — |
| 137 | #107 | — | 2 | 1 | no | — |
| 138 | #107 | — | 2 | 1 | no | — |
| 139 | #107 | — | 2 | 1 | no | — |
| 140 | #107 | — | 4 | 1 | no | — |
| 141 | #107 | C3×D4 | 1 | 1 | yes | characteristic |
| 142 | #107 | C3×D4 | 1 | 2 | yes | — |
| 143 | #107 | C3×D4 | 1 | 1 | yes | characteristic |
| 144 | #106 | — | 2 | 2 | no | — |
| 145 | #106 | — | 2 | 1 | no | — |
| 146 | #106 | — | 4 | 1 | no | — |
| 147 | #97 | — | 2 | 1 | no | — |
| 148 | C4×C8 | — | 2 | 1 | no | — |
| 149 | C23×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 150 | C3×Q8 | — | 16 | 2 | no | — |
| 151 | C2×C3×C4 | — | 2 | 1 | no | — |
| 152 | C2×C3×C4 | — | 2 | 1 | no | — |
| 153 | C2×C3×C4 | — | 4 | 1 | no | — |
| 154 | C2×C3×C4 | — | 4 | 1 | no | — |
| 155 | C2×C3×C4 | — | 4 | 2 | no | — |
| 156 | C2×C3×C4 | — | 8 | 1 | no | — |
| 157 | C2×C3×C4 | — | 16 | 1 | no | — |
| 158 | C2×C3×C4 | — | 16 | 1 | no | — |
| 159 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 160 | C2×C3×C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 161 | C2×C3×C4 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 162 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 163 | C2×C3×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 164 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 165 | C3×C8 | — | 2 | 2 | no | — |
| 166 | C3×C8 | — | 4 | 2 | no | — |
| 167 | C3×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 168 | C3×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 169 | C3×C8 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 170 | C3×C8 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 171 | C2×Q8 | — | 4 | 2 | no | — |
| 172 | C22×C4 | — | 4 | 1 | no | — |
| 173 | C22×C4 | — | 4 | 1 | no | — |
| 174 | C22×C4 | C2×C3×D4 | 1 | 1 | yes | characteristic |
| 175 | M16 | — | 2 | 1 | no | — |
| 176 | M16 | — | 2 | 2 | no | — |
| 177 | M16 | — | 4 | 1 | no | — |
| 178 | M16 | C3×Q16 | 1 | 2 | yes | — |
| 179 | C2×C8 | — | 2 | 1 | no | — |
| 180 | C2×C8 | — | 4 | 1 | no | — |
| 181 | C2×C8 | C2×C3×D4 | 1 | 1 | yes | characteristic |
| 182 | C2×C8 | C2×C3×D4 | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | C2×C3×Q8 | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | C2×C3×Q8 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | C2×C3×Q8 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | C3×#41 | 1 | 1 | yes | characteristic |
| 187 | C4⋊C4 | — | 2 | 1 | no | — |
| 188 | C4⋊C4 | — | 2 | 2 | no | — |
| 189 | C4⋊C4 | — | 2 | 1 | no | — |
| 190 | C4⋊C4 | — | 4 | 2 | no | — |
| 191 | C4⋊C4 | — | 4 | 2 | no | — |
| 192 | C4⋊C4 | — | 4 | 2 | no | — |
| 193 | C4⋊C4 | — | 4 | 1 | no | — |
| 194 | C4⋊C4 | — | 4 | 1 | no | — |
| 195 | C4⋊C4 | — | 4 | 1 | no | — |
| 196 | C4⋊C4 | C3×D8 | 1 | 1 | yes | characteristic |
| 197 | C4⋊C4 | C3×D8 | 1 | 1 | yes | characteristic |
| 198 | C22⋊C4 | — | 2 | 1 | no | — |
| 199 | C22⋊C4 | — | 4 | 1 | no | — |
| 200 | C42 | — | 2 | 1 | no | — |
| 201 | C42 | — | 4 | 1 | no | — |
| 202 | C16 | — | 2 | 2 | no | — |
| 203 | C16 | — | 2 | 2 | no | — |
| 204 | C22×C3 | — | 2 | 1 | no | — |
| 205 | C22×C3 | — | 2 | 1 | no | — |
| 206 | C22×C3 | #447 | 1 | 1 | yes | characteristic |
| 207 | C22×C3 | #477 | 1 | 1 | yes | characteristic |
| 208 | C22×C3 | #501 | 1 | 1 | yes | characteristic |
| 209 | C3×C4 | — | 4 | 1 | no | — |
| 210 | C3×C4 | — | 4 | 1 | no | — |
| 211 | C3×C4 | — | 8 | 1 | no | — |
| 212 | C3×C4 | — | 8 | 2 | no | — |
| 213 | C3×C4 | — | 8 | 1 | no | — |
| 214 | C3×C4 | — | 16 | 1 | no | — |
| 215 | C3×C4 | #395 | 1 | 1 | yes | characteristic |
| 216 | C3×C4 | #451 | 1 | 1 | yes | characteristic |
| 217 | C3×C4 | #474 | 1 | 1 | yes | characteristic |
| 218 | C3×C4 | #500 | 1 | 1 | yes | characteristic |
| 219 | C23 | C3×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 220 | Q8 | — | 16 | 2 | no | — |
| 221 | C2×C4 | — | 2 | 1 | no | — |
| 222 | C2×C4 | — | 2 | 1 | no | — |
| 223 | C2×C4 | — | 4 | 1 | no | — |
| 224 | C2×C4 | — | 4 | 1 | no | — |
| 225 | C2×C4 | — | 4 | 2 | no | — |
| 226 | C2×C4 | — | 8 | 1 | no | — |
| 227 | C2×C4 | — | 16 | 1 | no | — |
| 228 | C2×C4 | — | 16 | 1 | no | — |
| 229 | C2×C4 | C3×#120 | 1 | 1 | yes | characteristic |
| 230 | C2×C4 | C3×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | C3×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 232 | C2×C4 | C2×C3×D8 | 1 | 1 | yes | characteristic |
| 233 | C2×C4 | C2×C3×Q16 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | C3×#137 | 1 | 1 | yes | characteristic |
| 235 | C8 | — | 2 | 2 | no | — |
| 236 | C8 | — | 4 | 2 | no | — |
| 237 | C8 | C3×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 238 | C8 | C3×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 239 | C8 | C3×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 240 | C8 | C3×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 241 | C2×C3 | — | 2 | 1 | no | — |
| 242 | C2×C3 | — | 2 | 1 | no | — |
| 243 | C2×C3 | #2057 | 1 | 1 | yes | characteristic |
| 244 | C2×C3 | #2193 | 1 | 1 | yes | characteristic |
| 245 | C2×C3 | #2194 | 1 | 1 | yes | characteristic |
| 246 | C22 | — | 2 | 1 | no | — |
| 247 | C22 | — | 2 | 1 | no | — |
| 248 | C22 | C3×#447 | 1 | 1 | yes | characteristic |
| 249 | C22 | C3×#477 | 1 | 1 | yes | characteristic |
| 250 | C22 | C3×#501 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 4 | 1 | no | — |
| 252 | C4 | — | 4 | 1 | no | — |
| 253 | C4 | — | 8 | 1 | no | — |
| 254 | C4 | — | 8 | 2 | no | — |
| 255 | C4 | — | 8 | 1 | no | — |
| 256 | C4 | — | 16 | 1 | no | — |
| 257 | C4 | C3×#395 | 1 | 1 | yes | characteristic |
| 258 | C4 | C3×#451 | 1 | 1 | yes | characteristic |
| 259 | C4 | C3×#474 | 1 | 1 | yes | characteristic |
| 260 | C4 | C3×#500 | 1 | 1 | yes | characteristic |
| 261 | C3 | #13456 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 262 | C2 | — | 2 | 1 | no | — |
| 263 | C2 | — | 2 | 1 | no | — |
| 264 | C2 | C3×#2057 | 1 | 1 | yes | characteristic |
| 265 | C2 | C3×#2193 | 1 | 1 | yes | characteristic |
| 266 | C2 | C3×#2194 | 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