← #13136409
Subgroups — #13136409
Order 1728.
424 conjugacy classes of subgroups.
The subgroup lattice has 280 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C27×#124 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q8×Dic27 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720193 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11720185 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | D4×Dic27 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #11720128 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #11720128 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11720127 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #11720126 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C4×Dic54 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C4×Dic54 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #10589773 | — | 3 | 1 | no | maximal |
| 13 | C2×Q8×C27 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×D4×C27 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×Dic27 | C22 | 1 | 2 | yes | — |
| 16 | C2×Dic54 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×Dic54 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22⋊C4×C27 | C22 | 1 | 2 | yes | — |
| 19 | C22⋊C4×C27 | C22 | 1 | 2 | yes | — |
| 20 | C42×C27 | C22 | 1 | 1 | yes | characteristic |
| 21 | #88372 | C22 | 1 | 2 | yes | — |
| 22 | #88372 | C22 | 1 | 2 | yes | — |
| 23 | #88372 | C22 | 1 | 2 | yes | — |
| 24 | C4⋊Dic27 | C22 | 1 | 1 | yes | characteristic |
| 25 | C4⋊Dic27 | C22 | 1 | 2 | yes | — |
| 26 | C4⋊Dic27 | C22 | 1 | 1 | yes | characteristic |
| 27 | Dic27⋊C4 | C22 | 1 | 2 | yes | — |
| 28 | Dic27⋊C4 | C22 | 1 | 2 | yes | — |
| 29 | Dic27⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 30 | Dic27⋊C4 | C22 | 1 | 2 | yes | — |
| 31 | Dic27⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 32 | C4×Dic27 | C22 | 1 | 1 | yes | characteristic |
| 33 | C4×Dic27 | C22 | 1 | 1 | yes | characteristic |
| 34 | C4×Dic27 | C22 | 1 | 2 | yes | — |
| 35 | C4×Dic27 | C22 | 1 | 2 | yes | — |
| 36 | C9×#124 | D3 | 1 | 1 | yes | characteristic |
| 37 | Q8×Dic9 | — | 3 | 1 | no | — |
| 38 | #63506 | — | 3 | 1 | no | — |
| 39 | #63498 | — | 3 | 1 | no | — |
| 40 | D4×Dic9 | — | 3 | 1 | no | — |
| 41 | #63441 | — | 3 | 2 | no | — |
| 42 | #63441 | — | 3 | 2 | no | — |
| 43 | #63440 | — | 3 | 2 | no | — |
| 44 | #63439 | — | 3 | 2 | no | — |
| 45 | C4×Dic18 | — | 3 | 1 | no | — |
| 46 | C4×Dic18 | — | 3 | 1 | no | — |
| 47 | C23×C27 | C23 | 1 | 2 | yes | — |
| 48 | Q8×C27 | — | 2 | 2 | no | — |
| 49 | D4×C27 | — | 2 | 2 | no | — |
| 50 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×C27 | C23 | 1 | 2 | yes | — |
| 52 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×Dic27 | — | 2 | 4 | no | — |
| 55 | C2×Dic27 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×Dic27 | C23 | 1 | 2 | yes | — |
| 57 | C2×Dic27 | C23 | 1 | 1 | yes | characteristic |
| 58 | C2×Dic27 | C23 | 1 | 2 | yes | — |
| 59 | C2×Dic27 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×Dic27 | C23 | 1 | 1 | yes | characteristic |
| 61 | Dic54 | — | 2 | 2 | no | — |
| 62 | Dic54 | — | 2 | 2 | no | — |
| 63 | #5671 | — | 9 | 1 | no | — |
| 64 | C2×Q8×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 65 | C2×D4×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 66 | C22×Dic9 | — | 3 | 2 | no | — |
| 67 | C2×Dic18 | — | 3 | 1 | no | — |
| 68 | C2×Dic18 | — | 3 | 1 | no | — |
| 69 | C9×C22⋊C4 | C2×D3 | 1 | 2 | yes | — |
| 70 | C9×C22⋊C4 | C2×D3 | 1 | 2 | yes | — |
| 71 | C42×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 72 | #3676 | — | 3 | 2 | no | — |
| 73 | #3676 | — | 3 | 2 | no | — |
| 74 | #3676 | — | 3 | 2 | no | — |
| 75 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 76 | C4⋊Dic9 | — | 3 | 2 | no | — |
| 77 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 78 | Dic9⋊C4 | — | 3 | 2 | no | — |
| 79 | Dic9⋊C4 | — | 3 | 2 | no | — |
| 80 | Dic9⋊C4 | — | 3 | 1 | no | — |
| 81 | Dic9⋊C4 | — | 3 | 2 | no | — |
| 82 | Dic9⋊C4 | — | 3 | 1 | no | — |
| 83 | C4×Dic9 | — | 3 | 1 | no | — |
| 84 | C4×Dic9 | — | 3 | 1 | no | — |
| 85 | C4×Dic9 | — | 3 | 2 | no | — |
| 86 | C4×Dic9 | — | 3 | 2 | no | — |
| 87 | C22×C27 | — | 2 | 2 | no | — |
| 88 | C22×C27 | — | 2 | 2 | no | — |
| 89 | C22×C27 | — | 2 | 2 | no | — |
| 90 | C22×C27 | C24 | 1 | 1 | yes | characteristic |
| 91 | C4×C27 | — | 2 | 1 | no | — |
| 92 | C4×C27 | — | 2 | 2 | no | — |
| 93 | C4×C27 | — | 2 | 1 | no | — |
| 94 | C4×C27 | #41 | 1 | 2 | yes | — |
| 95 | Dic27 | — | 2 | 2 | no | — |
| 96 | Dic27 | — | 2 | 1 | no | — |
| 97 | Dic27 | — | 2 | 2 | no | — |
| 98 | Dic27 | — | 2 | 1 | no | — |
| 99 | Dic27 | — | 2 | 1 | no | — |
| 100 | Dic27 | #41 | 1 | 2 | yes | — |
| 101 | C3×#124 | D9 | 1 | 1 | yes | characteristic |
| 102 | Q8×Dic3 | — | 9 | 1 | no | — |
| 103 | #944 | — | 9 | 1 | no | — |
| 104 | #936 | — | 9 | 1 | no | — |
| 105 | D4×Dic3 | — | 9 | 1 | no | — |
| 106 | #879 | — | 9 | 2 | no | — |
| 107 | #879 | — | 9 | 2 | no | — |
| 108 | #878 | — | 9 | 2 | no | — |
| 109 | #877 | — | 9 | 2 | no | — |
| 110 | C4×Dic6 | — | 9 | 1 | no | — |
| 111 | C4×Dic6 | — | 9 | 1 | no | — |
| 112 | C23×C9 | C22×D3 | 1 | 2 | yes | — |
| 113 | Q8×C9 | — | 2 | 2 | no | — |
| 114 | D4×C9 | — | 2 | 2 | no | — |
| 115 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C9 | C22×D3 | 1 | 2 | yes | — |
| 117 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 119 | C2×Dic9 | — | 3 | 1 | no | — |
| 120 | C2×Dic9 | — | 3 | 1 | no | — |
| 121 | C2×Dic9 | — | 3 | 2 | no | — |
| 122 | C2×Dic9 | — | 3 | 1 | no | — |
| 123 | C2×Dic9 | — | 3 | 2 | no | — |
| 124 | C2×Dic9 | — | 3 | 1 | no | — |
| 125 | C2×Dic9 | — | 6 | 4 | no | — |
| 126 | Dic18 | — | 6 | 2 | no | — |
| 127 | Dic18 | — | 6 | 2 | no | — |
| 128 | #556 | — | 27 | 1 | no | Sylow |
| 129 | C2×C27 | — | 4 | 2 | no | — |
| 130 | C2×C27 | C2×#41 | 1 | 1 | yes | characteristic |
| 131 | C2×C27 | C2×#41 | 1 | 1 | yes | characteristic |
| 132 | C2×C27 | 2−1+4 | 1 | 1 | yes | characteristic |
| 133 | C2×C3×Q8 | C2×D9 | 1 | 1 | yes | characteristic |
| 134 | C2×C3×D4 | C2×D9 | 1 | 1 | yes | characteristic |
| 135 | C22×Dic3 | — | 9 | 2 | no | — |
| 136 | C2×Dic6 | — | 9 | 1 | no | — |
| 137 | C2×Dic6 | — | 9 | 1 | no | — |
| 138 | C3×C22⋊C4 | C2×D9 | 1 | 2 | yes | — |
| 139 | C3×C22⋊C4 | C2×D9 | 1 | 2 | yes | — |
| 140 | C3×C42 | C2×D9 | 1 | 1 | yes | characteristic |
| 141 | #217 | — | 9 | 2 | no | — |
| 142 | #217 | — | 9 | 2 | no | — |
| 143 | #217 | — | 9 | 2 | no | — |
| 144 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 145 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 146 | C4⋊Dic3 | — | 9 | 2 | no | — |
| 147 | Dic3⋊C4 | — | 9 | 1 | no | — |
| 148 | Dic3⋊C4 | — | 9 | 2 | no | — |
| 149 | Dic3⋊C4 | — | 9 | 1 | no | — |
| 150 | Dic3⋊C4 | — | 9 | 2 | no | — |
| 151 | Dic3⋊C4 | — | 9 | 2 | no | — |
| 152 | C4×Dic3 | — | 9 | 2 | no | — |
| 153 | C4×Dic3 | — | 9 | 1 | no | — |
| 154 | C4×Dic3 | — | 9 | 1 | no | — |
| 155 | C4×Dic3 | — | 9 | 2 | no | — |
| 156 | C22×C9 | — | 2 | 2 | no | — |
| 157 | C22×C9 | — | 2 | 2 | no | — |
| 158 | C22×C9 | — | 2 | 2 | no | — |
| 159 | C22×C9 | C23×D3 | 1 | 1 | yes | characteristic |
| 160 | C4×C9 | — | 2 | 2 | no | — |
| 161 | C4×C9 | — | 2 | 1 | no | — |
| 162 | C4×C9 | — | 2 | 1 | no | — |
| 163 | C4×C9 | #237 | 1 | 2 | yes | — |
| 164 | Dic9 | — | 3 | 2 | no | — |
| 165 | Dic9 | — | 6 | 2 | no | — |
| 166 | Dic9 | — | 6 | 1 | no | — |
| 167 | Dic9 | — | 6 | 2 | no | — |
| 168 | Dic9 | — | 6 | 1 | no | — |
| 169 | Dic9 | — | 6 | 1 | no | — |
| 170 | C4⋊Q8 | — | 27 | 1 | no | — |
| 171 | #126 | — | 27 | 2 | no | — |
| 172 | #126 | — | 27 | 2 | no | — |
| 173 | #124 | — | 27 | 1 | no | — |
| 174 | #124 | D27 | 1 | 1 | yes | characteristic |
| 175 | C22⋊Q8 | — | 27 | 2 | no | — |
| 176 | C4×Q8 | — | 27 | 1 | no | — |
| 177 | C4×Q8 | — | 27 | 1 | no | — |
| 178 | C4×Q8 | — | 27 | 1 | no | — |
| 179 | C4×D4 | — | 27 | 1 | no | — |
| 180 | #117 | — | 27 | 2 | no | — |
| 181 | C27 | #556 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 182 | C23×C3 | C22×D9 | 1 | 2 | yes | — |
| 183 | C3×Q8 | — | 2 | 2 | no | — |
| 184 | C3×D4 | — | 2 | 2 | no | — |
| 185 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 186 | C2×C3×C4 | C22×D9 | 1 | 2 | yes | — |
| 187 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 188 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 189 | C2×Dic3 | — | 9 | 1 | no | — |
| 190 | C2×Dic3 | — | 9 | 1 | no | — |
| 191 | C2×Dic3 | — | 9 | 2 | no | — |
| 192 | C2×Dic3 | — | 9 | 1 | no | — |
| 193 | C2×Dic3 | — | 9 | 2 | no | — |
| 194 | C2×Dic3 | — | 9 | 1 | no | — |
| 195 | C2×Dic3 | — | 18 | 4 | no | — |
| 196 | Dic6 | — | 18 | 2 | no | — |
| 197 | Dic6 | — | 18 | 2 | no | — |
| 198 | C2×C9 | — | 4 | 2 | no | — |
| 199 | C2×C9 | C2×#237 | 1 | 1 | yes | characteristic |
| 200 | C2×C9 | D3×#41 | 1 | 1 | yes | characteristic |
| 201 | C2×C9 | #1010 | 1 | 1 | yes | characteristic |
| 202 | C2×Q8 | — | 27 | 1 | no | — |
| 203 | C2×Q8 | — | 27 | 1 | no | — |
| 204 | C2×Q8 | C2×D27 | 1 | 1 | yes | characteristic |
| 205 | C2×D4 | C2×D27 | 1 | 1 | yes | characteristic |
| 206 | C22×C4 | — | 27 | 2 | no | — |
| 207 | C4⋊C4 | — | 27 | 1 | no | — |
| 208 | C4⋊C4 | — | 27 | 2 | no | — |
| 209 | C4⋊C4 | — | 27 | 1 | no | — |
| 210 | C4⋊C4 | — | 27 | 2 | no | — |
| 211 | C4⋊C4 | — | 27 | 2 | no | — |
| 212 | C4⋊C4 | — | 27 | 1 | no | — |
| 213 | C4⋊C4 | — | 27 | 1 | no | — |
| 214 | C4⋊C4 | — | 27 | 2 | no | — |
| 215 | C22⋊C4 | — | 27 | 2 | no | — |
| 216 | C22⋊C4 | — | 27 | 2 | no | — |
| 217 | C22⋊C4 | — | 27 | 2 | no | — |
| 218 | C22⋊C4 | C2×D27 | 1 | 2 | yes | — |
| 219 | C22⋊C4 | C2×D27 | 1 | 2 | yes | — |
| 220 | C42 | — | 27 | 2 | no | — |
| 221 | C42 | — | 27 | 2 | no | — |
| 222 | C42 | — | 27 | 1 | no | — |
| 223 | C42 | — | 27 | 1 | no | — |
| 224 | C42 | C2×D27 | 1 | 1 | yes | characteristic |
| 225 | C22×C3 | — | 2 | 2 | no | — |
| 226 | C22×C3 | — | 2 | 2 | no | — |
| 227 | C22×C3 | — | 2 | 2 | no | — |
| 228 | C22×C3 | C23×D9 | 1 | 1 | yes | characteristic |
| 229 | C3×C4 | — | 2 | 2 | no | — |
| 230 | C3×C4 | — | 2 | 1 | no | — |
| 231 | C3×C4 | — | 2 | 1 | no | — |
| 232 | C3×C4 | #3699 | 1 | 2 | yes | — |
| 233 | Dic3 | — | 9 | 2 | no | — |
| 234 | Dic3 | — | 18 | 2 | no | — |
| 235 | Dic3 | — | 18 | 1 | no | — |
| 236 | Dic3 | — | 18 | 2 | no | — |
| 237 | Dic3 | — | 18 | 1 | no | — |
| 238 | Dic3 | — | 18 | 1 | no | — |
| 239 | C9 | #5671 | 1 | 1 | yes | characteristic |
| 240 | C23 | C22×D27 | 1 | 2 | yes | — |
| 241 | Q8 | — | 2 | 2 | no | — |
| 242 | Q8 | — | 54 | 2 | no | — |
| 243 | Q8 | — | 54 | 2 | no | — |
| 244 | D4 | — | 2 | 2 | no | — |
| 245 | C2×C4 | — | 27 | 1 | no | — |
| 246 | C2×C4 | — | 27 | 1 | no | — |
| 247 | C2×C4 | — | 27 | 2 | no | — |
| 248 | C2×C4 | — | 27 | 1 | no | — |
| 249 | C2×C4 | — | 27 | 2 | no | — |
| 250 | C2×C4 | — | 27 | 1 | no | — |
| 251 | C2×C4 | — | 54 | 4 | no | — |
| 252 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 253 | C2×C4 | C22×D27 | 1 | 2 | yes | — |
| 254 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 256 | C2×C3 | — | 4 | 2 | no | — |
| 257 | C2×C3 | C2×#3699 | 1 | 1 | yes | characteristic |
| 258 | C2×C3 | #41×D9 | 1 | 1 | yes | characteristic |
| 259 | C2×C3 | #63716 | 1 | 1 | yes | characteristic |
| 260 | C22 | — | 2 | 2 | no | — |
| 261 | C22 | — | 2 | 2 | no | — |
| 262 | C22 | — | 2 | 2 | no | — |
| 263 | C22 | C23×D27 | 1 | 1 | yes | characteristic |
| 264 | C4 | — | 2 | 2 | no | — |
| 265 | C4 | — | 2 | 1 | no | — |
| 266 | C4 | — | 2 | 1 | no | — |
| 267 | C4 | — | 27 | 2 | no | — |
| 268 | C4 | — | 54 | 2 | no | — |
| 269 | C4 | — | 54 | 2 | no | — |
| 270 | C4 | — | 54 | 1 | no | — |
| 271 | C4 | — | 54 | 1 | no | — |
| 272 | C4 | — | 54 | 1 | no | — |
| 273 | C4 | #88401 | 1 | 2 | yes | — |
| 274 | C3 | #10589773 | 1 | 1 | yes | characteristic |
| 275 | C2 | — | 4 | 2 | no | — |
| 276 | C2 | C2×#88401 | 1 | 1 | yes | characteristic |
| 277 | C2 | #41×D27 | 1 | 1 | yes | characteristic |
| 278 | C2 | #11720676 | 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