← #13150152
Subgroups — #13150152
Order 1728.
774 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4×C3⋊Dic9 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C9×#926 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×#63488 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11721267 | C2 | 1 | 4 | yes | maximal |
| 5 | #11720971 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C4×#6189 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #11720937 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #10595050 | — | 3 | 1 | no | maximal |
| 9 | #10589619 | — | 3 | 1 | no | maximal |
| 10 | C22×C3×C4×C9 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×C3⋊Dic9 | C22 | 1 | 2 | yes | — |
| 12 | C2×#6189 | C22 | 1 | 4 | yes | — |
| 13 | C4×C3⋊Dic9 | C22 | 1 | 2 | yes | — |
| 14 | C4×C3⋊Dic9 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×C3⋊Dic9 | C22 | 1 | 1 | yes | characteristic |
| 16 | C9×#217 | C22 | 1 | 2 | yes | — |
| 17 | C3×#3676 | C22 | 1 | 2 | yes | — |
| 18 | C9×C4⋊Dic3 | C22 | 1 | 2 | yes | — |
| 19 | C4×C9×Dic3 | C22 | 1 | 2 | yes | — |
| 20 | C3×C4⋊Dic9 | C22 | 1 | 2 | yes | — |
| 21 | C3×C4×Dic9 | C22 | 1 | 2 | yes | — |
| 22 | Dic3⋊Dic9 | C22 | 1 | 4 | yes | — |
| 23 | #88442 | C22 | 1 | 4 | yes | — |
| 24 | Dic9⋊Dic3 | C22 | 1 | 4 | yes | — |
| 25 | C2×C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 26 | C3×#926 | — | 3 | 1 | no | — |
| 27 | C3×#926 | D3 | 1 | 1 | yes | characteristic |
| 28 | #63982 | — | 3 | 4 | no | — |
| 29 | #63919 | — | 3 | 2 | no | — |
| 30 | C4×#627 | — | 3 | 4 | no | — |
| 31 | #63872 | — | 3 | 2 | no | — |
| 32 | C9×#117 | — | 3 | 1 | no | — |
| 33 | #63488 | D3 | 1 | 1 | yes | characteristic |
| 34 | C2×C4×Dic9 | — | 3 | 1 | no | — |
| 35 | C2×C4×Dic9 | — | 6 | 1 | no | — |
| 36 | #63452 | — | 3 | 2 | no | — |
| 37 | #63450 | — | 3 | 2 | no | — |
| 38 | #63440 | — | 3 | 4 | no | — |
| 39 | C4×Dic18 | — | 3 | 4 | no | — |
| 40 | C23×C3×C9 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×C3×C4×C9 | Q8 | 1 | 4 | yes | complemented |
| 42 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×C3⋊Dic9 | — | 2 | 4 | no | — |
| 45 | C2×C3⋊Dic9 | C23 | 1 | 2 | yes | — |
| 46 | C2×C3⋊Dic9 | C23 | 1 | 2 | yes | — |
| 47 | C2×C9×Dic3 | C23 | 1 | 4 | yes | — |
| 48 | C2×C3×Dic9 | C23 | 1 | 4 | yes | — |
| 49 | #6189 | — | 2 | 8 | no | — |
| 50 | #5517 | — | 9 | 1 | no | — |
| 51 | #5517 | — | 9 | 1 | no | — |
| 52 | C22×C32×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 53 | C22×C3⋊Dic3 | — | 3 | 2 | no | — |
| 54 | C2×#627 | — | 3 | 4 | no | — |
| 55 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 56 | C4×C3⋊Dic3 | — | 3 | 2 | no | — |
| 57 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 58 | C3×#217 | — | 3 | 2 | no | — |
| 59 | C3×#217 | C2×D3 | 1 | 2 | yes | — |
| 60 | C3×C4⋊Dic3 | — | 3 | 2 | no | — |
| 61 | C3×C4⋊Dic3 | C2×D3 | 1 | 2 | yes | — |
| 62 | C3×C4×Dic3 | — | 3 | 2 | no | — |
| 63 | C3×C4×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 64 | #3724 | — | 3 | 4 | no | — |
| 65 | Dic3⋊Dic3 | — | 3 | 4 | no | — |
| 66 | Dic3⋊Dic3 | — | 3 | 4 | no | — |
| 67 | C22×C4×C9 | — | 2 | 1 | no | — |
| 68 | C22×C4×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 69 | C22×Dic9 | — | 3 | 2 | no | — |
| 70 | C22×Dic9 | — | 6 | 2 | no | — |
| 71 | C2×Dic18 | — | 3 | 4 | no | — |
| 72 | C9×C4⋊C4 | — | 3 | 2 | no | — |
| 73 | C9×C22⋊C4 | — | 3 | 2 | no | — |
| 74 | C42×C9 | — | 3 | 2 | no | — |
| 75 | #3676 | C2×D3 | 1 | 2 | yes | — |
| 76 | C4⋊Dic9 | — | 3 | 4 | no | — |
| 77 | C4⋊Dic9 | C2×D3 | 1 | 2 | yes | — |
| 78 | Dic9⋊C4 | — | 3 | 4 | no | — |
| 79 | Dic9⋊C4 | — | 3 | 4 | no | — |
| 80 | C4×Dic9 | — | 3 | 2 | no | — |
| 81 | C4×Dic9 | — | 3 | 1 | no | — |
| 82 | C4×Dic9 | — | 3 | 1 | no | — |
| 83 | C4×Dic9 | — | 6 | 2 | no | — |
| 84 | C4×Dic9 | — | 6 | 1 | no | — |
| 85 | C4×Dic9 | — | 6 | 1 | no | — |
| 86 | C4×Dic9 | C2×D3 | 1 | 2 | yes | — |
| 87 | C22×C3×C9 | — | 2 | 2 | no | — |
| 88 | C22×C3×C9 | C2×Q8 | 1 | 2 | yes | — |
| 89 | C22×C3×C9 | C24 | 1 | 1 | yes | characteristic |
| 90 | C3×C4×C9 | C2×Q8 | 1 | 2 | yes | — |
| 91 | C3×C4×C9 | C2×Q8 | 1 | 2 | yes | — |
| 92 | C3⋊Dic9 | — | 2 | 2 | no | — |
| 93 | C3⋊Dic9 | #41 | 1 | 4 | yes | — |
| 94 | C9×Dic3 | — | 2 | 4 | no | — |
| 95 | C3×Dic9 | — | 2 | 4 | no | — |
| 96 | C3×#117 | — | 3 | 1 | no | — |
| 97 | C3×#117 | — | 9 | 1 | no | — |
| 98 | #926 | — | 3 | 1 | no | — |
| 99 | #926 | D9 | 1 | 1 | yes | characteristic |
| 100 | C2×C4×Dic3 | — | 9 | 1 | no | — |
| 101 | C2×C4×Dic3 | — | 9 | 1 | no | — |
| 102 | C2×C4×Dic3 | — | 18 | 1 | no | — |
| 103 | #890 | — | 9 | 2 | no | — |
| 104 | #890 | — | 9 | 2 | no | — |
| 105 | #888 | — | 9 | 2 | no | — |
| 106 | #888 | — | 9 | 2 | no | — |
| 107 | #878 | — | 9 | 4 | no | — |
| 108 | #878 | — | 9 | 4 | no | — |
| 109 | C4×Dic6 | — | 9 | 4 | no | — |
| 110 | C4×Dic6 | — | 9 | 4 | no | — |
| 111 | C23×C32 | C22×D3 | 1 | 1 | yes | characteristic |
| 112 | C2×C32×C4 | Dic6 | 1 | 4 | yes | — |
| 113 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 114 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 115 | C2×C3⋊Dic3 | — | 3 | 2 | no | — |
| 116 | C2×C3⋊Dic3 | — | 3 | 2 | no | — |
| 117 | C2×C3⋊Dic3 | — | 6 | 4 | no | — |
| 118 | C2×C3×Dic3 | — | 3 | 4 | no | — |
| 119 | C2×C3×Dic3 | C22×D3 | 1 | 4 | yes | — |
| 120 | #627 | — | 6 | 8 | no | — |
| 121 | C23×C9 | — | 2 | 1 | no | — |
| 122 | C23×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C9 | — | 2 | 4 | no | — |
| 124 | C2×C4×C9 | — | 2 | 1 | no | — |
| 125 | C2×C4×C9 | — | 2 | 1 | no | — |
| 126 | C2×C4×C9 | — | 3 | 4 | no | — |
| 127 | C2×C4×C9 | Dic6 | 1 | 4 | yes | complemented |
| 128 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 130 | C2×Dic9 | — | 3 | 2 | no | — |
| 131 | C2×Dic9 | — | 3 | 2 | no | — |
| 132 | C2×Dic9 | — | 6 | 2 | no | — |
| 133 | C2×Dic9 | — | 6 | 2 | no | — |
| 134 | C2×Dic9 | — | 6 | 4 | no | — |
| 135 | C2×Dic9 | — | 6 | 8 | no | — |
| 136 | C2×Dic9 | C22×D3 | 1 | 4 | yes | — |
| 137 | Dic18 | — | 6 | 8 | no | — |
| 138 | #533 | — | 27 | 1 | no | Sylow |
| 139 | C2×C3×C9 | — | 2 | 2 | no | — |
| 140 | C2×C3×C9 | C22×Q8 | 1 | 1 | yes | characteristic |
| 141 | C2×C3×C9 | C2×#41 | 1 | 2 | yes | — |
| 142 | C22×C3×C4 | — | 2 | 1 | no | — |
| 143 | C22×C3×C4 | C2×D9 | 1 | 1 | yes | characteristic |
| 144 | C22×C3×C4 | D32 | 1 | 1 | yes | characteristic |
| 145 | C22×Dic3 | — | 9 | 2 | no | — |
| 146 | C22×Dic3 | — | 9 | 2 | no | — |
| 147 | C22×Dic3 | — | 18 | 2 | no | — |
| 148 | C2×Dic6 | — | 9 | 4 | no | — |
| 149 | C2×Dic6 | — | 9 | 4 | no | — |
| 150 | C3×C4⋊C4 | — | 3 | 2 | no | — |
| 151 | C3×C4⋊C4 | — | 9 | 2 | no | — |
| 152 | C3×C22⋊C4 | — | 3 | 2 | no | — |
| 153 | C3×C22⋊C4 | — | 9 | 2 | no | — |
| 154 | C3×C42 | — | 3 | 2 | no | — |
| 155 | C3×C42 | — | 9 | 2 | no | — |
| 156 | #217 | — | 3 | 2 | no | — |
| 157 | #217 | C2×D9 | 1 | 2 | yes | — |
| 158 | C4⋊Dic3 | — | 3 | 2 | no | — |
| 159 | C4⋊Dic3 | — | 9 | 4 | no | — |
| 160 | C4⋊Dic3 | — | 9 | 4 | no | — |
| 161 | C4⋊Dic3 | C2×D9 | 1 | 2 | yes | — |
| 162 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 163 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 164 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 165 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 166 | C4×Dic3 | — | 3 | 2 | no | — |
| 167 | C4×Dic3 | — | 9 | 2 | no | — |
| 168 | C4×Dic3 | — | 9 | 1 | no | — |
| 169 | C4×Dic3 | — | 9 | 1 | no | — |
| 170 | C4×Dic3 | — | 9 | 2 | no | — |
| 171 | C4×Dic3 | — | 9 | 1 | no | — |
| 172 | C4×Dic3 | — | 9 | 1 | no | — |
| 173 | C4×Dic3 | — | 18 | 1 | no | — |
| 174 | C4×Dic3 | — | 18 | 1 | no | — |
| 175 | C4×Dic3 | — | 18 | 2 | no | — |
| 176 | C4×Dic3 | C2×D9 | 1 | 2 | yes | — |
| 177 | C22×C32 | — | 2 | 2 | no | — |
| 178 | C22×C32 | C2×Dic6 | 1 | 2 | yes | — |
| 179 | C22×C32 | C23×D3 | 1 | 1 | yes | characteristic |
| 180 | C32×C4 | C2×Dic6 | 1 | 2 | yes | — |
| 181 | C32×C4 | C2×Dic6 | 1 | 2 | yes | — |
| 182 | C3⋊Dic3 | — | 3 | 4 | no | — |
| 183 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 184 | C3×Dic3 | — | 2 | 4 | no | — |
| 185 | C3×Dic3 | — | 6 | 4 | no | — |
| 186 | C22×C9 | — | 2 | 4 | no | — |
| 187 | C22×C9 | — | 2 | 2 | no | — |
| 188 | C22×C9 | — | 2 | 1 | no | — |
| 189 | C22×C9 | — | 2 | 2 | no | — |
| 190 | C22×C9 | C2×Dic6 | 1 | 2 | yes | — |
| 191 | C22×C9 | C23×D3 | 1 | 1 | yes | characteristic |
| 192 | C4×C9 | — | 2 | 2 | no | — |
| 193 | C4×C9 | — | 2 | 2 | no | — |
| 194 | C4×C9 | — | 6 | 4 | no | — |
| 195 | C4×C9 | C2×Dic6 | 1 | 2 | yes | — |
| 196 | C4×C9 | C2×Dic6 | 1 | 2 | yes | — |
| 197 | Dic9 | — | 2 | 4 | no | — |
| 198 | Dic9 | — | 3 | 4 | no | — |
| 199 | Dic9 | — | 6 | 4 | no | — |
| 200 | Dic9 | — | 6 | 4 | no | — |
| 201 | Dic9 | — | 6 | 2 | no | — |
| 202 | C4⋊Q8 | — | 27 | 2 | no | — |
| 203 | #125 | — | 27 | 2 | no | — |
| 204 | C22⋊Q8 | — | 27 | 4 | no | — |
| 205 | C4×Q8 | — | 27 | 4 | no | — |
| 206 | #117 | — | 3 | 1 | no | — |
| 207 | #117 | — | 9 | 1 | no | — |
| 208 | C2×C42 | — | 27 | 1 | no | — |
| 209 | C3×C9 | #533 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 210 | C23×C3 | — | 2 | 1 | no | — |
| 211 | C23×C3 | C22×D9 | 1 | 1 | yes | characteristic |
| 212 | C23×C3 | C2×D32 | 1 | 1 | yes | characteristic |
| 213 | C2×C3×C4 | — | 2 | 4 | no | — |
| 214 | C2×C3×C4 | — | 2 | 1 | no | — |
| 215 | C2×C3×C4 | — | 2 | 1 | no | — |
| 216 | C2×C3×C4 | — | 3 | 4 | no | — |
| 217 | C2×C3×C4 | — | 9 | 4 | no | — |
| 218 | C2×C3×C4 | Dic18 | 1 | 4 | yes | complemented |
| 219 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 220 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 221 | C2×C3×C4 | #627 | 1 | 4 | yes | — |
| 222 | C2×C3×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 223 | C2×C3×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 224 | C2×Dic3 | — | 3 | 4 | no | — |
| 225 | C2×Dic3 | — | 9 | 2 | no | — |
| 226 | C2×Dic3 | — | 9 | 2 | no | — |
| 227 | C2×Dic3 | — | 9 | 2 | no | — |
| 228 | C2×Dic3 | — | 9 | 2 | no | — |
| 229 | C2×Dic3 | — | 18 | 2 | no | — |
| 230 | C2×Dic3 | — | 18 | 2 | no | — |
| 231 | C2×Dic3 | — | 18 | 4 | no | — |
| 232 | C2×Dic3 | — | 18 | 4 | no | — |
| 233 | C2×Dic3 | — | 18 | 8 | no | — |
| 234 | C2×Dic3 | C22×D9 | 1 | 4 | yes | — |
| 235 | Dic6 | — | 18 | 8 | no | — |
| 236 | Dic6 | — | 18 | 8 | no | — |
| 237 | C2×C32 | — | 2 | 2 | no | — |
| 238 | C2×C32 | C22×Dic6 | 1 | 1 | yes | characteristic |
| 239 | C2×C32 | D3×#41 | 1 | 2 | yes | — |
| 240 | C2×C9 | — | 2 | 4 | no | — |
| 241 | C2×C9 | — | 2 | 2 | no | — |
| 242 | C2×C9 | — | 2 | 2 | no | — |
| 243 | C2×C9 | — | 2 | 1 | no | — |
| 244 | C2×C9 | C22×Dic6 | 1 | 1 | yes | characteristic |
| 245 | C2×C9 | D3×#41 | 1 | 2 | yes | — |
| 246 | C2×Q8 | — | 27 | 4 | no | — |
| 247 | C22×C4 | — | 27 | 2 | no | — |
| 248 | C22×C4 | D3×D9 | 1 | 1 | yes | characteristic |
| 249 | C4⋊C4 | — | 3 | 2 | no | — |
| 250 | C4⋊C4 | — | 9 | 2 | no | — |
| 251 | C4⋊C4 | — | 27 | 4 | no | — |
| 252 | C4⋊C4 | — | 27 | 4 | no | — |
| 253 | C4⋊C4 | — | 27 | 4 | no | — |
| 254 | C22⋊C4 | — | 3 | 2 | no | — |
| 255 | C22⋊C4 | — | 9 | 2 | no | — |
| 256 | C42 | — | 3 | 2 | no | — |
| 257 | C42 | — | 9 | 2 | no | — |
| 258 | C42 | — | 27 | 1 | no | — |
| 259 | C42 | — | 27 | 1 | no | — |
| 260 | C42 | — | 27 | 2 | no | — |
| 261 | C22×C3 | — | 2 | 4 | no | — |
| 262 | C22×C3 | — | 2 | 2 | no | — |
| 263 | C22×C3 | — | 2 | 2 | no | — |
| 264 | C22×C3 | — | 2 | 1 | no | — |
| 265 | C22×C3 | — | 2 | 2 | no | — |
| 266 | C22×C3 | C2×Dic18 | 1 | 2 | yes | — |
| 267 | C22×C3 | C23×D9 | 1 | 1 | yes | characteristic |
| 268 | C22×C3 | C2×#627 | 1 | 2 | yes | — |
| 269 | C22×C3 | C22×D32 | 1 | 1 | yes | characteristic |
| 270 | C3×C4 | — | 2 | 2 | no | — |
| 271 | C3×C4 | — | 2 | 2 | no | — |
| 272 | C3×C4 | — | 6 | 4 | no | — |
| 273 | C3×C4 | — | 18 | 4 | no | — |
| 274 | C3×C4 | C2×Dic18 | 1 | 2 | yes | — |
| 275 | C3×C4 | C2×Dic18 | 1 | 2 | yes | — |
| 276 | C3×C4 | C2×#627 | 1 | 2 | yes | — |
| 277 | C3×C4 | C2×#627 | 1 | 2 | yes | — |
| 278 | Dic3 | — | 2 | 4 | no | — |
| 279 | Dic3 | — | 6 | 4 | no | — |
| 280 | Dic3 | — | 9 | 4 | no | — |
| 281 | Dic3 | — | 9 | 4 | no | — |
| 282 | Dic3 | — | 18 | 4 | no | — |
| 283 | Dic3 | — | 18 | 4 | no | — |
| 284 | Dic3 | — | 18 | 2 | no | — |
| 285 | Dic3 | — | 18 | 2 | no | — |
| 286 | C32 | #5517 | 1 | 1 | yes | characteristic |
| 287 | C9 | — | 2 | 1 | no | — |
| 288 | C9 | #5517 | 1 | 1 | yes | characteristic, complemented |
| 289 | C23 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 290 | Q8 | — | 54 | 8 | no | — |
| 291 | C2×C4 | — | 3 | 4 | no | — |
| 292 | C2×C4 | — | 9 | 4 | no | — |
| 293 | C2×C4 | — | 27 | 2 | no | — |
| 294 | C2×C4 | — | 27 | 2 | no | — |
| 295 | C2×C4 | — | 54 | 4 | no | — |
| 296 | C2×C4 | #6189 | 1 | 4 | yes | complemented |
| 297 | C2×C4 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 299 | C2×C3 | — | 2 | 4 | no | — |
| 300 | C2×C3 | — | 2 | 2 | no | — |
| 301 | C2×C3 | — | 2 | 2 | no | — |
| 302 | C2×C3 | — | 2 | 1 | no | — |
| 303 | C2×C3 | — | 2 | 2 | no | — |
| 304 | C2×C3 | C22×Dic18 | 1 | 1 | yes | characteristic |
| 305 | C2×C3 | #41×D9 | 1 | 2 | yes | — |
| 306 | C2×C3 | #64306 | 1 | 2 | yes | — |
| 307 | C2×C3 | C22×#627 | 1 | 1 | yes | characteristic |
| 308 | C22 | — | 2 | 2 | no | — |
| 309 | C22 | C2×#6189 | 1 | 2 | yes | — |
| 310 | C22 | C22×D3×D9 | 1 | 1 | yes | characteristic |
| 311 | C4 | — | 6 | 4 | no | — |
| 312 | C4 | — | 18 | 4 | no | — |
| 313 | C4 | — | 27 | 4 | no | — |
| 314 | C4 | — | 54 | 2 | no | — |
| 315 | C4 | C2×#6189 | 1 | 2 | yes | — |
| 316 | C4 | C2×#6189 | 1 | 2 | yes | — |
| 317 | C3 | — | 2 | 1 | no | — |
| 318 | C3 | #10589619 | 1 | 1 | yes | characteristic, complemented |
| 319 | C3 | #10595050 | 1 | 1 | yes | characteristic |
| 320 | C2 | — | 2 | 2 | no | — |
| 321 | C2 | #11722475 | 1 | 2 | yes | — |
| 322 | C2 | C22×#6189 | 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