← #13153871
Subgroups — #13153871
Order 1728.
1030 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×C9×#117 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11721815 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C4⋊(C3⋊Dic9) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11721746 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11721744 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11721724 | C2 | 1 | 4 | yes | maximal |
| 7 | #11721707 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11721706 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #10595990 | — | 3 | 1 | no | maximal |
| 10 | #10589621 | — | 3 | 3 | no | maximal |
| 11 | C22×C3×C4×C9 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×C3⋊Dic9 | C22 | 1 | 2 | yes | — |
| 13 | C2×#6226 | C22 | 1 | 4 | yes | — |
| 14 | C3×C9×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 15 | C3×C9×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×C9×C22⋊C4 | C22 | 1 | 2 | yes | — |
| 17 | C3×C42×C9 | C22 | 1 | 2 | yes | — |
| 18 | #88551 | C22 | 1 | 2 | yes | — |
| 19 | C4⋊(C3⋊Dic9) | C22 | 1 | 2 | yes | — |
| 20 | C4⋊(C3⋊Dic9) | C22 | 1 | 4 | yes | — |
| 21 | C4⋊(C3⋊Dic9) | C22 | 1 | 2 | yes | — |
| 22 | C4⋊(C3⋊Dic9) | C22 | 1 | 1 | yes | characteristic |
| 23 | C4⋊(C3⋊Dic9) | C22 | 1 | 1 | yes | characteristic |
| 24 | #88534 | C22 | 1 | 4 | yes | — |
| 25 | #88534 | C22 | 1 | 4 | yes | — |
| 26 | C4×C3⋊Dic9 | C22 | 1 | 2 | yes | — |
| 27 | C32×#117 | D3 | 1 | 1 | yes | characteristic |
| 28 | #64135 | — | 3 | 1 | no | — |
| 29 | C2×C4⋊(C3⋊Dic3) | — | 3 | 1 | no | — |
| 30 | #64099 | — | 3 | 2 | no | — |
| 31 | #64097 | — | 3 | 2 | no | — |
| 32 | #64087 | — | 3 | 4 | no | — |
| 33 | #64079 | — | 3 | 2 | no | — |
| 34 | #64078 | — | 3 | 2 | no | — |
| 35 | C9×#117 | D3 | 1 | 3 | yes | — |
| 36 | #63488 | — | 3 | 3 | no | — |
| 37 | C2×C4⋊Dic9 | — | 3 | 3 | no | — |
| 38 | #63452 | — | 3 | 6 | no | — |
| 39 | #63450 | — | 3 | 6 | no | — |
| 40 | #63440 | — | 3 | 12 | no | — |
| 41 | #63432 | — | 3 | 6 | no | — |
| 42 | #63431 | — | 3 | 6 | no | — |
| 43 | C23×C3×C9 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×C3×C4×C9 | Q8 | 1 | 4 | yes | complemented |
| 45 | C2×C3×C4×C9 | C23 | 1 | 2 | yes | — |
| 46 | C2×C3×C4×C9 | C23 | 1 | 2 | yes | — |
| 47 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C3⋊Dic9 | — | 4 | 2 | no | — |
| 50 | C2×C3⋊Dic9 | C23 | 1 | 2 | yes | — |
| 51 | C2×C3⋊Dic9 | C23 | 1 | 4 | yes | — |
| 52 | C2×C3⋊Dic9 | C23 | 1 | 2 | yes | — |
| 53 | #6226 | — | 4 | 4 | no | — |
| 54 | #5519 | — | 9 | 3 | no | — |
| 55 | #5519 | — | 9 | 1 | no | — |
| 56 | C22×C32×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | C22×C3⋊Dic3 | — | 3 | 2 | no | — |
| 58 | C2×#634 | — | 3 | 4 | no | — |
| 59 | C32×C4⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 60 | C32×C4⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | C32×C22⋊C4 | C2×D3 | 1 | 2 | yes | — |
| 62 | C32×C42 | C2×D3 | 1 | 2 | yes | — |
| 63 | #3757 | — | 3 | 2 | no | — |
| 64 | C4⋊(C3⋊Dic3) | — | 3 | 1 | no | — |
| 65 | C4⋊(C3⋊Dic3) | — | 3 | 2 | no | — |
| 66 | C4⋊(C3⋊Dic3) | — | 3 | 4 | no | — |
| 67 | C4⋊(C3⋊Dic3) | — | 3 | 1 | no | — |
| 68 | C4⋊(C3⋊Dic3) | — | 3 | 2 | no | — |
| 69 | #3750 | — | 3 | 4 | no | — |
| 70 | #3750 | — | 3 | 4 | no | — |
| 71 | C4×C3⋊Dic3 | — | 3 | 2 | no | — |
| 72 | C22×C4×C9 | C2×D3 | 1 | 3 | yes | — |
| 73 | C22×Dic9 | — | 3 | 6 | no | — |
| 74 | C2×Dic18 | — | 3 | 12 | no | — |
| 75 | C9×C4⋊C4 | C2×D3 | 1 | 3 | yes | — |
| 76 | C9×C4⋊C4 | C2×D3 | 1 | 3 | yes | — |
| 77 | C9×C22⋊C4 | C2×D3 | 1 | 6 | yes | — |
| 78 | C42×C9 | C2×D3 | 1 | 6 | yes | — |
| 79 | #3676 | — | 3 | 6 | no | — |
| 80 | C4⋊Dic9 | — | 3 | 3 | no | — |
| 81 | C4⋊Dic9 | — | 3 | 3 | no | — |
| 82 | C4⋊Dic9 | — | 3 | 6 | no | — |
| 83 | C4⋊Dic9 | — | 3 | 12 | no | — |
| 84 | C4⋊Dic9 | — | 3 | 6 | no | — |
| 85 | Dic9⋊C4 | — | 3 | 12 | no | — |
| 86 | Dic9⋊C4 | — | 3 | 12 | no | — |
| 87 | C4×Dic9 | — | 3 | 6 | no | — |
| 88 | C22×C3×C9 | — | 2 | 1 | no | — |
| 89 | C22×C3×C9 | — | 2 | 1 | no | — |
| 90 | C22×C3×C9 | C2×Q8 | 1 | 2 | yes | — |
| 91 | C22×C3×C9 | C24 | 1 | 1 | yes | characteristic |
| 92 | C3×C4×C9 | — | 2 | 2 | no | — |
| 93 | C3×C4×C9 | — | 2 | 2 | no | — |
| 94 | C3×C4×C9 | C2×Q8 | 1 | 2 | yes | — |
| 95 | C3×C4×C9 | C2×Q8 | 1 | 2 | yes | — |
| 96 | C3⋊Dic9 | — | 2 | 2 | no | — |
| 97 | C3⋊Dic9 | — | 2 | 4 | no | — |
| 98 | C3⋊Dic9 | — | 2 | 2 | no | — |
| 99 | C3×#117 | D9 | 1 | 3 | yes | — |
| 100 | C3×#117 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 101 | #926 | — | 9 | 3 | no | — |
| 102 | #926 | — | 9 | 1 | no | — |
| 103 | C2×C4⋊Dic3 | — | 9 | 3 | no | — |
| 104 | C2×C4⋊Dic3 | — | 9 | 1 | no | — |
| 105 | #890 | — | 9 | 6 | no | — |
| 106 | #890 | — | 9 | 2 | no | — |
| 107 | #888 | — | 9 | 6 | no | — |
| 108 | #888 | — | 9 | 2 | no | — |
| 109 | #878 | — | 9 | 12 | no | — |
| 110 | #878 | — | 9 | 4 | no | — |
| 111 | #870 | — | 9 | 6 | no | — |
| 112 | #870 | — | 9 | 2 | no | — |
| 113 | #869 | — | 9 | 6 | no | — |
| 114 | #869 | — | 9 | 2 | no | — |
| 115 | C23×C32 | C22×D3 | 1 | 1 | yes | characteristic |
| 116 | C2×C32×C4 | Dic6 | 1 | 4 | yes | — |
| 117 | C2×C32×C4 | C22×D3 | 1 | 2 | yes | — |
| 118 | C2×C32×C4 | C22×D3 | 1 | 2 | yes | — |
| 119 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 120 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 121 | C2×C3⋊Dic3 | — | 3 | 2 | no | — |
| 122 | C2×C3⋊Dic3 | — | 3 | 2 | no | — |
| 123 | C2×C3⋊Dic3 | — | 3 | 4 | no | — |
| 124 | C2×C3⋊Dic3 | — | 12 | 2 | no | — |
| 125 | #634 | — | 12 | 4 | no | — |
| 126 | C23×C9 | C22×D3 | 1 | 3 | yes | — |
| 127 | C2×C4×C9 | Dic6 | 1 | 12 | yes | complemented |
| 128 | C2×C4×C9 | C22×D3 | 1 | 6 | yes | — |
| 129 | C2×C4×C9 | C22×D3 | 1 | 6 | yes | — |
| 130 | C2×C4×C9 | C22×D3 | 1 | 3 | yes | — |
| 131 | C2×C4×C9 | C22×D3 | 1 | 3 | yes | — |
| 132 | C2×Dic9 | — | 3 | 6 | no | — |
| 133 | C2×Dic9 | — | 3 | 6 | no | — |
| 134 | C2×Dic9 | — | 3 | 12 | no | — |
| 135 | C2×Dic9 | — | 12 | 6 | no | — |
| 136 | Dic18 | — | 12 | 12 | no | — |
| 137 | #544 | — | 27 | 1 | no | Sylow |
| 138 | C2×C3×C9 | — | 2 | 2 | no | — |
| 139 | C2×C3×C9 | C22×Q8 | 1 | 1 | yes | characteristic |
| 140 | C2×C3×C9 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 141 | C2×C3×C9 | 2−1+4 | 1 | 1 | yes | characteristic |
| 142 | C22×C3×C4 | C2×D9 | 1 | 3 | yes | — |
| 143 | C22×C3×C4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 144 | C22×Dic3 | — | 9 | 6 | no | — |
| 145 | C22×Dic3 | — | 9 | 2 | no | — |
| 146 | C2×Dic6 | — | 9 | 12 | no | — |
| 147 | C2×Dic6 | — | 9 | 4 | no | — |
| 148 | C3×C4⋊C4 | C2×D9 | 1 | 3 | yes | — |
| 149 | C3×C4⋊C4 | C2×D9 | 1 | 3 | yes | — |
| 150 | C3×C4⋊C4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 151 | C3×C4⋊C4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 152 | C3×C22⋊C4 | C2×D9 | 1 | 6 | yes | — |
| 153 | C3×C22⋊C4 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 154 | C3×C42 | C2×D9 | 1 | 6 | yes | — |
| 155 | C3×C42 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 156 | #217 | — | 9 | 6 | no | — |
| 157 | #217 | — | 9 | 2 | no | — |
| 158 | C4⋊Dic3 | — | 9 | 3 | no | — |
| 159 | C4⋊Dic3 | — | 9 | 6 | no | — |
| 160 | C4⋊Dic3 | — | 9 | 12 | no | — |
| 161 | C4⋊Dic3 | — | 9 | 3 | no | — |
| 162 | C4⋊Dic3 | — | 9 | 6 | no | — |
| 163 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 164 | C4⋊Dic3 | — | 9 | 2 | no | — |
| 165 | C4⋊Dic3 | — | 9 | 4 | no | — |
| 166 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 167 | C4⋊Dic3 | — | 9 | 2 | no | — |
| 168 | Dic3⋊C4 | — | 9 | 12 | no | — |
| 169 | Dic3⋊C4 | — | 9 | 12 | no | — |
| 170 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 171 | Dic3⋊C4 | — | 9 | 4 | no | — |
| 172 | C4×Dic3 | — | 9 | 6 | no | — |
| 173 | C4×Dic3 | — | 9 | 2 | no | — |
| 174 | C22×C32 | — | 2 | 1 | no | — |
| 175 | C22×C32 | — | 2 | 1 | no | — |
| 176 | C22×C32 | C2×Dic6 | 1 | 2 | yes | — |
| 177 | C22×C32 | C23×D3 | 1 | 1 | yes | characteristic |
| 178 | C32×C4 | — | 2 | 2 | no | — |
| 179 | C32×C4 | — | 2 | 2 | no | — |
| 180 | C32×C4 | C2×Dic6 | 1 | 2 | yes | — |
| 181 | C32×C4 | C2×Dic6 | 1 | 2 | yes | — |
| 182 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 183 | C3⋊Dic3 | — | 6 | 4 | no | — |
| 184 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 185 | C22×C9 | — | 2 | 3 | no | — |
| 186 | C22×C9 | — | 2 | 3 | no | — |
| 187 | C22×C9 | C2×Dic6 | 1 | 6 | yes | — |
| 188 | C22×C9 | C23×D3 | 1 | 3 | yes | — |
| 189 | C4×C9 | — | 2 | 6 | no | — |
| 190 | C4×C9 | — | 2 | 6 | no | — |
| 191 | C4×C9 | C2×Dic6 | 1 | 6 | yes | — |
| 192 | C4×C9 | C2×Dic6 | 1 | 6 | yes | — |
| 193 | Dic9 | — | 6 | 6 | no | — |
| 194 | Dic9 | — | 6 | 12 | no | — |
| 195 | Dic9 | — | 6 | 6 | no | — |
| 196 | C4⋊Q8 | — | 27 | 2 | no | — |
| 197 | C4⋊Q8 | — | 27 | 2 | no | — |
| 198 | #125 | — | 27 | 2 | no | — |
| 199 | #125 | — | 27 | 2 | no | — |
| 200 | C22⋊Q8 | — | 27 | 4 | no | — |
| 201 | #117 | — | 27 | 1 | no | — |
| 202 | #117 | C3⋊D9 | 1 | 1 | yes | characteristic |
| 203 | C2×C4⋊C4 | — | 27 | 1 | no | — |
| 204 | C3×C9 | #544 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 205 | C23×C3 | C22×D9 | 1 | 3 | yes | — |
| 206 | C23×C3 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 207 | C2×C3×C4 | Dic18 | 1 | 12 | yes | complemented |
| 208 | C2×C3×C4 | C22×D9 | 1 | 6 | yes | — |
| 209 | C2×C3×C4 | C22×D9 | 1 | 6 | yes | — |
| 210 | C2×C3×C4 | C22×D9 | 1 | 3 | yes | — |
| 211 | C2×C3×C4 | C22×D9 | 1 | 3 | yes | — |
| 212 | C2×C3×C4 | #634 | 1 | 4 | yes | — |
| 213 | C2×C3×C4 | C22×C3⋊D3 | 1 | 2 | yes | — |
| 214 | C2×C3×C4 | C22×C3⋊D3 | 1 | 2 | yes | — |
| 215 | C2×C3×C4 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 216 | C2×C3×C4 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 217 | C2×Dic3 | — | 9 | 2 | no | — |
| 218 | C2×Dic3 | — | 9 | 2 | no | — |
| 219 | C2×Dic3 | — | 9 | 4 | no | — |
| 220 | C2×Dic3 | — | 9 | 6 | no | — |
| 221 | C2×Dic3 | — | 9 | 6 | no | — |
| 222 | C2×Dic3 | — | 9 | 12 | no | — |
| 223 | C2×Dic3 | — | 36 | 2 | no | — |
| 224 | C2×Dic3 | — | 36 | 6 | no | — |
| 225 | Dic6 | — | 36 | 4 | no | — |
| 226 | Dic6 | — | 36 | 12 | no | — |
| 227 | C2×C32 | — | 2 | 2 | no | — |
| 228 | C2×C32 | C22×Dic6 | 1 | 1 | yes | characteristic |
| 229 | C2×C32 | #1009 | 1 | 1 | yes | characteristic |
| 230 | C2×C32 | #1010 | 1 | 1 | yes | characteristic |
| 231 | C2×C9 | — | 2 | 6 | no | — |
| 232 | C2×C9 | C22×Dic6 | 1 | 3 | yes | — |
| 233 | C2×C9 | #1009 | 1 | 3 | yes | — |
| 234 | C2×C9 | #1010 | 1 | 3 | yes | — |
| 235 | C2×Q8 | — | 27 | 4 | no | — |
| 236 | C22×C4 | — | 27 | 2 | no | — |
| 237 | C22×C4 | C2×C3⋊D9 | 1 | 1 | yes | characteristic |
| 238 | C4⋊C4 | — | 27 | 2 | no | — |
| 239 | C4⋊C4 | — | 27 | 2 | no | — |
| 240 | C4⋊C4 | — | 27 | 1 | no | — |
| 241 | C4⋊C4 | — | 27 | 1 | no | — |
| 242 | C4⋊C4 | — | 27 | 4 | no | — |
| 243 | C4⋊C4 | — | 27 | 4 | no | — |
| 244 | C4⋊C4 | — | 27 | 4 | no | — |
| 245 | C4⋊C4 | C2×C3⋊D9 | 1 | 1 | yes | characteristic |
| 246 | C4⋊C4 | C2×C3⋊D9 | 1 | 1 | yes | characteristic |
| 247 | C22⋊C4 | — | 27 | 2 | no | — |
| 248 | C22⋊C4 | C2×C3⋊D9 | 1 | 2 | yes | — |
| 249 | C42 | — | 27 | 2 | no | — |
| 250 | C42 | C2×C3⋊D9 | 1 | 2 | yes | — |
| 251 | C22×C3 | — | 2 | 1 | no | — |
| 252 | C22×C3 | — | 2 | 1 | no | — |
| 253 | C22×C3 | — | 2 | 3 | no | — |
| 254 | C22×C3 | — | 2 | 3 | no | — |
| 255 | C22×C3 | C2×Dic18 | 1 | 6 | yes | — |
| 256 | C22×C3 | C23×D9 | 1 | 3 | yes | — |
| 257 | C22×C3 | C2×#634 | 1 | 2 | yes | — |
| 258 | C22×C3 | C23×C3⋊D3 | 1 | 1 | yes | characteristic |
| 259 | C3×C4 | — | 2 | 2 | no | — |
| 260 | C3×C4 | — | 2 | 2 | no | — |
| 261 | C3×C4 | — | 2 | 6 | no | — |
| 262 | C3×C4 | — | 2 | 6 | no | — |
| 263 | C3×C4 | C2×Dic18 | 1 | 6 | yes | — |
| 264 | C3×C4 | C2×Dic18 | 1 | 6 | yes | — |
| 265 | C3×C4 | C2×#634 | 1 | 2 | yes | — |
| 266 | C3×C4 | C2×#634 | 1 | 2 | yes | — |
| 267 | Dic3 | — | 18 | 2 | no | — |
| 268 | Dic3 | — | 18 | 4 | no | — |
| 269 | Dic3 | — | 18 | 2 | no | — |
| 270 | Dic3 | — | 18 | 6 | no | — |
| 271 | Dic3 | — | 18 | 12 | no | — |
| 272 | Dic3 | — | 18 | 6 | no | — |
| 273 | C32 | #5519 | 1 | 1 | yes | characteristic |
| 274 | C9 | #5519 | 1 | 3 | yes | complemented |
| 275 | C23 | C22×C3⋊D9 | 1 | 1 | yes | characteristic |
| 276 | Q8 | — | 108 | 4 | no | — |
| 277 | C2×C4 | — | 27 | 2 | no | — |
| 278 | C2×C4 | — | 27 | 2 | no | — |
| 279 | C2×C4 | — | 27 | 4 | no | — |
| 280 | C2×C4 | — | 108 | 2 | no | — |
| 281 | C2×C4 | #6226 | 1 | 4 | yes | complemented |
| 282 | C2×C4 | C22×C3⋊D9 | 1 | 2 | yes | — |
| 283 | C2×C4 | C22×C3⋊D9 | 1 | 2 | yes | — |
| 284 | C2×C4 | C22×C3⋊D9 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C22×C3⋊D9 | 1 | 1 | yes | characteristic |
| 286 | C2×C3 | — | 2 | 2 | no | — |
| 287 | C2×C3 | — | 2 | 6 | no | — |
| 288 | C2×C3 | C22×Dic18 | 1 | 3 | yes | — |
| 289 | C2×C3 | #63715 | 1 | 3 | yes | — |
| 290 | C2×C3 | #63716 | 1 | 3 | yes | — |
| 291 | C2×C3 | C22×#634 | 1 | 1 | yes | characteristic |
| 292 | C2×C3 | #64366 | 1 | 1 | yes | characteristic |
| 293 | C2×C3 | #64367 | 1 | 1 | yes | characteristic |
| 294 | C22 | — | 2 | 1 | no | — |
| 295 | C22 | — | 2 | 1 | no | — |
| 296 | C22 | C2×#6226 | 1 | 2 | yes | — |
| 297 | C22 | C23×C3⋊D9 | 1 | 1 | yes | characteristic |
| 298 | C4 | — | 2 | 2 | no | — |
| 299 | C4 | — | 2 | 2 | no | — |
| 300 | C4 | — | 54 | 4 | no | — |
| 301 | C4 | — | 54 | 2 | no | — |
| 302 | C4 | — | 54 | 2 | no | — |
| 303 | C4 | C2×#6226 | 1 | 2 | yes | — |
| 304 | C4 | C2×#6226 | 1 | 2 | yes | — |
| 305 | C3 | #10589621 | 1 | 3 | yes | complemented |
| 306 | C3 | #10595990 | 1 | 1 | yes | characteristic |
| 307 | C2 | — | 2 | 2 | no | — |
| 308 | C2 | C22×#6226 | 1 | 1 | yes | characteristic |
| 309 | C2 | #11722672 | 1 | 1 | yes | characteristic |
| 310 | C2 | #11722673 | 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