Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
478 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×D4×C22⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊(C3⋊((C22⋊C9)⋊C2)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊((C22⋊C9)⋊C8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10592916 | — | 3 | 1 | no | maximal |
| 5 | #10589517 | — | 3 | 3 | no | maximal |
| 6 | C22×C3×C22⋊C9 | — | 2 | 1 | no | — |
| 7 | C2×C3⋊((C22⋊C9)⋊C2) | — | 2 | 1 | no | — |
| 8 | C3×C4×C22⋊C9 | C22 | 1 | 1 | yes | characteristic |
| 9 | #88544 | — | 4 | 1 | no | maximal |
| 10 | C22×C32×D4 | D3 | 1 | 1 | yes | characteristic |
| 11 | C2×#3753 | — | 3 | 1 | no | — |
| 12 | C2×#3753 | — | 3 | 1 | no | — |
| 13 | #64139 | — | 3 | 1 | no | — |
| 14 | D4×C22⋊C9 | D3 | 1 | 3 | yes | complemented |
| 15 | C4⋊((C22⋊C9)⋊C2) | — | 3 | 3 | no | — |
| 16 | #63659 | — | 3 | 1 | no | — |
| 17 | #63659 | — | 3 | 1 | no | — |
| 18 | #63657 | — | 3 | 1 | no | — |
| 19 | (C22⋊C9)⋊C8 | — | 3 | 3 | no | — |
| 20 | C2×C3×C22⋊C9 | — | 4 | 1 | no | — |
| 21 | C2×C3×C22⋊C9 | D4 | 1 | 1 | yes | characteristic |
| 22 | C3⋊((C22⋊C9)⋊C2) | — | 4 | 1 | no | — |
| 23 | C3×D4×C9 | — | 4 | 1 | no | — |
| 24 | C4⋊(C3⋊D9) | — | 4 | 1 | no | — |
| 25 | C3⋊(C9⋊C8) | — | 4 | 1 | no | — |
| 26 | #5217 | — | 9 | 3 | no | — |
| 27 | #5217 | — | 9 | 1 | no | — |
| 28 | C24×C32 | — | 2 | 1 | no | — |
| 29 | C2×C32×D4 | — | 3 | 1 | no | — |
| 30 | C2×C32×D4 | — | 3 | 1 | no | — |
| 31 | C2×C32×D4 | — | 6 | 1 | no | — |
| 32 | C22×C32×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | C2×#638 | — | 6 | 1 | no | — |
| 34 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 35 | C22×C22⋊C9 | — | 2 | 3 | no | — |
| 36 | C2×(C22⋊C9)⋊C2 | — | 6 | 3 | no | — |
| 37 | #3753 | — | 6 | 2 | no | — |
| 38 | #3753 | — | 6 | 1 | no | — |
| 39 | #3753 | — | 6 | 1 | no | — |
| 40 | #3752 | — | 6 | 1 | no | — |
| 41 | C4⋊(C3⋊Dic3) | — | 3 | 1 | no | — |
| 42 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 43 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 44 | C4×C22⋊C9 | C2×D3 | 1 | 3 | yes | — |
| 45 | #3673 | — | 12 | 3 | no | — |
| 46 | C22×C3×C9 | — | 8 | 1 | no | — |
| 47 | C2×C3⋊D9 | — | 8 | 1 | no | — |
| 48 | C3×C22⋊C9 | D8 | 1 | 1 | yes | characteristic, complemented |
| 49 | C3×C4×C9 | — | 4 | 1 | no | — |
| 50 | C22×C3×D4 | D9 | 1 | 3 | yes | complemented |
| 51 | C22×C3×D4 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 52 | C2×#213 | — | 9 | 3 | no | — |
| 53 | C2×#213 | — | 9 | 1 | no | — |
| 54 | C2×#213 | — | 9 | 3 | no | — |
| 55 | C2×#213 | — | 9 | 1 | no | — |
| 56 | #930 | — | 9 | 3 | no | — |
| 57 | #930 | — | 9 | 1 | no | — |
| 58 | #832 | — | 9 | 3 | no | — |
| 59 | #832 | — | 9 | 1 | no | — |
| 60 | #832 | — | 9 | 3 | no | — |
| 61 | #832 | — | 9 | 1 | no | — |
| 62 | #830 | — | 9 | 3 | no | — |
| 63 | #830 | — | 9 | 1 | no | — |
| 64 | C23×C32 | — | 4 | 1 | no | — |
| 65 | C23×C32 | — | 6 | 1 | no | — |
| 66 | C23×C32 | — | 6 | 1 | no | — |
| 67 | C23×C32 | — | 12 | 1 | no | — |
| 68 | C23×C32 | #67 | 1 | 1 | yes | characteristic |
| 69 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 70 | C32×D4 | — | 3 | 2 | no | — |
| 71 | C32×D4 | — | 3 | 1 | no | — |
| 72 | C32×D4 | — | 6 | 1 | no | — |
| 73 | C32×D4 | S4 | 1 | 1 | yes | characteristic |
| 74 | C2×C32×C4 | — | 3 | 1 | no | — |
| 75 | C2×C32×C4 | — | 3 | 1 | no | — |
| 76 | #638 | — | 12 | 1 | no | — |
| 77 | #638 | — | 12 | 1 | no | — |
| 78 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 79 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 80 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 81 | C2×C22⋊C9 | — | 4 | 3 | no | — |
| 82 | C2×C22⋊C9 | #67 | 1 | 3 | yes | — |
| 83 | (C22⋊C9)⋊C2 | — | 12 | 3 | no | — |
| 84 | C3⋊(C3⋊C8) | — | 6 | 1 | no | — |
| 85 | C3⋊(C3⋊C8) | — | 6 | 1 | no | — |
| 86 | D4×C9 | — | 4 | 3 | no | — |
| 87 | D36 | — | 12 | 3 | no | — |
| 88 | C9⋊C8 | — | 12 | 3 | no | — |
| 89 | #447 | — | 27 | 1 | no | Sylow |
| 90 | C2×C3×C9 | — | 4 | 1 | no | — |
| 91 | C2×C3×C9 | — | 16 | 1 | no | — |
| 92 | C3⋊D9 | — | 16 | 1 | no | — |
| 93 | C24×C3 | — | 2 | 3 | no | — |
| 94 | C24×C3 | — | 2 | 1 | no | — |
| 95 | C2×C3×D4 | — | 3 | 3 | no | — |
| 96 | C2×C3×D4 | — | 3 | 1 | no | — |
| 97 | C2×C3×D4 | — | 3 | 3 | no | — |
| 98 | C2×C3×D4 | — | 3 | 1 | no | — |
| 99 | C2×C3×D4 | — | 6 | 3 | no | — |
| 100 | C2×C3×D4 | — | 6 | 1 | no | — |
| 101 | C22×C3×C4 | C2×D9 | 1 | 3 | yes | — |
| 102 | C22×C3×C4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 103 | C2×#67 | — | 18 | 3 | no | — |
| 104 | C2×#67 | — | 18 | 1 | no | — |
| 105 | C2×D12 | — | 9 | 1 | no | — |
| 106 | C2×D12 | — | 9 | 3 | no | — |
| 107 | #213 | — | 18 | 2 | no | — |
| 108 | #213 | — | 18 | 6 | no | — |
| 109 | #213 | — | 18 | 1 | no | — |
| 110 | #213 | — | 18 | 3 | no | — |
| 111 | #213 | — | 18 | 3 | no | — |
| 112 | #213 | — | 18 | 1 | no | — |
| 113 | (C2×D3)⋊C4 | — | 18 | 3 | no | — |
| 114 | (C2×D3)⋊C4 | — | 18 | 1 | no | — |
| 115 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 116 | C4⋊Dic3 | — | 9 | 3 | no | — |
| 117 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 118 | C2×C3⋊C8 | — | 9 | 3 | no | — |
| 119 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 120 | C2×C3⋊C8 | — | 9 | 3 | no | — |
| 121 | C22×C32 | — | 2 | 1 | no | — |
| 122 | C22×C32 | — | 3 | 1 | no | — |
| 123 | C22×C32 | — | 3 | 1 | no | — |
| 124 | C22×C32 | — | 6 | 1 | no | — |
| 125 | C22×C32 | — | 12 | 1 | no | — |
| 126 | C22×C32 | — | 12 | 1 | no | — |
| 127 | C22×C32 | — | 12 | 1 | no | — |
| 128 | C22×C32 | — | 12 | 1 | no | — |
| 129 | C22×C32 | #213 | 1 | 1 | yes | characteristic |
| 130 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 131 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 132 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 133 | C32×C4 | — | 3 | 1 | no | — |
| 134 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 135 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 136 | C22×C9 | — | 8 | 3 | no | — |
| 137 | C2×D9 | — | 24 | 3 | no | — |
| 138 | C22⋊C9 | #213 | 1 | 3 | yes | complemented |
| 139 | C4×C9 | — | 4 | 3 | no | — |
| 140 | C22×D4 | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 141 | C2×D8 | — | 27 | 1 | no | — |
| 142 | C2×D8 | — | 27 | 1 | no | — |
| 143 | #121 | — | 27 | 1 | no | — |
| 144 | #102 | — | 27 | 1 | no | — |
| 145 | #102 | — | 27 | 1 | no | — |
| 146 | C22⋊C8 | — | 27 | 1 | no | — |
| 147 | C3×C9 | — | 4 | 1 | no | Sylow |
| 148 | C23×C3 | — | 4 | 3 | no | — |
| 149 | C23×C3 | — | 4 | 1 | no | — |
| 150 | C23×C3 | — | 6 | 1 | no | — |
| 151 | C23×C3 | — | 6 | 3 | no | — |
| 152 | C23×C3 | — | 6 | 1 | no | — |
| 153 | C23×C3 | — | 6 | 3 | no | — |
| 154 | C23×C3 | — | 12 | 3 | no | — |
| 155 | C23×C3 | — | 12 | 1 | no | — |
| 156 | C23×C3 | #611 | 1 | 3 | yes | — |
| 157 | C23×C3 | #638 | 1 | 1 | yes | characteristic |
| 158 | C22×D3 | — | 18 | 1 | no | — |
| 159 | C22×D3 | — | 18 | 3 | no | — |
| 160 | C3×D4 | — | 3 | 2 | no | — |
| 161 | C3×D4 | — | 3 | 6 | no | — |
| 162 | C3×D4 | — | 3 | 1 | no | — |
| 163 | C3×D4 | — | 3 | 3 | no | — |
| 164 | C3×D4 | — | 6 | 1 | no | — |
| 165 | C3×D4 | — | 6 | 3 | no | — |
| 166 | C3×D4 | (C22⋊C9)⋊C2 | 1 | 3 | yes | complemented |
| 167 | C3×D4 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 168 | C2×C3×C4 | — | 3 | 1 | no | — |
| 169 | C2×C3×C4 | — | 3 | 3 | no | — |
| 170 | C2×C3×C4 | — | 3 | 3 | no | — |
| 171 | C2×C3×C4 | — | 3 | 1 | no | — |
| 172 | #67 | — | 36 | 3 | no | — |
| 173 | #67 | — | 36 | 1 | no | — |
| 174 | #67 | — | 36 | 3 | no | — |
| 175 | #67 | — | 36 | 1 | no | — |
| 176 | C2×Dic3 | — | 18 | 1 | no | — |
| 177 | C2×Dic3 | — | 18 | 3 | no | — |
| 178 | D12 | — | 18 | 1 | no | — |
| 179 | D12 | — | 18 | 3 | no | — |
| 180 | D12 | — | 18 | 1 | no | — |
| 181 | D12 | — | 18 | 3 | no | — |
| 182 | C3⋊C8 | — | 18 | 1 | no | — |
| 183 | C3⋊C8 | — | 18 | 3 | no | — |
| 184 | C3⋊C8 | — | 18 | 1 | no | — |
| 185 | C3⋊C8 | — | 18 | 3 | no | — |
| 186 | C2×C32 | — | 3 | 1 | no | — |
| 187 | C2×C32 | — | 3 | 1 | no | — |
| 188 | C2×C32 | — | 4 | 1 | no | — |
| 189 | C2×C32 | — | 12 | 1 | no | — |
| 190 | C2×C32 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 191 | C3⋊D3 | — | 24 | 1 | no | — |
| 192 | C2×C9 | — | 4 | 3 | no | — |
| 193 | C2×C9 | — | 16 | 3 | no | — |
| 194 | D9 | — | 48 | 3 | no | — |
| 195 | C24 | — | 2 | 1 | no | — |
| 196 | C2×D4 | — | 3 | 1 | no | — |
| 197 | C2×D4 | — | 3 | 1 | no | — |
| 198 | C2×D4 | — | 6 | 1 | no | — |
| 199 | C2×D4 | — | 27 | 1 | no | — |
| 200 | C2×D4 | — | 54 | 1 | no | — |
| 201 | C22×C4 | C2×C3⋊D9 | 1 | 1 | yes | characteristic |
| 202 | D8 | — | 54 | 2 | no | — |
| 203 | D8 | — | 54 | 1 | no | — |
| 204 | D8 | — | 54 | 1 | no | — |
| 205 | C2×C8 | — | 27 | 1 | no | — |
| 206 | C2×C8 | — | 27 | 1 | no | — |
| 207 | C4⋊C4 | — | 27 | 1 | no | — |
| 208 | C22⋊C4 | — | 54 | 1 | no | — |
| 209 | C22×C3 | — | 2 | 1 | no | — |
| 210 | C22×C3 | — | 2 | 3 | no | — |
| 211 | C22×C3 | — | 3 | 3 | no | — |
| 212 | C22×C3 | — | 3 | 1 | no | — |
| 213 | C22×C3 | — | 3 | 1 | no | — |
| 214 | C22×C3 | — | 3 | 3 | no | — |
| 215 | C22×C3 | — | 6 | 1 | no | — |
| 216 | C22×C3 | — | 6 | 3 | no | — |
| 217 | C22×C3 | — | 12 | 3 | no | — |
| 218 | C22×C3 | — | 12 | 1 | no | — |
| 219 | C22×C3 | — | 12 | 3 | no | — |
| 220 | C22×C3 | — | 12 | 1 | no | — |
| 221 | C22×C3 | — | 12 | 3 | no | — |
| 222 | C22×C3 | — | 12 | 1 | no | — |
| 223 | C22×C3 | — | 12 | 3 | no | — |
| 224 | C22×C3 | — | 12 | 1 | no | — |
| 225 | C22×C3 | #3673 | 1 | 3 | yes | complemented |
| 226 | C22×C3 | #3753 | 1 | 1 | yes | characteristic |
| 227 | C2×D3 | — | 36 | 3 | no | — |
| 228 | C2×D3 | — | 36 | 1 | no | — |
| 229 | C2×D3 | — | 36 | 3 | no | — |
| 230 | C2×D3 | — | 36 | 1 | no | — |
| 231 | C2×D3 | — | 36 | 1 | no | — |
| 232 | C2×D3 | — | 36 | 3 | no | — |
| 233 | C3×C4 | — | 3 | 1 | no | — |
| 234 | C3×C4 | — | 3 | 3 | no | — |
| 235 | C3×C4 | C2×(C22⋊C9)⋊C2 | 1 | 3 | yes | — |
| 236 | C3×C4 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 237 | Dic3 | — | 36 | 3 | no | — |
| 238 | Dic3 | — | 36 | 1 | no | — |
| 239 | C32 | #5415 | 1 | 1 | yes | characteristic |
| 240 | C9 | — | 4 | 3 | no | — |
| 241 | C23 | — | 4 | 1 | no | — |
| 242 | C23 | — | 6 | 1 | no | — |
| 243 | C23 | — | 6 | 1 | no | — |
| 244 | C23 | — | 12 | 1 | no | — |
| 245 | C23 | — | 54 | 1 | no | — |
| 246 | C23 | #6233 | 1 | 1 | yes | characteristic |
| 247 | D4 | — | 3 | 2 | no | — |
| 248 | D4 | — | 3 | 1 | no | — |
| 249 | D4 | — | 6 | 1 | no | — |
| 250 | D4 | — | 54 | 1 | no | — |
| 251 | D4 | — | 54 | 1 | no | — |
| 252 | D4 | — | 108 | 1 | no | — |
| 253 | D4 | — | 108 | 1 | no | — |
| 254 | D4 | C3⋊((C22⋊C9)⋊C2) | 1 | 1 | yes | characteristic, complemented |
| 255 | C2×C4 | — | 3 | 1 | no | — |
| 256 | C2×C4 | — | 3 | 1 | no | — |
| 257 | C2×C4 | — | 54 | 1 | no | — |
| 258 | C8 | — | 54 | 1 | no | — |
| 259 | C8 | — | 54 | 1 | no | — |
| 260 | C2×C3 | — | 3 | 3 | no | — |
| 261 | C2×C3 | — | 3 | 3 | no | — |
| 262 | C2×C3 | — | 3 | 1 | no | — |
| 263 | C2×C3 | — | 3 | 1 | no | — |
| 264 | C2×C3 | — | 4 | 3 | no | — |
| 265 | C2×C3 | — | 4 | 1 | no | — |
| 266 | C2×C3 | — | 12 | 3 | no | — |
| 267 | C2×C3 | — | 12 | 1 | no | — |
| 268 | C2×C3 | #63694 | 1 | 3 | yes | — |
| 269 | C2×C3 | #64269 | 1 | 1 | yes | characteristic |
| 270 | D3 | — | 72 | 3 | no | — |
| 271 | D3 | — | 72 | 1 | no | — |
| 272 | C22 | — | 2 | 1 | no | — |
| 273 | C22 | — | 3 | 1 | no | — |
| 274 | C22 | — | 3 | 1 | no | — |
| 275 | C22 | — | 6 | 1 | no | — |
| 276 | C22 | — | 12 | 1 | no | — |
| 277 | C22 | — | 12 | 1 | no | — |
| 278 | C22 | — | 12 | 1 | no | — |
| 279 | C22 | — | 12 | 1 | no | — |
| 280 | C22 | — | 108 | 1 | no | — |
| 281 | C22 | — | 108 | 1 | no | — |
| 282 | C22 | — | 108 | 1 | no | — |
| 283 | C22 | #88544 | 1 | 1 | yes | characteristic, complemented |
| 284 | C4 | — | 3 | 1 | no | — |
| 285 | C4 | — | 108 | 1 | no | — |
| 286 | C4 | C2×C3⋊((C22⋊C9)⋊C2) | 1 | 1 | yes | characteristic |
| 287 | C3 | #10589517 | 1 | 3 | yes | complemented |
| 288 | C3 | #10593649 | 1 | 1 | yes | characteristic |
| 289 | C2 | — | 3 | 1 | no | — |
| 290 | C2 | — | 3 | 1 | no | — |
| 291 | C2 | — | 4 | 1 | no | — |
| 292 | C2 | — | 12 | 1 | no | — |
| 293 | C2 | — | 216 | 1 | no | — |
| 294 | C2 | #11722396 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×D4×C22⋊C9 | C2 | semidirect |
| 2. C4⋊(C3⋊((C22⋊C9)⋊C2)) | C2 | semidirect |
| 3. C3⋊((C22⋊C9)⋊C8) | C2 | semidirect |
| 14. D4×C22⋊C9 | D3 | semidirect |
| 48. C3×C22⋊C9 | D8 | semidirect |
| 50. C22×C3×D4 | D9 | semidirect |
| 138. C22⋊C9 | #213 | semidirect |
| 140. C22×D4 | C3⋊D9 | semidirect |
| 166. C3×D4 | (C22⋊C9)⋊C2 | semidirect |
| 225. C22×C3 | #3673 | semidirect |
| 254. D4 | C3⋊((C22⋊C9)⋊C2) | semidirect |
| 283. C22 | #88544 | semidirect |
| 287. C3 | #10589517 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D4×C22⋊C9 | 2 |
| C4⋊(C3⋊((C22⋊C9)⋊C2)) | 2 |
| C3⋊((C22⋊C9)⋊C8) | 2 |
| #10592916 | 3 |
| #10589517 | 3 |
| #88544 | 4 |