Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
311 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 45 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 | D4×C9×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | A4×D36 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | A4×C9⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | A4×#213 | — | 3 | 1 | no | maximal |
| 5 | C22×#3673 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×C9×A4 | — | 2 | 1 | no | — |
| 7 | C2×A4×D9 | — | 2 | 1 | no | — |
| 8 | C4×C9×A4 | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×#3673 | — | 4 | 1 | no | maximal |
| 10 | C3×D4×A4 | D3 | 1 | 1 | yes | characteristic |
| 11 | A4×D12 | — | 3 | 1 | no | — |
| 12 | A4×C3⋊C8 | — | 3 | 1 | no | — |
| 13 | C22×D4×C9 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×D36 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 15 | D4×C22⋊C9 | — | 2 | 1 | no | — |
| 16 | C2×#3673 | — | 3 | 1 | no | — |
| 17 | C2×#3673 | — | 3 | 1 | no | — |
| 18 | C2×#3673 | — | 3 | 1 | no | — |
| 19 | C2×#3673 | — | 3 | 1 | no | — |
| 20 | C22×C9⋊C8 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 21 | C2×C9×A4 | — | 4 | 1 | no | — |
| 22 | C2×C9×A4 | D4 | 1 | 1 | yes | characteristic |
| 23 | A4×D9 | — | 4 | 1 | no | — |
| 24 | C3×D4×C9 | — | 4 | 1 | no | — |
| 25 | C3×D36 | — | 4 | 1 | no | — |
| 26 | C3×C9⋊C8 | — | 4 | 1 | no | — |
| 27 | C22×#213 | — | 3 | 1 | no | — |
| 28 | A4×D8 | — | 9 | 1 | no | — |
| 29 | C22×C3×A4 | — | 2 | 1 | no | — |
| 30 | C2×D3×A4 | — | 6 | 1 | no | — |
| 31 | C3×C4×A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 32 | C24×C9 | — | 2 | 1 | no | — |
| 33 | C23×D9 | — | 2 | 1 | no | — |
| 34 | C22×C22⋊C9 | — | 2 | 2 | no | — |
| 35 | C3×#213 | — | 12 | 1 | no | — |
| 36 | C2×D4×C9 | — | 3 | 1 | no | — |
| 37 | C2×D4×C9 | — | 3 | 1 | no | — |
| 38 | C2×D4×C9 | — | 6 | 1 | no | — |
| 39 | C22×C4×C9 | C22×C3 | 1 | 1 | yes | characteristic |
| 40 | C2×D36 | — | 3 | 1 | no | — |
| 41 | C2×D36 | — | 3 | 1 | no | — |
| 42 | C2×D36 | — | 6 | 1 | no | — |
| 43 | C4×C22⋊C9 | — | 2 | 1 | no | — |
| 44 | #3673 | — | 3 | 2 | no | — |
| 45 | #3673 | — | 3 | 1 | no | — |
| 46 | #3673 | — | 3 | 1 | no | — |
| 47 | #3673 | — | 3 | 1 | no | — |
| 48 | #3673 | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 49 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 50 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 51 | C22×C3×C9 | — | 8 | 1 | no | — |
| 52 | C2×C3×D9 | — | 8 | 1 | no | — |
| 53 | C9×A4 | D8 | 1 | 1 | yes | characteristic, complemented |
| 54 | C3×C4×C9 | — | 4 | 1 | no | — |
| 55 | C22×C3×D4 | C3×D3 | 1 | 1 | yes | characteristic |
| 56 | C22×D12 | — | 3 | 1 | no | — |
| 57 | D4×A4 | — | 2 | 1 | no | — |
| 58 | D4×A4 | — | 9 | 1 | no | — |
| 59 | D4×A4 | D9 | 1 | 1 | yes | characteristic, complemented |
| 60 | C2×#213 | — | 9 | 1 | no | — |
| 61 | C2×#213 | — | 9 | 1 | no | — |
| 62 | C2×#213 | — | 9 | 1 | no | — |
| 63 | C2×#213 | — | 9 | 1 | no | — |
| 64 | C22×C3⋊C8 | — | 3 | 1 | no | — |
| 65 | C8×A4 | — | 9 | 1 | no | — |
| 66 | C2×C3×A4 | — | 4 | 1 | no | — |
| 67 | C2×C3×A4 | #67 | 1 | 1 | yes | characteristic |
| 68 | D3×A4 | — | 12 | 1 | no | — |
| 69 | C32×D4 | — | 4 | 1 | no | — |
| 70 | C3×D12 | — | 12 | 1 | no | — |
| 71 | C23×C9 | — | 4 | 1 | no | — |
| 72 | C23×C9 | — | 6 | 1 | no | — |
| 73 | C23×C9 | — | 6 | 1 | no | — |
| 74 | C23×C9 | — | 12 | 1 | no | — |
| 75 | C23×C9 | C3×D4 | 1 | 1 | yes | characteristic |
| 76 | C22×D9 | — | 4 | 1 | no | — |
| 77 | C22×D9 | — | 6 | 1 | no | — |
| 78 | C22×D9 | — | 6 | 1 | no | — |
| 79 | C22×D9 | — | 12 | 1 | no | — |
| 80 | C2×C22⋊C9 | — | 2 | 1 | no | — |
| 81 | C2×C22⋊C9 | — | 4 | 2 | no | — |
| 82 | C3×C3⋊C8 | — | 12 | 1 | no | — |
| 83 | D4×C9 | — | 3 | 1 | no | — |
| 84 | D4×C9 | — | 6 | 1 | no | — |
| 85 | D4×C9 | — | 6 | 1 | no | — |
| 86 | D4×C9 | — | 8 | 1 | no | — |
| 87 | D4×C9 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 88 | C2×C4×C9 | — | 3 | 1 | no | — |
| 89 | C2×C4×C9 | — | 3 | 1 | no | — |
| 90 | D36 | — | 3 | 1 | no | — |
| 91 | D36 | — | 6 | 1 | no | — |
| 92 | D36 | — | 6 | 1 | no | — |
| 93 | D36 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 94 | C9⋊C8 | — | 3 | 1 | no | — |
| 95 | C9⋊C8 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 96 | C22×D8 | — | 9 | 1 | no | Sylow |
| 97 | C2×C3×C9 | — | 4 | 1 | no | — |
| 98 | C2×C3×C9 | — | 16 | 1 | no | — |
| 99 | C3×D9 | — | 16 | 1 | no | — |
| 100 | C24×C3 | — | 2 | 1 | no | — |
| 101 | C23×D3 | — | 6 | 1 | no | — |
| 102 | C22×A4 | — | 2 | 2 | no | — |
| 103 | C22×A4 | — | 2 | 1 | no | — |
| 104 | C22×A4 | — | 18 | 1 | no | — |
| 105 | C2×C3×D4 | — | 3 | 1 | no | — |
| 106 | C2×C3×D4 | — | 3 | 1 | no | — |
| 107 | C2×C3×D4 | — | 6 | 1 | no | — |
| 108 | C22×C3×C4 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 109 | C2×D12 | — | 9 | 1 | no | — |
| 110 | C2×D12 | — | 9 | 1 | no | — |
| 111 | C2×D12 | — | 18 | 1 | no | — |
| 112 | C4×A4 | — | 2 | 1 | no | — |
| 113 | C4×A4 | C2×D9 | 1 | 1 | yes | characteristic |
| 114 | C3×D8 | — | 36 | 1 | no | — |
| 115 | #213 | — | 3 | 1 | no | — |
| 116 | #213 | — | 9 | 1 | no | — |
| 117 | #213 | — | 9 | 1 | no | — |
| 118 | #213 | — | 9 | 2 | no | — |
| 119 | #213 | — | 9 | 1 | no | — |
| 120 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 121 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 122 | C22×C32 | — | 8 | 1 | no | — |
| 123 | C2×C3×D3 | — | 24 | 1 | no | — |
| 124 | C3×A4 | #213 | 1 | 1 | yes | characteristic |
| 125 | C32×C4 | — | 4 | 1 | no | — |
| 126 | C22×C9 | — | 2 | 1 | no | — |
| 127 | C22×C9 | — | 3 | 1 | no | — |
| 128 | C22×C9 | — | 3 | 1 | no | — |
| 129 | C22×C9 | — | 6 | 1 | no | — |
| 130 | C22×C9 | — | 8 | 2 | no | — |
| 131 | C22×C9 | — | 12 | 1 | no | — |
| 132 | C22×C9 | — | 12 | 1 | no | — |
| 133 | C22×C9 | — | 12 | 1 | no | — |
| 134 | C22×C9 | — | 12 | 1 | no | — |
| 135 | C22×C9 | C3×D8 | 1 | 1 | yes | characteristic, complemented |
| 136 | C2×D9 | — | 2 | 1 | no | — |
| 137 | C2×D9 | — | 6 | 1 | no | — |
| 138 | C2×D9 | — | 12 | 1 | no | — |
| 139 | C2×D9 | — | 12 | 1 | no | — |
| 140 | C2×D9 | — | 12 | 1 | no | — |
| 141 | C2×D9 | — | 12 | 1 | no | — |
| 142 | C22⋊C9 | — | 2 | 1 | no | — |
| 143 | C4×C9 | — | 3 | 1 | no | — |
| 144 | C4×C9 | — | 8 | 1 | no | — |
| 145 | C4×C9 | C22×A4 | 1 | 1 | yes | characteristic |
| 146 | C22×D4 | — | 9 | 1 | no | — |
| 147 | C22×D4 | C3×D9 | 1 | 1 | yes | characteristic, complemented |
| 148 | C2×D8 | — | 27 | 1 | no | — |
| 149 | C2×D8 | — | 27 | 1 | no | — |
| 150 | C2×D8 | — | 27 | 1 | no | — |
| 151 | C2×D8 | — | 27 | 1 | no | — |
| 152 | C22×C8 | — | 9 | 1 | no | — |
| 153 | C3×C9 | — | 4 | 1 | no | Sylow |
| 154 | C23×C3 | — | 4 | 1 | no | — |
| 155 | C23×C3 | — | 6 | 1 | no | — |
| 156 | C23×C3 | — | 6 | 1 | no | — |
| 157 | C23×C3 | — | 12 | 1 | no | — |
| 158 | C23×C3 | C3×#67 | 1 | 1 | yes | characteristic |
| 159 | C22×D3 | — | 12 | 1 | no | — |
| 160 | C22×D3 | — | 18 | 1 | no | — |
| 161 | C22×D3 | — | 18 | 1 | no | — |
| 162 | C22×D3 | — | 36 | 1 | no | — |
| 163 | C2×A4 | — | 2 | 1 | no | — |
| 164 | C2×A4 | — | 4 | 2 | no | — |
| 165 | C2×A4 | — | 4 | 1 | no | — |
| 166 | C2×A4 | — | 36 | 1 | no | — |
| 167 | C2×A4 | #611 | 1 | 1 | yes | characteristic |
| 168 | C3×D4 | — | 3 | 1 | no | — |
| 169 | C3×D4 | — | 4 | 1 | no | — |
| 170 | C3×D4 | — | 6 | 1 | no | — |
| 171 | C3×D4 | — | 6 | 1 | no | — |
| 172 | C3×D4 | — | 8 | 1 | no | — |
| 173 | C3×D4 | — | 36 | 1 | no | — |
| 174 | C3×D4 | D3×A4 | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | — | 3 | 1 | no | — |
| 176 | C2×C3×C4 | — | 3 | 1 | no | — |
| 177 | D12 | — | 3 | 1 | no | — |
| 178 | D12 | — | 9 | 1 | no | — |
| 179 | D12 | — | 18 | 1 | no | — |
| 180 | D12 | — | 18 | 1 | no | — |
| 181 | C3×C8 | — | 36 | 1 | no | — |
| 182 | C3⋊C8 | — | 3 | 1 | no | — |
| 183 | C3⋊C8 | — | 9 | 1 | no | — |
| 184 | C2×C32 | — | 4 | 1 | no | — |
| 185 | C2×C32 | — | 16 | 1 | no | — |
| 186 | C3×D3 | — | 48 | 1 | no | — |
| 187 | C2×C9 | — | 3 | 1 | no | — |
| 188 | C2×C9 | — | 3 | 1 | no | — |
| 189 | C2×C9 | — | 4 | 1 | no | — |
| 190 | C2×C9 | — | 8 | 1 | no | — |
| 191 | C2×C9 | — | 12 | 1 | no | — |
| 192 | C2×C9 | — | 16 | 2 | no | — |
| 193 | C2×C9 | D4×A4 | 1 | 1 | yes | characteristic |
| 194 | D9 | — | 4 | 1 | no | — |
| 195 | D9 | — | 12 | 1 | no | — |
| 196 | C24 | — | 2 | 1 | no | — |
| 197 | C24 | — | 18 | 1 | no | — |
| 198 | C2×D4 | — | 3 | 1 | no | — |
| 199 | C2×D4 | — | 3 | 1 | no | — |
| 200 | C2×D4 | — | 6 | 1 | no | — |
| 201 | C2×D4 | — | 27 | 1 | no | — |
| 202 | C2×D4 | — | 27 | 1 | no | — |
| 203 | C2×D4 | — | 54 | 1 | no | — |
| 204 | C22×C4 | C2×C3×D9 | 1 | 1 | yes | characteristic |
| 205 | D8 | — | 9 | 1 | no | — |
| 206 | D8 | — | 27 | 1 | no | — |
| 207 | D8 | — | 27 | 1 | no | — |
| 208 | D8 | — | 27 | 2 | no | — |
| 209 | D8 | — | 27 | 1 | no | — |
| 210 | C2×C8 | — | 27 | 1 | no | — |
| 211 | C2×C8 | — | 27 | 1 | no | — |
| 212 | C22×C3 | — | 2 | 1 | no | — |
| 213 | C22×C3 | — | 3 | 1 | no | — |
| 214 | C22×C3 | — | 3 | 1 | no | — |
| 215 | C22×C3 | — | 6 | 1 | no | — |
| 216 | C22×C3 | — | 8 | 1 | no | — |
| 217 | C22×C3 | — | 8 | 2 | no | — |
| 218 | C22×C3 | — | 12 | 1 | no | — |
| 219 | C22×C3 | — | 12 | 1 | no | — |
| 220 | C22×C3 | — | 12 | 1 | no | — |
| 221 | C22×C3 | — | 12 | 1 | no | — |
| 222 | C22×C3 | — | 72 | 1 | no | — |
| 223 | C22×C3 | C3×#213 | 1 | 1 | yes | characteristic |
| 224 | C2×D3 | — | 6 | 1 | no | — |
| 225 | C2×D3 | — | 18 | 1 | no | — |
| 226 | C2×D3 | — | 36 | 1 | no | — |
| 227 | C2×D3 | — | 36 | 1 | no | — |
| 228 | C2×D3 | — | 36 | 1 | no | — |
| 229 | C2×D3 | — | 36 | 1 | no | — |
| 230 | A4 | — | 2 | 1 | no | — |
| 231 | A4 | #3673 | 1 | 1 | yes | characteristic, direct product factor |
| 232 | C3×C4 | — | 3 | 1 | no | — |
| 233 | C3×C4 | — | 4 | 1 | no | — |
| 234 | C3×C4 | — | 8 | 1 | no | — |
| 235 | C3×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 236 | C32 | — | 4 | 1 | no | — |
| 237 | C9 | — | 8 | 1 | no | — |
| 238 | C9 | A4×D8 | 1 | 1 | yes | characteristic, complemented |
| 239 | C23 | — | 4 | 1 | no | — |
| 240 | C23 | — | 6 | 1 | no | — |
| 241 | C23 | — | 6 | 1 | no | — |
| 242 | C23 | — | 12 | 1 | no | — |
| 243 | C23 | — | 36 | 1 | no | — |
| 244 | C23 | — | 54 | 1 | no | — |
| 245 | C23 | — | 54 | 1 | no | — |
| 246 | C23 | — | 108 | 1 | no | — |
| 247 | C23 | C3×#611 | 1 | 1 | yes | characteristic |
| 248 | D4 | — | 3 | 1 | no | — |
| 249 | D4 | — | 6 | 1 | no | — |
| 250 | D4 | — | 6 | 1 | no | — |
| 251 | D4 | — | 9 | 1 | no | — |
| 252 | D4 | — | 27 | 1 | no | — |
| 253 | D4 | — | 54 | 1 | no | — |
| 254 | D4 | — | 54 | 1 | no | — |
| 255 | D4 | A4×D9 | 1 | 1 | yes | characteristic, complemented |
| 256 | C2×C4 | — | 3 | 1 | no | — |
| 257 | C2×C4 | — | 3 | 1 | no | — |
| 258 | C8 | — | 9 | 1 | no | — |
| 259 | C8 | — | 27 | 1 | no | — |
| 260 | C2×C3 | — | 3 | 1 | no | — |
| 261 | C2×C3 | — | 3 | 1 | no | — |
| 262 | C2×C3 | — | 4 | 1 | no | — |
| 263 | C2×C3 | — | 4 | 1 | no | — |
| 264 | C2×C3 | — | 8 | 1 | no | — |
| 265 | C2×C3 | — | 12 | 1 | no | — |
| 266 | C2×C3 | — | 16 | 1 | no | — |
| 267 | C2×C3 | — | 16 | 2 | no | — |
| 268 | C2×C3 | — | 144 | 1 | no | — |
| 269 | C2×C3 | A4×#67 | 1 | 1 | yes | characteristic |
| 270 | D3 | — | 12 | 1 | no | — |
| 271 | D3 | — | 36 | 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 | — | 18 | 1 | no | — |
| 281 | C22 | — | 54 | 1 | no | — |
| 282 | C22 | — | 108 | 1 | no | — |
| 283 | C22 | — | 108 | 1 | no | — |
| 284 | C22 | — | 108 | 1 | no | — |
| 285 | C22 | — | 108 | 1 | no | — |
| 286 | C22 | C3×#3673 | 1 | 1 | yes | characteristic, complemented |
| 287 | C4 | — | 3 | 1 | no | — |
| 288 | C4 | C2×A4×D9 | 1 | 1 | yes | characteristic |
| 289 | C3 | — | 4 | 1 | no | — |
| 290 | C3 | — | 8 | 1 | no | — |
| 291 | C3 | A4×#213 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 3 | 1 | no | — |
| 293 | C2 | — | 3 | 1 | no | — |
| 294 | C2 | — | 4 | 1 | no | — |
| 295 | C2 | — | 12 | 1 | no | — |
| 296 | C2 | — | 36 | 1 | no | — |
| 297 | C2 | — | 108 | 1 | no | — |
| 298 | C2 | A4×#611 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×C9×A4 | C2 | semidirect |
| 2. A4×D36 | C2 | semidirect |
| 3. A4×C9⋊C8 | C2 | semidirect |
| 5. C22×#3673 | C3 | semidirect |
| 13. C22×D4×C9 | C2×C3 | semidirect |
| 14. C22×D36 | C2×C3 | semidirect |
| 20. C22×C9⋊C8 | C2×C3 | semidirect |
| 48. #3673 | A4 | direct (N × H) |
| 53. C9×A4 | D8 | semidirect |
| 59. D4×A4 | D9 | semidirect |
| 87. D4×C9 | C2×A4 | semidirect |
| 93. D36 | C2×A4 | semidirect |
| 95. C9⋊C8 | C2×A4 | semidirect |
| 135. C22×C9 | C3×D8 | semidirect |
| 147. C22×D4 | C3×D9 | semidirect |
| 231. A4 | #3673 | direct (N × H) |
| 238. C9 | A4×D8 | semidirect |
| 255. D4 | A4×D9 | semidirect |
| 286. C22 | C3×#3673 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C9×A4 | 2 |
| A4×D36 | 2 |
| A4×C9⋊C8 | 2 |
| A4×#213 | 3 |
| C22×#3673 | 3 |
| C3×#3673 | 4 |