Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
434 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 29 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 | D9⋊(C22⋊A4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11721330 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720760 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D3×#120⋊C3 | — | 3 | 1 | no | maximal |
| 5 | D9×#120 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C9⋊(C22⋊A4) | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×D9⋊A4 | — | 4 | 1 | no | maximal |
| 8 | D3×C22⋊A4 | — | 3 | 1 | no | — |
| 9 | C24×D9 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C3×#120⋊C3 | D3 | 1 | 1 | yes | characteristic |
| 11 | C3⋊(#120⋊C3) | — | 3 | 1 | no | — |
| 12 | C2×D4×D9 | — | 3 | 1 | no | — |
| 13 | C9×#120 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #63515 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 15 | #63499 | — | 3 | 1 | no | — |
| 16 | #63444 | — | 3 | 1 | no | — |
| 17 | C22⋊C4×D9 | — | 3 | 1 | no | — |
| 18 | #120⋊C9 | — | 2 | 1 | no | — |
| 19 | C2×C9⋊A4 | — | 4 | 1 | no | — |
| 20 | D9⋊A4 | — | 4 | 1 | no | — |
| 21 | D9⋊A4 | — | 4 | 1 | no | — |
| 22 | D9⋊A4 | — | 8 | 2 | no | — |
| 23 | D3×#120 | — | 3 | 1 | no | — |
| 24 | C2×#120⋊C3 | — | 9 | 1 | no | — |
| 25 | C3×C22⋊A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 26 | C2×D3×A4 | — | 12 | 1 | no | — |
| 27 | C24×C9 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 28 | C23×D9 | — | 3 | 1 | no | — |
| 29 | C23×D9 | — | 3 | 1 | no | — |
| 30 | C23×D9 | — | 6 | 2 | no | — |
| 31 | C23×D9 | — | 6 | 2 | no | — |
| 32 | C23×D9 | A4 | 1 | 1 | yes | characteristic, complemented |
| 33 | C22⋊(C22⋊C9) | — | 2 | 1 | no | — |
| 34 | C2×D4×C9 | — | 3 | 1 | no | — |
| 35 | C2×#611 | — | 3 | 1 | no | — |
| 36 | C2×#611 | — | 3 | 1 | no | — |
| 37 | D4×D9 | — | 6 | 2 | no | — |
| 38 | D4×D9 | — | 6 | 2 | no | — |
| 39 | C2×D36 | — | 3 | 1 | no | — |
| 40 | C2×C4×D9 | — | 3 | 1 | no | — |
| 41 | C9×C22⋊C4 | — | 3 | 1 | no | — |
| 42 | #3676 | — | 3 | 1 | no | — |
| 43 | (C2×D9)⋊C4 | — | 3 | 2 | no | — |
| 44 | C2×AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 45 | C9⋊A4 | — | 4 | 1 | no | — |
| 46 | C9⋊A4 | — | 8 | 2 | no | — |
| 47 | C24×D3 | — | 3 | 1 | no | — |
| 48 | C2×C22⋊A4 | — | 9 | 1 | no | — |
| 49 | C2×D3×D4 | — | 9 | 1 | no | — |
| 50 | C3×#120 | C3×D3 | 1 | 1 | yes | characteristic |
| 51 | #953 | — | 3 | 1 | no | — |
| 52 | #937 | — | 9 | 1 | no | — |
| 53 | #882 | — | 9 | 1 | no | — |
| 54 | D3×C22⋊C4 | — | 9 | 1 | no | — |
| 55 | #120⋊C3 | — | 3 | 1 | no | — |
| 56 | #120⋊C3 | — | 9 | 1 | no | — |
| 57 | C2×C3×A4 | — | 4 | 1 | no | — |
| 58 | D3×A4 | — | 12 | 1 | no | — |
| 59 | D3×A4 | — | 12 | 1 | no | — |
| 60 | D3×A4 | — | 24 | 2 | no | — |
| 61 | C23×C9 | — | 3 | 1 | no | — |
| 62 | C23×C9 | — | 6 | 2 | no | — |
| 63 | C23×C9 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 64 | C22×D9 | — | 2 | 2 | no | — |
| 65 | C22×D9 | — | 3 | 2 | no | — |
| 66 | C22×D9 | — | 3 | 2 | no | — |
| 67 | C22×D9 | — | 3 | 1 | no | — |
| 68 | C22×D9 | — | 6 | 1 | no | — |
| 69 | C22×D9 | — | 6 | 2 | no | — |
| 70 | C22×D9 | — | 6 | 2 | no | — |
| 71 | C22×D9 | — | 6 | 4 | no | — |
| 72 | C22×D9 | — | 6 | 1 | no | — |
| 73 | C22×D9 | — | 6 | 2 | no | — |
| 74 | C22×D9 | — | 6 | 2 | no | — |
| 75 | C22×D9 | — | 6 | 4 | no | — |
| 76 | C22×D9 | — | 6 | 2 | no | — |
| 77 | C22×D9 | — | 6 | 2 | no | — |
| 78 | C22×D9 | C2×A4 | 1 | 1 | yes | characteristic |
| 79 | C22×D9 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 80 | C2×C22⋊C9 | — | 8 | 1 | no | — |
| 81 | D4×C9 | — | 6 | 2 | no | — |
| 82 | C2×C4×C9 | — | 3 | 1 | no | — |
| 83 | #611 | — | 6 | 2 | no | — |
| 84 | #611 | — | 6 | 2 | no | — |
| 85 | C2×Dic9 | — | 3 | 1 | no | — |
| 86 | D36 | — | 6 | 2 | no | — |
| 87 | C4×D9 | — | 6 | 1 | no | — |
| 88 | C4×D9 | — | 6 | 1 | no | — |
| 89 | C2×#120 | — | 9 | 1 | no | Sylow |
| 90 | C2×3−1+2 | — | 16 | 1 | no | — |
| 91 | AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 92 | AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 93 | C24×C3 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 94 | C23×D3 | — | 3 | 1 | no | — |
| 95 | C23×D3 | — | 9 | 1 | no | — |
| 96 | C23×D3 | — | 9 | 1 | no | — |
| 97 | C23×D3 | — | 18 | 2 | no | — |
| 98 | C23×D3 | — | 18 | 2 | no | — |
| 99 | C22⋊A4 | — | 3 | 1 | no | — |
| 100 | C22×A4 | — | 36 | 1 | no | — |
| 101 | C2×C3×D4 | — | 3 | 1 | no | — |
| 102 | C2×#67 | — | 9 | 1 | no | — |
| 103 | C2×#67 | — | 9 | 1 | no | — |
| 104 | D3×D4 | — | 18 | 2 | no | — |
| 105 | D3×D4 | — | 18 | 2 | no | — |
| 106 | C2×D12 | — | 9 | 1 | no | — |
| 107 | C2×C4×D3 | — | 9 | 1 | no | — |
| 108 | C3×C22⋊C4 | — | 3 | 1 | no | — |
| 109 | #217 | — | 9 | 1 | no | — |
| 110 | (C2×D3)⋊C4 | — | 9 | 2 | no | — |
| 111 | C2×C3×D3 | — | 48 | 1 | no | — |
| 112 | C3×A4 | — | 4 | 1 | no | — |
| 113 | C3×A4 | — | 8 | 2 | no | — |
| 114 | C22×C9 | — | 2 | 2 | no | — |
| 115 | C22×C9 | — | 3 | 2 | no | — |
| 116 | C22×C9 | — | 6 | 1 | no | — |
| 117 | C22×C9 | — | 6 | 2 | no | — |
| 118 | C22×C9 | — | 6 | 2 | no | — |
| 119 | C22×C9 | C22×A4 | 1 | 1 | yes | characteristic |
| 120 | C2×D9 | — | 3 | 1 | no | — |
| 121 | C2×D9 | — | 3 | 1 | no | — |
| 122 | C2×D9 | — | 3 | 2 | no | — |
| 123 | C2×D9 | — | 4 | 1 | no | — |
| 124 | C2×D9 | — | 6 | 2 | no | — |
| 125 | C2×D9 | — | 6 | 4 | no | — |
| 126 | C2×D9 | — | 6 | 1 | no | — |
| 127 | C2×D9 | — | 6 | 2 | no | — |
| 128 | C2×D9 | — | 6 | 4 | no | — |
| 129 | C2×D9 | — | 6 | 2 | no | — |
| 130 | C2×D9 | — | 6 | 2 | no | — |
| 131 | C2×D9 | — | 6 | 2 | no | — |
| 132 | C2×D9 | — | 12 | 1 | no | — |
| 133 | C22⋊C9 | — | 8 | 1 | no | — |
| 134 | C22⋊C9 | — | 16 | 2 | no | — |
| 135 | C4×C9 | — | 6 | 1 | no | — |
| 136 | Dic9 | — | 6 | 1 | no | — |
| 137 | C25 | — | 9 | 1 | no | — |
| 138 | C22×D4 | — | 27 | 1 | no | — |
| 139 | #120 | — | 9 | 1 | no | — |
| 140 | #120 | — | 27 | 1 | no | — |
| 141 | #120 | — | 27 | 1 | no | — |
| 142 | #120 | AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 143 | C2×C22⋊C4 | — | 27 | 1 | no | — |
| 144 | 3−1+2 | — | 16 | 1 | no | Sylow |
| 145 | C23×C3 | — | 3 | 1 | no | — |
| 146 | C23×C3 | — | 6 | 2 | no | — |
| 147 | C23×C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 148 | C22×D3 | — | 3 | 1 | no | — |
| 149 | C22×D3 | — | 3 | 1 | no | — |
| 150 | C22×D3 | — | 6 | 2 | no | — |
| 151 | C22×D3 | — | 9 | 2 | no | — |
| 152 | C22×D3 | — | 9 | 2 | no | — |
| 153 | C22×D3 | — | 9 | 1 | no | — |
| 154 | C22×D3 | — | 18 | 1 | no | — |
| 155 | C22×D3 | — | 18 | 2 | no | — |
| 156 | C22×D3 | — | 18 | 2 | no | — |
| 157 | C22×D3 | — | 18 | 4 | no | — |
| 158 | C22×D3 | — | 18 | 1 | no | — |
| 159 | C22×D3 | — | 18 | 2 | no | — |
| 160 | C22×D3 | — | 18 | 2 | no | — |
| 161 | C22×D3 | — | 18 | 4 | no | — |
| 162 | C22×D3 | — | 18 | 2 | no | — |
| 163 | C22×D3 | — | 18 | 2 | no | — |
| 164 | C2×A4 | — | 12 | 1 | no | — |
| 165 | C2×A4 | — | 36 | 1 | no | — |
| 166 | C2×A4 | — | 36 | 1 | no | — |
| 167 | C2×A4 | — | 72 | 2 | no | — |
| 168 | C3×D4 | — | 6 | 2 | no | — |
| 169 | C2×C3×C4 | — | 3 | 1 | no | — |
| 170 | #67 | — | 18 | 2 | no | — |
| 171 | #67 | — | 18 | 2 | no | — |
| 172 | C2×Dic3 | — | 9 | 1 | no | — |
| 173 | D12 | — | 18 | 2 | no | — |
| 174 | C4×D3 | — | 18 | 1 | no | — |
| 175 | C4×D3 | — | 18 | 1 | no | — |
| 176 | C2×C32 | — | 16 | 1 | no | — |
| 177 | C3×D3 | — | 48 | 1 | no | — |
| 178 | C3×D3 | — | 48 | 1 | no | — |
| 179 | C2×C9 | — | 3 | 1 | no | — |
| 180 | C2×C9 | — | 4 | 1 | no | — |
| 181 | C2×C9 | — | 6 | 2 | no | — |
| 182 | C2×C9 | — | 32 | 1 | no | — |
| 183 | D9 | — | 3 | 1 | no | — |
| 184 | D9 | — | 4 | 1 | no | — |
| 185 | D9 | — | 6 | 2 | no | — |
| 186 | D9 | #120⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 187 | C24 | — | 9 | 1 | no | — |
| 188 | C24 | — | 27 | 1 | no | — |
| 189 | C24 | — | 27 | 1 | no | — |
| 190 | C24 | — | 54 | 2 | no | — |
| 191 | C24 | — | 54 | 2 | no | — |
| 192 | C24 | C2×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 193 | C2×D4 | — | 3 | 1 | no | — |
| 194 | C2×D4 | — | 27 | 1 | no | — |
| 195 | C2×D4 | — | 27 | 1 | no | — |
| 196 | C2×D4 | — | 27 | 1 | no | — |
| 197 | C2×D4 | — | 54 | 2 | no | — |
| 198 | C2×D4 | — | 54 | 2 | no | — |
| 199 | C22×C4 | — | 27 | 1 | no | — |
| 200 | C22⋊C4 | — | 3 | 1 | no | — |
| 201 | C22⋊C4 | — | 27 | 2 | no | — |
| 202 | C22⋊C4 | — | 27 | 1 | no | — |
| 203 | C22×C3 | — | 2 | 2 | no | — |
| 204 | C22×C3 | — | 3 | 2 | no | — |
| 205 | C22×C3 | — | 6 | 1 | no | — |
| 206 | C22×C3 | — | 6 | 2 | no | — |
| 207 | C22×C3 | — | 6 | 2 | no | — |
| 208 | C22×C3 | — | 144 | 1 | no | — |
| 209 | C22×C3 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 210 | C2×D3 | — | 9 | 1 | no | — |
| 211 | C2×D3 | — | 9 | 1 | no | — |
| 212 | C2×D3 | — | 9 | 2 | no | — |
| 213 | C2×D3 | — | 12 | 1 | no | — |
| 214 | C2×D3 | — | 18 | 2 | no | — |
| 215 | C2×D3 | — | 18 | 2 | no | — |
| 216 | C2×D3 | — | 18 | 4 | no | — |
| 217 | C2×D3 | — | 18 | 1 | no | — |
| 218 | C2×D3 | — | 18 | 2 | no | — |
| 219 | C2×D3 | — | 18 | 4 | no | — |
| 220 | C2×D3 | — | 18 | 2 | no | — |
| 221 | C2×D3 | — | 18 | 2 | no | — |
| 222 | C2×D3 | — | 36 | 1 | no | — |
| 223 | A4 | — | 12 | 1 | no | — |
| 224 | A4 | — | 24 | 2 | no | — |
| 225 | C3×C4 | — | 6 | 1 | no | — |
| 226 | Dic3 | — | 18 | 1 | no | — |
| 227 | C32 | — | 16 | 1 | no | — |
| 228 | C9 | — | 32 | 1 | no | — |
| 229 | C9 | C2×#120⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 230 | C23 | — | 3 | 1 | no | — |
| 231 | C23 | — | 6 | 2 | no | — |
| 232 | C23 | — | 9 | 1 | no | — |
| 233 | C23 | — | 9 | 1 | no | — |
| 234 | C23 | — | 18 | 2 | no | — |
| 235 | C23 | — | 27 | 2 | no | — |
| 236 | C23 | — | 27 | 2 | no | — |
| 237 | C23 | — | 27 | 1 | no | — |
| 238 | C23 | — | 54 | 2 | no | — |
| 239 | C23 | — | 54 | 2 | no | — |
| 240 | C23 | — | 54 | 1 | no | — |
| 241 | C23 | — | 54 | 2 | no | — |
| 242 | C23 | — | 54 | 2 | no | — |
| 243 | C23 | — | 54 | 4 | no | — |
| 244 | C23 | — | 54 | 1 | no | — |
| 245 | C23 | — | 54 | 2 | no | — |
| 246 | C23 | — | 54 | 2 | no | — |
| 247 | C23 | — | 54 | 4 | no | — |
| 248 | C23 | D9⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 249 | D4 | — | 6 | 2 | no | — |
| 250 | D4 | — | 54 | 2 | no | — |
| 251 | D4 | — | 54 | 2 | no | — |
| 252 | D4 | — | 54 | 2 | no | — |
| 253 | C2×C4 | — | 3 | 1 | no | — |
| 254 | C2×C4 | — | 27 | 1 | no | — |
| 255 | C2×C4 | — | 54 | 1 | no | — |
| 256 | C2×C4 | — | 54 | 1 | no | — |
| 257 | C2×C3 | — | 3 | 1 | no | — |
| 258 | C2×C3 | — | 4 | 1 | no | — |
| 259 | C2×C3 | — | 6 | 2 | no | — |
| 260 | C2×C3 | — | 48 | 1 | no | — |
| 261 | C2×C3 | — | 144 | 1 | no | — |
| 262 | C2×C3 | — | 144 | 1 | no | — |
| 263 | D3 | — | 3 | 1 | no | — |
| 264 | D3 | — | 9 | 1 | no | — |
| 265 | D3 | — | 12 | 1 | no | — |
| 266 | D3 | — | 18 | 2 | no | — |
| 267 | C22 | — | 2 | 2 | no | — |
| 268 | C22 | — | 3 | 2 | no | — |
| 269 | C22 | — | 6 | 1 | no | — |
| 270 | C22 | — | 6 | 2 | no | — |
| 271 | C22 | — | 6 | 2 | no | — |
| 272 | C22 | — | 27 | 1 | no | — |
| 273 | C22 | — | 27 | 1 | no | — |
| 274 | C22 | — | 27 | 2 | no | — |
| 275 | C22 | — | 36 | 1 | no | — |
| 276 | C22 | — | 54 | 1 | no | — |
| 277 | C22 | — | 54 | 2 | no | — |
| 278 | C22 | — | 54 | 2 | no | — |
| 279 | C22 | — | 54 | 2 | no | — |
| 280 | C22 | — | 54 | 4 | no | — |
| 281 | C22 | — | 54 | 2 | no | — |
| 282 | C22 | — | 54 | 2 | no | — |
| 283 | C22 | — | 54 | 4 | no | — |
| 284 | C22 | — | 108 | 1 | no | — |
| 285 | C22 | C2×D9⋊A4 | 1 | 1 | yes | characteristic |
| 286 | C4 | — | 6 | 1 | no | — |
| 287 | C4 | — | 54 | 1 | no | — |
| 288 | C3 | — | 48 | 1 | no | — |
| 289 | C3 | D3×#120⋊C3 | 1 | 1 | yes | characteristic |
| 290 | C2 | — | 3 | 1 | no | — |
| 291 | C2 | — | 4 | 1 | no | — |
| 292 | C2 | — | 6 | 2 | no | — |
| 293 | C2 | — | 9 | 1 | no | — |
| 294 | C2 | — | 27 | 1 | no | — |
| 295 | C2 | — | 36 | 1 | no | — |
| 296 | C2 | — | 54 | 2 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C9×#120 | C2×C3 |
| LCS | G ▷ C24×C9 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D9⋊(C22⋊A4) | C2 | semidirect |
| 2. #11721330 | C2 | semidirect |
| 3. #11720760 | C2 | semidirect |
| 5. D9×#120 | C3 | semidirect |
| 6. C9⋊(C22⋊A4) | C22 | semidirect |
| 9. C24×D9 | C2×C3 | semidirect |
| 13. C9×#120 | C2×C3 | semidirect |
| 14. #63515 | C2×C3 | semidirect |
| 27. C24×C9 | C22×C3 | semidirect |
| 32. C23×D9 | A4 | semidirect |
| 63. C23×C9 | C2×A4 | semidirect |
| 79. C22×D9 | C2×A4 | semidirect |
| 142. #120 | AGL(1,ℤ/9) | semidirect |
| 186. D9 | #120⋊C3 | semidirect |
| 192. C24 | C2×AGL(1,ℤ/9) | semidirect |
| 229. C9 | C2×#120⋊C3 | semidirect |
| 248. C23 | D9⋊A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D9⋊(C22⋊A4) | 2 |
| #11721330 | 2 |
| #11720760 | 2 |
| D3×#120⋊C3 | 3 |
| D9×#120 | 3 |
| C2×D9⋊A4 | 4 |