Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
349 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 55 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 | C4⋊(C9⋊S4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C9⋊(A4⋊C8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593649 | — | 3 | 1 | no | maximal |
| 5 | #10589517 | — | 3 | 2 | no | maximal |
| 6 | #10588898 | — | 3 | 1 | no | maximal |
| 7 | C22×C9×A4 | — | 2 | 1 | no | — |
| 8 | C2×C9⋊S4 | — | 2 | 1 | no | — |
| 9 | C4×C9×A4 | C22 | 1 | 1 | yes | characteristic |
| 10 | #88544 | — | 4 | 1 | no | maximal |
| 11 | C3×D4×A4 | D3 | 1 | 1 | yes | characteristic |
| 12 | C4⋊(C3⋊S4) | — | 3 | 1 | no | — |
| 13 | C3⋊(A4⋊C8) | — | 3 | 1 | no | — |
| 14 | C22×D4×C9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | D4×C22⋊C9 | D3 | 1 | 2 | yes | complemented |
| 16 | C4⋊((C22⋊C9)⋊C2) | — | 3 | 2 | no | — |
| 17 | C2×#3673 | — | 3 | 1 | no | — |
| 18 | C2×#3673 | — | 3 | 1 | no | — |
| 19 | #63492 | — | 3 | 1 | no | — |
| 20 | (C22⋊C9)⋊C8 | — | 3 | 2 | no | — |
| 21 | #63392 | — | 3 | 1 | no | — |
| 22 | #63392 | — | 3 | 1 | no | — |
| 23 | #63389 | — | 3 | 1 | no | — |
| 24 | C2×C9×A4 | — | 4 | 1 | no | — |
| 25 | C2×C9×A4 | D4 | 1 | 1 | yes | characteristic |
| 26 | C9⋊S4 | — | 4 | 1 | no | — |
| 27 | C3×D4×C9 | — | 4 | 1 | no | — |
| 28 | C4⋊(C3⋊D9) | — | 4 | 1 | no | — |
| 29 | C3⋊(C9⋊C8) | — | 4 | 1 | no | — |
| 30 | #5415 | — | 9 | 3 | no | — |
| 31 | #5217 | — | 9 | 1 | no | — |
| 32 | C22×C3×A4 | — | 2 | 1 | no | — |
| 33 | C2×C3⋊S4 | — | 6 | 1 | no | — |
| 34 | C3×C4×A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 35 | C24×C9 | — | 2 | 1 | no | — |
| 36 | C22×C22⋊C9 | — | 2 | 2 | no | — |
| 37 | C2×(C22⋊C9)⋊C2 | — | 6 | 2 | no | — |
| 38 | #3753 | — | 12 | 1 | no | — |
| 39 | C2×D4×C9 | — | 3 | 1 | no | — |
| 40 | C2×D4×C9 | — | 3 | 1 | no | — |
| 41 | C2×D4×C9 | — | 6 | 1 | no | — |
| 42 | C22×C4×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | C2×#611 | — | 6 | 1 | no | — |
| 44 | C2×D36 | — | 3 | 1 | no | — |
| 45 | C4×C22⋊C9 | C2×D3 | 1 | 2 | yes | — |
| 46 | #3673 | — | 6 | 2 | no | — |
| 47 | #3673 | — | 6 | 1 | no | — |
| 48 | #3673 | — | 6 | 1 | no | — |
| 49 | #3673 | — | 12 | 2 | no | — |
| 50 | (C2×D9)⋊C4 | — | 6 | 1 | no | — |
| 51 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 52 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 53 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 54 | C22×C3×C9 | — | 8 | 1 | no | — |
| 55 | C2×C3⋊D9 | — | 8 | 1 | no | — |
| 56 | C9×A4 | D8 | 1 | 1 | yes | characteristic, complemented |
| 57 | C3×C4×C9 | — | 4 | 1 | no | — |
| 58 | C22×C3×D4 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 59 | D4×A4 | D9 | 1 | 3 | yes | complemented |
| 60 | C4⋊S4 | — | 9 | 3 | no | — |
| 61 | C2×#213 | — | 9 | 1 | no | — |
| 62 | C2×#213 | — | 9 | 1 | no | — |
| 63 | #930 | — | 9 | 1 | no | — |
| 64 | A4⋊C8 | — | 9 | 3 | no | — |
| 65 | #832 | — | 9 | 1 | no | — |
| 66 | #832 | — | 9 | 1 | no | — |
| 67 | #830 | — | 9 | 1 | no | — |
| 68 | C2×C3×A4 | — | 4 | 1 | no | — |
| 69 | C2×C3×A4 | #67 | 1 | 1 | yes | characteristic |
| 70 | C3⋊S4 | — | 12 | 1 | no | — |
| 71 | C32×D4 | — | 4 | 1 | no | — |
| 72 | C4⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 73 | C23×C9 | — | 4 | 1 | no | — |
| 74 | C23×C9 | — | 6 | 1 | no | — |
| 75 | C23×C9 | — | 6 | 1 | no | — |
| 76 | C23×C9 | — | 12 | 1 | no | — |
| 77 | C23×C9 | #67 | 1 | 1 | yes | characteristic |
| 78 | C22×D9 | — | 6 | 1 | no | — |
| 79 | C2×C22⋊C9 | — | 4 | 2 | no | — |
| 80 | C2×C22⋊C9 | #67 | 1 | 2 | yes | — |
| 81 | (C22⋊C9)⋊C2 | — | 12 | 2 | no | — |
| 82 | C3⋊(C3⋊C8) | — | 12 | 1 | no | — |
| 83 | D4×C9 | — | 3 | 2 | no | — |
| 84 | D4×C9 | — | 3 | 1 | no | — |
| 85 | D4×C9 | — | 4 | 2 | no | — |
| 86 | D4×C9 | — | 6 | 1 | no | — |
| 87 | D4×C9 | S4 | 1 | 1 | yes | characteristic, complemented |
| 88 | C2×C4×C9 | — | 3 | 1 | no | — |
| 89 | C2×C4×C9 | — | 3 | 1 | no | — |
| 90 | #611 | — | 12 | 1 | no | — |
| 91 | #611 | — | 12 | 1 | no | — |
| 92 | C2×Dic9 | — | 6 | 1 | no | — |
| 93 | D36 | — | 6 | 1 | no | — |
| 94 | D36 | — | 6 | 1 | no | — |
| 95 | D36 | — | 12 | 2 | no | — |
| 96 | C9⋊C8 | — | 6 | 1 | no | — |
| 97 | C9⋊C8 | — | 6 | 1 | no | — |
| 98 | C9⋊C8 | — | 12 | 2 | no | — |
| 99 | #447 | — | 27 | 1 | no | Sylow |
| 100 | C2×C3×C9 | — | 4 | 1 | no | — |
| 101 | C2×C3×C9 | — | 16 | 1 | no | — |
| 102 | C3⋊D9 | — | 16 | 1 | no | — |
| 103 | C24×C3 | — | 2 | 1 | no | — |
| 104 | C22×A4 | — | 2 | 3 | no | — |
| 105 | C2×S4 | — | 18 | 3 | no | — |
| 106 | C2×C3×D4 | — | 3 | 1 | no | — |
| 107 | C2×C3×D4 | — | 3 | 1 | no | — |
| 108 | C2×C3×D4 | — | 6 | 1 | no | — |
| 109 | C22×C3×C4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 110 | C2×#67 | — | 18 | 1 | no | — |
| 111 | C2×D12 | — | 9 | 1 | no | — |
| 112 | C4×A4 | C2×D9 | 1 | 3 | yes | — |
| 113 | #213 | — | 18 | 2 | no | — |
| 114 | #213 | — | 18 | 1 | no | — |
| 115 | #213 | — | 18 | 1 | no | — |
| 116 | #213 | — | 36 | 3 | no | — |
| 117 | (C2×D3)⋊C4 | — | 18 | 1 | no | — |
| 118 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 119 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 120 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 121 | C22×C32 | — | 8 | 1 | no | — |
| 122 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 123 | C3×A4 | #213 | 1 | 1 | yes | characteristic |
| 124 | C32×C4 | — | 4 | 1 | no | — |
| 125 | C22×C9 | — | 2 | 1 | no | — |
| 126 | C22×C9 | — | 3 | 1 | no | — |
| 127 | C22×C9 | — | 3 | 1 | no | — |
| 128 | C22×C9 | — | 6 | 1 | no | — |
| 129 | C22×C9 | — | 8 | 2 | no | — |
| 130 | C22×C9 | — | 12 | 1 | no | — |
| 131 | C22×C9 | — | 12 | 1 | no | — |
| 132 | C22×C9 | — | 12 | 1 | no | — |
| 133 | C22×C9 | — | 12 | 1 | no | — |
| 134 | C22×C9 | #213 | 1 | 1 | yes | characteristic, complemented |
| 135 | C2×D9 | — | 12 | 1 | no | — |
| 136 | C2×D9 | — | 12 | 1 | no | — |
| 137 | C2×D9 | — | 12 | 1 | no | — |
| 138 | C2×D9 | — | 24 | 2 | no | — |
| 139 | C22⋊C9 | #213 | 1 | 2 | yes | complemented |
| 140 | C4×C9 | — | 3 | 1 | no | — |
| 141 | C4×C9 | — | 4 | 2 | no | — |
| 142 | C4×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 143 | Dic9 | — | 12 | 1 | no | — |
| 144 | C22×D4 | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 145 | C2×D8 | — | 27 | 1 | no | — |
| 146 | C2×D8 | — | 27 | 1 | no | — |
| 147 | #121 | — | 27 | 1 | no | — |
| 148 | #102 | — | 27 | 1 | no | — |
| 149 | #102 | — | 27 | 1 | no | — |
| 150 | C22⋊C8 | — | 27 | 1 | no | — |
| 151 | C3×C9 | — | 4 | 1 | no | Sylow |
| 152 | C23×C3 | — | 4 | 1 | no | — |
| 153 | C23×C3 | — | 6 | 1 | no | — |
| 154 | C23×C3 | — | 6 | 1 | no | — |
| 155 | C23×C3 | — | 12 | 1 | no | — |
| 156 | C23×C3 | #638 | 1 | 1 | yes | characteristic |
| 157 | C22×D3 | — | 18 | 1 | no | — |
| 158 | C2×A4 | — | 4 | 3 | no | — |
| 159 | C2×A4 | #611 | 1 | 3 | yes | — |
| 160 | S4 | — | 36 | 3 | no | — |
| 161 | C3×D4 | — | 3 | 2 | no | — |
| 162 | C3×D4 | — | 3 | 1 | no | — |
| 163 | C3×D4 | — | 4 | 3 | no | — |
| 164 | C3×D4 | — | 6 | 1 | no | — |
| 165 | C3×D4 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 166 | C2×C3×C4 | — | 3 | 1 | no | — |
| 167 | C2×C3×C4 | — | 3 | 1 | no | — |
| 168 | #67 | — | 36 | 1 | no | — |
| 169 | #67 | — | 36 | 1 | no | — |
| 170 | C2×Dic3 | — | 18 | 1 | no | — |
| 171 | D12 | — | 18 | 1 | no | — |
| 172 | D12 | — | 18 | 1 | no | — |
| 173 | D12 | — | 36 | 3 | no | — |
| 174 | C3⋊C8 | — | 18 | 1 | no | — |
| 175 | C3⋊C8 | — | 18 | 1 | no | — |
| 176 | C3⋊C8 | — | 36 | 3 | no | — |
| 177 | C2×C32 | — | 4 | 1 | no | — |
| 178 | C2×C32 | — | 16 | 1 | no | — |
| 179 | C3⋊D3 | — | 48 | 1 | no | — |
| 180 | C2×C9 | — | 3 | 1 | no | — |
| 181 | C2×C9 | — | 3 | 1 | no | — |
| 182 | C2×C9 | — | 4 | 1 | no | — |
| 183 | C2×C9 | — | 4 | 2 | no | — |
| 184 | C2×C9 | — | 12 | 1 | no | — |
| 185 | C2×C9 | — | 16 | 2 | no | — |
| 186 | C2×C9 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 187 | D9 | — | 24 | 1 | no | — |
| 188 | D9 | — | 48 | 2 | no | — |
| 189 | C24 | — | 2 | 1 | no | — |
| 190 | C2×D4 | — | 3 | 1 | no | — |
| 191 | C2×D4 | — | 3 | 1 | no | — |
| 192 | C2×D4 | — | 6 | 1 | no | — |
| 193 | C2×D4 | — | 27 | 1 | no | — |
| 194 | C2×D4 | — | 54 | 1 | no | — |
| 195 | C22×C4 | C2×C3⋊D9 | 1 | 1 | yes | characteristic |
| 196 | D8 | — | 54 | 2 | no | — |
| 197 | D8 | — | 54 | 1 | no | — |
| 198 | D8 | — | 54 | 1 | no | — |
| 199 | C2×C8 | — | 27 | 1 | no | — |
| 200 | C2×C8 | — | 27 | 1 | no | — |
| 201 | C4⋊C4 | — | 27 | 1 | no | — |
| 202 | C22⋊C4 | — | 54 | 1 | no | — |
| 203 | C22×C3 | — | 2 | 1 | no | — |
| 204 | C22×C3 | — | 3 | 1 | no | — |
| 205 | C22×C3 | — | 3 | 1 | no | — |
| 206 | C22×C3 | — | 6 | 1 | no | — |
| 207 | C22×C3 | — | 8 | 3 | no | — |
| 208 | C22×C3 | — | 12 | 1 | no | — |
| 209 | C22×C3 | — | 12 | 1 | no | — |
| 210 | C22×C3 | — | 12 | 1 | no | — |
| 211 | C22×C3 | — | 12 | 1 | no | — |
| 212 | C22×C3 | #3753 | 1 | 1 | yes | characteristic |
| 213 | C2×D3 | — | 36 | 1 | no | — |
| 214 | C2×D3 | — | 36 | 1 | no | — |
| 215 | C2×D3 | — | 36 | 1 | no | — |
| 216 | C2×D3 | — | 72 | 3 | no | — |
| 217 | A4 | #3673 | 1 | 3 | yes | complemented |
| 218 | C3×C4 | — | 3 | 1 | no | — |
| 219 | C3×C4 | — | 4 | 3 | no | — |
| 220 | C3×C4 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 221 | Dic3 | — | 36 | 1 | no | — |
| 222 | C32 | — | 4 | 1 | no | — |
| 223 | C9 | — | 4 | 2 | no | — |
| 224 | C9 | #5415 | 1 | 1 | yes | characteristic, complemented |
| 225 | C23 | — | 4 | 1 | no | — |
| 226 | C23 | — | 6 | 1 | no | — |
| 227 | C23 | — | 6 | 1 | no | — |
| 228 | C23 | — | 12 | 1 | no | — |
| 229 | C23 | — | 54 | 1 | no | — |
| 230 | C23 | #6233 | 1 | 1 | yes | characteristic |
| 231 | D4 | — | 3 | 2 | no | — |
| 232 | D4 | — | 3 | 1 | no | — |
| 233 | D4 | — | 6 | 1 | no | — |
| 234 | D4 | — | 54 | 1 | no | — |
| 235 | D4 | — | 54 | 1 | no | — |
| 236 | D4 | — | 108 | 1 | no | — |
| 237 | D4 | — | 108 | 1 | no | — |
| 238 | D4 | C9⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 239 | C2×C4 | — | 3 | 1 | no | — |
| 240 | C2×C4 | — | 3 | 1 | no | — |
| 241 | C2×C4 | — | 54 | 1 | no | — |
| 242 | C8 | — | 54 | 1 | no | — |
| 243 | C8 | — | 54 | 1 | no | — |
| 244 | C2×C3 | — | 3 | 1 | no | — |
| 245 | C2×C3 | — | 3 | 1 | no | — |
| 246 | C2×C3 | — | 4 | 1 | no | — |
| 247 | C2×C3 | — | 4 | 3 | no | — |
| 248 | C2×C3 | — | 12 | 1 | no | — |
| 249 | C2×C3 | — | 16 | 3 | no | — |
| 250 | C2×C3 | #64269 | 1 | 1 | yes | characteristic |
| 251 | D3 | — | 72 | 1 | no | — |
| 252 | D3 | — | 144 | 3 | no | — |
| 253 | C22 | — | 2 | 1 | no | — |
| 254 | C22 | — | 3 | 1 | no | — |
| 255 | C22 | — | 3 | 1 | no | — |
| 256 | C22 | — | 6 | 1 | no | — |
| 257 | C22 | — | 12 | 1 | no | — |
| 258 | C22 | — | 12 | 1 | no | — |
| 259 | C22 | — | 12 | 1 | no | — |
| 260 | C22 | — | 12 | 1 | no | — |
| 261 | C22 | — | 108 | 1 | no | — |
| 262 | C22 | — | 108 | 1 | no | — |
| 263 | C22 | — | 108 | 1 | no | — |
| 264 | C22 | #88544 | 1 | 1 | yes | characteristic, complemented |
| 265 | C4 | — | 3 | 1 | no | — |
| 266 | C4 | — | 108 | 1 | no | — |
| 267 | C4 | C2×C9⋊S4 | 1 | 1 | yes | characteristic |
| 268 | C3 | — | 4 | 3 | no | — |
| 269 | C3 | #10593649 | 1 | 1 | yes | characteristic |
| 270 | C2 | — | 3 | 1 | no | — |
| 271 | C2 | — | 3 | 1 | no | — |
| 272 | C2 | — | 4 | 1 | no | — |
| 273 | C2 | — | 12 | 1 | no | — |
| 274 | C2 | — | 216 | 1 | no | — |
| 275 | C2 | #11722376 | 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. C4⋊(C9⋊S4) | C2 | semidirect |
| 3. C9⋊(A4⋊C8) | C2 | semidirect |
| 14. C22×D4×C9 | D3 | semidirect |
| 15. D4×C22⋊C9 | D3 | semidirect |
| 56. C9×A4 | D8 | semidirect |
| 59. D4×A4 | D9 | semidirect |
| 87. D4×C9 | S4 | semidirect |
| 134. C22×C9 | #213 | semidirect |
| 139. C22⋊C9 | #213 | semidirect |
| 144. C22×D4 | C3⋊D9 | semidirect |
| 217. A4 | #3673 | semidirect |
| 224. C9 | #5415 | semidirect |
| 238. D4 | C9⋊S4 | semidirect |
| 264. C22 | #88544 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C9×A4 | 2 |
| C4⋊(C9⋊S4) | 2 |
| C9⋊(A4⋊C8) | 2 |
| #10593649 | 3 |
| #10589517 | 3 |
| #10588898 | 3 |
| #88544 | 4 |