Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
439 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 53 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×C22⋊C27 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11720654 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×(C22⋊C27)⋊C4 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C4×(C22⋊C27)⋊C2 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | (C22⋊C27)⋊Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10589908 | — | 3 | 1 | no | maximal |
| 7 | C22×C22⋊C27 | C22 | 1 | 2 | yes | — |
| 8 | C2×(C22⋊C27)⋊C2 | C22 | 1 | 1 | yes | characteristic |
| 9 | #88401 | — | 4 | 1 | no | maximal |
| 10 | C4×C22⋊C27 | C22 | 1 | 1 | yes | characteristic |
| 11 | (C22⋊C27)⋊C4 | C22 | 1 | 2 | yes | — |
| 12 | (C22⋊C27)⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×D4×C9 | D3 | 1 | 1 | yes | characteristic |
| 14 | C2×#3699 | — | 3 | 1 | no | — |
| 15 | #63515 | — | 3 | 2 | no | — |
| 16 | C2×#3676 | — | 3 | 2 | no | — |
| 17 | #63501 | — | 3 | 2 | no | — |
| 18 | #63500 | — | 3 | 1 | no | — |
| 19 | #63498 | — | 3 | 1 | no | — |
| 20 | #63497 | — | 3 | 2 | no | — |
| 21 | D4×Dic9 | — | 3 | 1 | no | — |
| 22 | C4×#611 | — | 3 | 1 | no | — |
| 23 | #63486 | — | 3 | 1 | no | — |
| 24 | C2×C22⋊C27 | — | 2 | 2 | no | — |
| 25 | C2×C22⋊C27 | C23 | 1 | 1 | yes | characteristic |
| 26 | (C22⋊C27)⋊C2 | — | 2 | 1 | no | — |
| 27 | D4×C27 | — | 4 | 1 | no | — |
| 28 | #6171 | — | 4 | 2 | no | — |
| 29 | C2×Dic27 | — | 4 | 2 | no | — |
| 30 | C4×D27 | — | 4 | 1 | no | — |
| 31 | Dic54 | — | 4 | 1 | no | — |
| 32 | #5806 | — | 9 | 1 | no | — |
| 33 | C24×C9 | C2×D3 | 1 | 2 | yes | — |
| 34 | C2×D4×C9 | — | 3 | 1 | no | — |
| 35 | C2×D4×C9 | — | 3 | 2 | no | — |
| 36 | C2×D4×C9 | — | 3 | 1 | no | — |
| 37 | C22×C4×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 38 | C2×#611 | — | 3 | 2 | no | — |
| 39 | C2×#611 | — | 3 | 1 | no | — |
| 40 | C2×#611 | — | 3 | 2 | no | — |
| 41 | C22×Dic9 | — | 3 | 2 | no | — |
| 42 | C22×Dic9 | — | 3 | 2 | no | — |
| 43 | #3699 | — | 6 | 2 | no | — |
| 44 | #3699 | — | 6 | 1 | no | — |
| 45 | #3699 | — | 6 | 1 | no | — |
| 46 | C2×C4×D9 | — | 3 | 1 | no | — |
| 47 | C2×Dic18 | — | 3 | 1 | no | — |
| 48 | #3676 | — | 3 | 2 | no | — |
| 49 | #3676 | — | 3 | 2 | no | — |
| 50 | #3676 | — | 3 | 2 | no | — |
| 51 | #3676 | — | 3 | 2 | no | — |
| 52 | #3676 | — | 3 | 1 | no | — |
| 53 | #3676 | — | 3 | 2 | no | — |
| 54 | (C2×D9)⋊C4 | — | 3 | 1 | no | — |
| 55 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 56 | Dic9⋊C4 | — | 3 | 1 | no | — |
| 57 | Dic9⋊C4 | — | 3 | 2 | no | — |
| 58 | C4×Dic9 | — | 3 | 1 | no | — |
| 59 | C22×C27 | — | 4 | 2 | no | — |
| 60 | C2×D27 | — | 4 | 1 | no | — |
| 61 | C22⋊C27 | #41 | 1 | 1 | yes | characteristic, complemented |
| 62 | C4×C27 | — | 4 | 1 | no | — |
| 63 | Dic27 | — | 4 | 2 | no | — |
| 64 | Dic27 | — | 4 | 1 | no | — |
| 65 | C22×C3×D4 | D9 | 1 | 1 | yes | characteristic |
| 66 | C2×#237 | — | 9 | 1 | no | — |
| 67 | #953 | — | 9 | 2 | no | — |
| 68 | C2×#217 | — | 9 | 2 | no | — |
| 69 | #939 | — | 9 | 2 | no | — |
| 70 | #938 | — | 9 | 1 | no | — |
| 71 | #936 | — | 9 | 1 | no | — |
| 72 | #935 | — | 9 | 2 | no | — |
| 73 | D4×Dic3 | — | 9 | 1 | no | — |
| 74 | C4×#67 | — | 9 | 1 | no | — |
| 75 | #924 | — | 9 | 1 | no | — |
| 76 | C23×C9 | — | 2 | 2 | no | — |
| 77 | C23×C9 | — | 3 | 2 | no | — |
| 78 | C23×C9 | — | 3 | 2 | no | — |
| 79 | C23×C9 | — | 6 | 2 | no | — |
| 80 | C23×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 81 | C22×D9 | — | 3 | 1 | no | — |
| 82 | D4×C9 | — | 3 | 2 | no | — |
| 83 | D4×C9 | — | 3 | 1 | no | — |
| 84 | D4×C9 | — | 6 | 1 | no | — |
| 85 | D4×C9 | S4 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C9 | — | 3 | 1 | no | — |
| 87 | C2×C4×C9 | — | 3 | 1 | no | — |
| 88 | #611 | — | 6 | 2 | no | — |
| 89 | #611 | — | 6 | 1 | no | — |
| 90 | #611 | — | 6 | 1 | no | — |
| 91 | #611 | — | 6 | 2 | no | — |
| 92 | #611 | — | 12 | 2 | no | — |
| 93 | C2×Dic9 | — | 3 | 1 | no | — |
| 94 | C2×Dic9 | — | 3 | 1 | no | — |
| 95 | C2×Dic9 | — | 3 | 2 | no | — |
| 96 | C2×Dic9 | — | 3 | 2 | no | — |
| 97 | C2×Dic9 | — | 3 | 1 | no | — |
| 98 | C2×Dic9 | — | 6 | 2 | no | — |
| 99 | C2×Dic9 | — | 6 | 2 | no | — |
| 100 | C2×Dic9 | — | 12 | 2 | no | — |
| 101 | C4×D9 | — | 6 | 1 | no | — |
| 102 | C4×D9 | — | 6 | 1 | no | — |
| 103 | Dic18 | — | 6 | 1 | no | — |
| 104 | Dic18 | — | 6 | 1 | no | — |
| 105 | #546 | — | 27 | 1 | no | Sylow |
| 106 | C2×C27 | — | 4 | 1 | no | — |
| 107 | C2×C27 | — | 8 | 2 | no | — |
| 108 | D27 | — | 8 | 1 | no | — |
| 109 | C24×C3 | C2×D9 | 1 | 2 | yes | — |
| 110 | C2×C3×D4 | — | 3 | 1 | no | — |
| 111 | C2×C3×D4 | — | 3 | 2 | no | — |
| 112 | C2×C3×D4 | — | 3 | 1 | no | — |
| 113 | C22×C3×C4 | C2×D9 | 1 | 1 | yes | characteristic |
| 114 | C2×#67 | — | 9 | 2 | no | — |
| 115 | C2×#67 | — | 9 | 1 | no | — |
| 116 | C2×#67 | — | 9 | 2 | no | — |
| 117 | C22×Dic3 | — | 9 | 2 | no | — |
| 118 | C22×Dic3 | — | 9 | 2 | no | — |
| 119 | #237 | — | 18 | 2 | no | — |
| 120 | #237 | — | 18 | 1 | no | — |
| 121 | #237 | — | 18 | 1 | no | — |
| 122 | C2×C4×D3 | — | 9 | 1 | no | — |
| 123 | C2×Dic6 | — | 9 | 1 | no | — |
| 124 | #217 | — | 9 | 2 | no | — |
| 125 | #217 | — | 9 | 2 | no | — |
| 126 | #217 | — | 9 | 2 | no | — |
| 127 | #217 | — | 9 | 2 | no | — |
| 128 | #217 | — | 9 | 2 | no | — |
| 129 | #217 | — | 9 | 1 | no | — |
| 130 | (C2×D3)⋊C4 | — | 9 | 1 | no | — |
| 131 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 132 | Dic3⋊C4 | — | 9 | 2 | no | — |
| 133 | Dic3⋊C4 | — | 9 | 1 | no | — |
| 134 | C4×Dic3 | — | 9 | 1 | no | — |
| 135 | C22×C9 | — | 3 | 1 | no | — |
| 136 | C22×C9 | — | 3 | 1 | no | — |
| 137 | C22×C9 | — | 3 | 2 | no | — |
| 138 | C22×C9 | — | 6 | 2 | no | — |
| 139 | C22×C9 | — | 6 | 2 | no | — |
| 140 | C22×C9 | — | 6 | 2 | no | — |
| 141 | C22×C9 | — | 6 | 2 | no | — |
| 142 | C22×C9 | #237 | 1 | 1 | yes | characteristic |
| 143 | C22×C9 | C2×S4 | 1 | 2 | yes | — |
| 144 | C2×D9 | — | 6 | 1 | no | — |
| 145 | C2×D9 | — | 6 | 1 | no | — |
| 146 | C2×D9 | — | 6 | 1 | no | — |
| 147 | C4×C9 | — | 3 | 1 | no | — |
| 148 | C4×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 149 | Dic9 | — | 6 | 1 | no | — |
| 150 | Dic9 | — | 6 | 2 | no | — |
| 151 | Dic9 | — | 6 | 1 | no | — |
| 152 | Dic9 | — | 6 | 2 | no | — |
| 153 | Dic9 | — | 6 | 1 | no | — |
| 154 | C2×#41 | — | 27 | 1 | no | — |
| 155 | C22×D4 | D27 | 1 | 1 | yes | characteristic, complemented |
| 156 | #124 | — | 27 | 1 | no | — |
| 157 | #123 | — | 27 | 2 | no | — |
| 158 | C22⋊Q8 | — | 27 | 1 | no | — |
| 159 | #121 | — | 27 | 1 | no | — |
| 160 | #121 | — | 27 | 2 | no | — |
| 161 | #120 | — | 27 | 2 | no | — |
| 162 | C4×D4 | — | 27 | 1 | no | — |
| 163 | C4×D4 | — | 27 | 1 | no | — |
| 164 | C2×C22⋊C4 | — | 27 | 2 | no | — |
| 165 | C27 | — | 4 | 1 | no | Sylow |
| 166 | C23×C3 | — | 2 | 2 | no | — |
| 167 | C23×C3 | — | 3 | 2 | no | — |
| 168 | C23×C3 | — | 3 | 2 | no | — |
| 169 | C23×C3 | — | 6 | 2 | no | — |
| 170 | C23×C3 | C22×D9 | 1 | 1 | yes | characteristic |
| 171 | C22×D3 | — | 9 | 1 | no | — |
| 172 | C3×D4 | — | 3 | 2 | no | — |
| 173 | C3×D4 | — | 3 | 1 | no | — |
| 174 | C3×D4 | — | 6 | 1 | no | — |
| 175 | C3×D4 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 176 | C2×C3×C4 | — | 3 | 1 | no | — |
| 177 | C2×C3×C4 | — | 3 | 1 | no | — |
| 178 | #67 | — | 18 | 2 | no | — |
| 179 | #67 | — | 18 | 1 | no | — |
| 180 | #67 | — | 18 | 1 | no | — |
| 181 | #67 | — | 18 | 2 | no | — |
| 182 | #67 | — | 36 | 2 | no | — |
| 183 | C2×Dic3 | — | 9 | 1 | no | — |
| 184 | C2×Dic3 | — | 9 | 1 | no | — |
| 185 | C2×Dic3 | — | 9 | 2 | no | — |
| 186 | C2×Dic3 | — | 9 | 2 | no | — |
| 187 | C2×Dic3 | — | 9 | 1 | no | — |
| 188 | C2×Dic3 | — | 18 | 2 | no | — |
| 189 | C2×Dic3 | — | 18 | 2 | no | — |
| 190 | C2×Dic3 | — | 36 | 2 | no | — |
| 191 | C4×D3 | — | 18 | 1 | no | — |
| 192 | C4×D3 | — | 18 | 1 | no | — |
| 193 | Dic6 | — | 18 | 1 | no | — |
| 194 | Dic6 | — | 18 | 1 | no | — |
| 195 | C2×C9 | — | 2 | 2 | no | — |
| 196 | C2×C9 | — | 3 | 1 | no | — |
| 197 | C2×C9 | — | 3 | 1 | no | — |
| 198 | C2×C9 | — | 6 | 2 | no | — |
| 199 | C2×C9 | C22×S4 | 1 | 1 | yes | characteristic |
| 200 | D9 | — | 12 | 1 | no | — |
| 201 | C24 | C2×D27 | 1 | 2 | yes | — |
| 202 | #41 | — | 54 | 2 | no | — |
| 203 | #41 | — | 54 | 1 | no | — |
| 204 | #41 | — | 54 | 1 | no | — |
| 205 | C2×Q8 | — | 27 | 1 | no | — |
| 206 | C2×D4 | — | 3 | 1 | no | — |
| 207 | C2×D4 | — | 3 | 2 | no | — |
| 208 | C2×D4 | — | 3 | 1 | no | — |
| 209 | C2×D4 | — | 27 | 2 | no | — |
| 210 | C2×D4 | — | 27 | 1 | no | — |
| 211 | C2×D4 | — | 27 | 2 | no | — |
| 212 | C22×C4 | — | 27 | 1 | no | — |
| 213 | C22×C4 | — | 27 | 2 | no | — |
| 214 | C22×C4 | — | 27 | 2 | no | — |
| 215 | C22×C4 | C2×D27 | 1 | 1 | yes | characteristic |
| 216 | C4⋊C4 | — | 27 | 1 | no | — |
| 217 | C4⋊C4 | — | 27 | 1 | no | — |
| 218 | C4⋊C4 | — | 27 | 2 | no | — |
| 219 | C22⋊C4 | — | 27 | 1 | no | — |
| 220 | C22⋊C4 | — | 27 | 2 | no | — |
| 221 | C22⋊C4 | — | 27 | 2 | no | — |
| 222 | C22⋊C4 | — | 27 | 2 | no | — |
| 223 | C22⋊C4 | — | 27 | 2 | no | — |
| 224 | C22⋊C4 | — | 27 | 1 | no | — |
| 225 | C22⋊C4 | — | 27 | 2 | no | — |
| 226 | C42 | — | 27 | 1 | no | — |
| 227 | C22×C3 | — | 3 | 1 | no | — |
| 228 | C22×C3 | — | 3 | 1 | no | — |
| 229 | C22×C3 | — | 3 | 2 | no | — |
| 230 | C22×C3 | — | 6 | 2 | no | — |
| 231 | C22×C3 | — | 6 | 2 | no | — |
| 232 | C22×C3 | — | 6 | 2 | no | — |
| 233 | C22×C3 | — | 6 | 2 | no | — |
| 234 | C22×C3 | #3699 | 1 | 1 | yes | characteristic |
| 235 | C22×C3 | C2×(C22⋊C9)⋊C2 | 1 | 2 | yes | — |
| 236 | C2×D3 | — | 18 | 1 | no | — |
| 237 | C2×D3 | — | 18 | 1 | no | — |
| 238 | C2×D3 | — | 18 | 1 | no | — |
| 239 | C3×C4 | — | 3 | 1 | no | — |
| 240 | C3×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 241 | Dic3 | — | 18 | 1 | no | — |
| 242 | Dic3 | — | 18 | 2 | no | — |
| 243 | Dic3 | — | 18 | 1 | no | — |
| 244 | Dic3 | — | 18 | 2 | no | — |
| 245 | Dic3 | — | 18 | 1 | no | — |
| 246 | C9 | #5914 | 1 | 1 | yes | characteristic |
| 247 | C23 | — | 2 | 2 | no | — |
| 248 | C23 | — | 3 | 2 | no | — |
| 249 | C23 | — | 3 | 2 | no | — |
| 250 | C23 | — | 6 | 2 | no | — |
| 251 | C23 | — | 27 | 1 | no | — |
| 252 | C23 | C22×D27 | 1 | 1 | yes | characteristic |
| 253 | Q8 | — | 54 | 1 | no | — |
| 254 | Q8 | — | 54 | 1 | no | — |
| 255 | D4 | — | 3 | 2 | no | — |
| 256 | D4 | — | 3 | 1 | no | — |
| 257 | D4 | — | 6 | 1 | no | — |
| 258 | D4 | — | 54 | 2 | no | — |
| 259 | D4 | — | 54 | 1 | no | — |
| 260 | D4 | — | 54 | 2 | no | — |
| 261 | D4 | — | 54 | 1 | no | — |
| 262 | D4 | — | 108 | 2 | no | — |
| 263 | D4 | (C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 264 | C2×C4 | — | 3 | 1 | no | — |
| 265 | C2×C4 | — | 3 | 1 | no | — |
| 266 | C2×C4 | — | 27 | 1 | no | — |
| 267 | C2×C4 | — | 27 | 1 | no | — |
| 268 | C2×C4 | — | 27 | 2 | no | — |
| 269 | C2×C4 | — | 27 | 1 | no | — |
| 270 | C2×C4 | — | 27 | 2 | no | — |
| 271 | C2×C4 | — | 54 | 1 | no | — |
| 272 | C2×C4 | — | 54 | 2 | no | — |
| 273 | C2×C4 | — | 54 | 2 | no | — |
| 274 | C2×C4 | — | 54 | 1 | no | — |
| 275 | C2×C4 | — | 108 | 2 | no | — |
| 276 | C2×C3 | — | 2 | 2 | no | — |
| 277 | C2×C3 | — | 3 | 1 | no | — |
| 278 | C2×C3 | — | 3 | 1 | no | — |
| 279 | C2×C3 | — | 6 | 2 | no | — |
| 280 | C2×C3 | C22×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 281 | D3 | — | 36 | 1 | no | — |
| 282 | C22 | — | 3 | 1 | no | — |
| 283 | C22 | — | 3 | 1 | no | — |
| 284 | C22 | — | 3 | 2 | no | — |
| 285 | C22 | — | 6 | 2 | no | — |
| 286 | C22 | — | 6 | 2 | no | — |
| 287 | C22 | — | 6 | 2 | no | — |
| 288 | C22 | — | 6 | 2 | no | — |
| 289 | C22 | — | 54 | 1 | no | — |
| 290 | C22 | — | 54 | 1 | no | — |
| 291 | C22 | — | 54 | 1 | no | — |
| 292 | C22 | #88401 | 1 | 1 | yes | characteristic, complemented |
| 293 | C22 | C2×(C22⋊C27)⋊C2 | 1 | 2 | yes | — |
| 294 | C4 | — | 3 | 1 | no | — |
| 295 | C4 | — | 54 | 1 | no | — |
| 296 | C4 | — | 54 | 2 | no | — |
| 297 | C4 | — | 54 | 1 | no | — |
| 298 | C4 | — | 54 | 1 | no | — |
| 299 | C4 | — | 54 | 2 | no | — |
| 300 | C4 | C2×(C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 301 | C3 | #10593089 | 1 | 1 | yes | characteristic |
| 302 | C2 | — | 2 | 2 | no | — |
| 303 | C2 | — | 3 | 1 | no | — |
| 304 | C2 | — | 3 | 1 | no | — |
| 305 | C2 | — | 6 | 2 | no | — |
| 306 | C2 | — | 108 | 1 | no | — |
| 307 | C2 | C22×(C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×C22⋊C27 | C2 | semidirect |
| 2. #11720654 | C2 | semidirect |
| 3. C2×(C22⋊C27)⋊C4 | C2 | semidirect |
| 4. C4×(C22⋊C27)⋊C2 | C2 | semidirect |
| 5. (C22⋊C27)⋊Q8 | C2 | semidirect |
| 61. C22⋊C27 | #41 | semidirect |
| 155. C22×D4 | D27 | semidirect |
| 263. D4 | (C22⋊C27)⋊C2 | semidirect |
| 292. C22 | #88401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C22⋊C27 | 2 |
| #11720654 | 2 |
| C2×(C22⋊C27)⋊C4 | 2 |
| C4×(C22⋊C27)⋊C2 | 2 |
| (C22⋊C27)⋊Q8 | 2 |
| #10589908 | 3 |
| #88401 | 4 |