Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
740 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #72225579 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #72225406 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72225342 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #72216535 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #72216251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #498569 | — | 2 | 2 | no | — |
| 7 | #491465 | C22 | 1 | 1 | yes | characteristic |
| 8 | #152874 | — | 2 | 2 | no | — |
| 9 | C2×#7459 | C22 | 1 | 2 | yes | — |
| 10 | #141812 | — | 2 | 2 | no | — |
| 11 | #141262 | — | 2 | 2 | no | — |
| 12 | #137550 | — | 2 | 2 | no | — |
| 13 | #135781 | C22 | 1 | 1 | yes | characteristic |
| 14 | #129142 | C22 | 1 | 2 | yes | — |
| 15 | C2×#7052 | C22 | 1 | 1 | yes | characteristic |
| 16 | #94366 | — | 2 | 2 | no | — |
| 17 | #93867 | C4 | 1 | 2 | yes | complemented |
| 18 | #93866 | C4 | 1 | 2 | yes | complemented |
| 19 | #33713 | — | 2 | 2 | no | — |
| 20 | #33465 | — | 4 | 2 | no | — |
| 21 | #23196 | — | 2 | 2 | no | — |
| 22 | #19603 | D4 | 1 | 2 | yes | — |
| 23 | #13638 | — | 4 | 2 | no | — |
| 24 | C2×#1422 | — | 2 | 2 | no | — |
| 25 | #13259 | — | 2 | 4 | no | — |
| 26 | #13243 | — | 4 | 2 | no | — |
| 27 | #13016 | — | 4 | 4 | no | — |
| 28 | #12909 | — | 2 | 2 | no | — |
| 29 | #12670 | D4 | 1 | 2 | yes | — |
| 30 | #12664 | — | 2 | 2 | no | — |
| 31 | #12518 | — | 2 | 2 | no | — |
| 32 | #12517 | Q8 | 1 | 2 | yes | — |
| 33 | #12516 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | #12515 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C2×#1373 | D4 | 1 | 2 | yes | — |
| 36 | #11922 | — | 2 | 2 | no | — |
| 37 | #11671 | — | 4 | 4 | no | — |
| 38 | C2×C8×C16 | — | 2 | 1 | no | — |
| 39 | C2×C8×C16 | C23 | 1 | 1 | yes | characteristic |
| 40 | #8823 | — | 2 | 1 | no | — |
| 41 | #7479 | — | 4 | 2 | no | — |
| 42 | #7459 | — | 2 | 2 | no | — |
| 43 | #7459 | C2×C4 | 1 | 4 | yes | — |
| 44 | #7179 | — | 2 | 2 | no | — |
| 45 | #7052 | — | 2 | 2 | no | — |
| 46 | C2×#508 | — | 4 | 2 | no | — |
| 47 | C2×#508 | — | 4 | 2 | no | — |
| 48 | #3169 | — | 4 | 2 | no | — |
| 49 | #3147 | — | 2 | 2 | no | — |
| 50 | #2957 | — | 4 | 4 | no | — |
| 51 | C2×M64 | — | 4 | 2 | no | — |
| 52 | #2249 | — | 4 | 2 | no | — |
| 53 | #2249 | — | 4 | 2 | no | — |
| 54 | #2248 | — | 2 | 4 | no | — |
| 55 | #2247 | — | 2 | 2 | no | — |
| 56 | #2246 | — | 2 | 2 | no | — |
| 57 | #2179 | — | 8 | 4 | no | — |
| 58 | #2178 | — | 8 | 4 | no | — |
| 59 | C2×#368 | — | 4 | 2 | no | — |
| 60 | C2×#368 | — | 4 | 2 | no | — |
| 61 | C2×#368 | D8 | 1 | 2 | yes | — |
| 62 | #2158 | D8 | 1 | 2 | yes | — |
| 63 | C2×#364 | — | 2 | 2 | no | — |
| 64 | #2145 | — | 4 | 4 | no | — |
| 65 | #2143 | — | 4 | 4 | no | — |
| 66 | #2108 | — | 4 | 2 | no | — |
| 67 | C2×C4×C16 | — | 2 | 2 | no | — |
| 68 | C2×C4×C16 | C2×D4 | 1 | 2 | yes | — |
| 69 | #1976 | — | 4 | 4 | no | — |
| 70 | #1884 | — | 8 | 2 | no | — |
| 71 | #1848 | — | 4 | 1 | no | — |
| 72 | #1772 | — | 2 | 2 | no | — |
| 73 | #1713 | C4⋊C4 | 1 | 2 | yes | — |
| 74 | #1712 | C4⋊C4 | 1 | 2 | yes | — |
| 75 | #1709 | — | 2 | 2 | no | — |
| 76 | #1706 | — | 4 | 2 | no | — |
| 77 | #1567 | — | 4 | 2 | no | — |
| 78 | C2×C82 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | #1426 | — | 8 | 4 | no | — |
| 80 | #1422 | — | 2 | 4 | no | — |
| 81 | #1422 | — | 4 | 2 | no | — |
| 82 | #1382 | — | 2 | 2 | no | — |
| 83 | #1379 | — | 4 | 2 | no | — |
| 84 | #1373 | — | 2 | 2 | no | — |
| 85 | #1373 | — | 2 | 2 | no | — |
| 86 | #1359 | — | 8 | 2 | no | — |
| 87 | #1331 | — | 8 | 8 | no | — |
| 88 | C8×C16 | — | 4 | 1 | no | — |
| 89 | C8×C16 | — | 4 | 1 | no | — |
| 90 | C8×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C8×C16 | C2×Q8 | 1 | 1 | yes | characteristic |
| 92 | C8×C16 | #41 | 1 | 2 | yes | — |
| 93 | C2×#135 | — | 4 | 4 | no | — |
| 94 | #568 | — | 8 | 2 | no | — |
| 95 | #532 | — | 4 | 2 | no | — |
| 96 | #508 | — | 8 | 2 | no | — |
| 97 | #508 | — | 8 | 2 | no | — |
| 98 | #508 | — | 8 | 2 | no | — |
| 99 | #508 | — | 8 | 2 | no | — |
| 100 | #508 | — | 16 | 2 | no | — |
| 101 | #508 | — | 16 | 2 | no | — |
| 102 | C2×Q32 | — | 4 | 2 | no | — |
| 103 | C2×Q32 | — | 4 | 2 | no | — |
| 104 | C2×SD32 | — | 4 | 4 | no | — |
| 105 | C2×SD32 | — | 4 | 4 | no | — |
| 106 | C2×D16 | — | 4 | 2 | no | — |
| 107 | C2×D16 | — | 4 | 2 | no | — |
| 108 | C2×M32 | — | 4 | 2 | no | — |
| 109 | C22×C16 | — | 4 | 2 | no | — |
| 110 | C22×C16 | C2×D8 | 1 | 2 | yes | — |
| 111 | C22×C16 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 112 | #500 | — | 4 | 2 | no | — |
| 113 | #495 | — | 4 | 4 | no | — |
| 114 | #494 | — | 2 | 2 | no | — |
| 115 | #493 | — | 2 | 2 | no | — |
| 116 | #493 | — | 4 | 2 | no | — |
| 117 | #492 | — | 4 | 2 | no | — |
| 118 | #486 | — | 8 | 2 | no | — |
| 119 | #466 | — | 8 | 4 | no | — |
| 120 | #444 | — | 16 | 4 | no | — |
| 121 | #443 | — | 8 | 8 | no | — |
| 122 | C4×D8 | — | 8 | 4 | no | — |
| 123 | C2×#108 | — | 2 | 2 | no | — |
| 124 | C2×#108 | — | 8 | 2 | no | — |
| 125 | #427 | — | 2 | 2 | no | — |
| 126 | #423 | — | 8 | 1 | no | — |
| 127 | #421 | — | 8 | 4 | no | — |
| 128 | #410 | — | 8 | 2 | no | — |
| 129 | #405 | — | 4 | 2 | no | — |
| 130 | C2×C4×C8 | — | 2 | 1 | no | — |
| 131 | C2×C4×C8 | #120 | 1 | 1 | yes | characteristic |
| 132 | M64 | — | 4 | 4 | no | — |
| 133 | M64 | — | 8 | 2 | no | — |
| 134 | C2×C32 | — | 8 | 2 | no | — |
| 135 | #368 | — | 2 | 2 | no | — |
| 136 | #368 | — | 8 | 2 | no | — |
| 137 | #368 | — | 8 | 2 | no | — |
| 138 | #368 | — | 16 | 2 | no | — |
| 139 | #368 | SD32 | 1 | 4 | yes | — |
| 140 | #367 | — | 2 | 2 | no | — |
| 141 | #366 | — | 2 | 2 | no | — |
| 142 | #364 | — | 4 | 2 | no | — |
| 143 | #364 | — | 4 | 2 | no | — |
| 144 | #360 | — | 16 | 4 | no | — |
| 145 | #359 | — | 8 | 8 | no | — |
| 146 | #346 | — | 8 | 2 | no | — |
| 147 | C4×C16 | — | 2 | 4 | no | — |
| 148 | C4×C16 | — | 4 | 2 | no | — |
| 149 | C4×C16 | — | 8 | 2 | no | — |
| 150 | C4×C16 | C4×D4 | 1 | 2 | yes | — |
| 151 | C4×C16 | C22⋊Q8 | 1 | 2 | yes | — |
| 152 | C4×C16 | #135 | 1 | 2 | yes | — |
| 153 | C4×C16 | #137 | 1 | 2 | yes | — |
| 154 | #339 | — | 4 | 2 | no | — |
| 155 | #335 | — | 8 | 4 | no | — |
| 156 | C82 | — | 2 | 1 | no | — |
| 157 | C82 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 158 | C82 | #123 | 1 | 1 | yes | characteristic |
| 159 | C2×#41 | — | 8 | 2 | no | — |
| 160 | #135 | — | 8 | 4 | no | — |
| 161 | #135 | — | 8 | 4 | no | — |
| 162 | #135 | — | 8 | 4 | no | — |
| 163 | #135 | — | 8 | 4 | no | — |
| 164 | C2×Q16 | — | 4 | 4 | no | — |
| 165 | C2×SD16 | — | 8 | 4 | no | — |
| 166 | C2×D8 | — | 4 | 4 | no | — |
| 167 | C2×D8 | — | 8 | 4 | no | — |
| 168 | #131 | — | 16 | 8 | no | — |
| 169 | C2×M16 | — | 4 | 2 | no | — |
| 170 | C2×M16 | — | 16 | 1 | no | — |
| 171 | C2×M16 | — | 32 | 2 | no | — |
| 172 | C22×C8 | — | 2 | 2 | no | — |
| 173 | C22×C8 | #447 | 1 | 2 | yes | — |
| 174 | C4⋊Q8 | — | 4 | 2 | no | — |
| 175 | #127 | — | 4 | 2 | no | — |
| 176 | #124 | — | 8 | 2 | no | — |
| 177 | #121 | — | 16 | 2 | no | — |
| 178 | C4×D4 | — | 16 | 2 | no | — |
| 179 | #117 | — | 4 | 2 | no | — |
| 180 | C2×C42 | #453 | 1 | 1 | yes | characteristic |
| 181 | Q32 | — | 8 | 2 | no | — |
| 182 | Q32 | — | 8 | 2 | no | — |
| 183 | Q32 | — | 16 | 2 | no | — |
| 184 | SD32 | — | 8 | 4 | no | — |
| 185 | SD32 | — | 8 | 4 | no | — |
| 186 | SD32 | — | 16 | 4 | no | — |
| 187 | D16 | — | 8 | 2 | no | — |
| 188 | D16 | — | 8 | 2 | no | — |
| 189 | D16 | — | 16 | 2 | no | — |
| 190 | M32 | — | 8 | 2 | no | — |
| 191 | M32 | — | 8 | 2 | no | — |
| 192 | C2×C16 | — | 2 | 2 | no | — |
| 193 | C2×C16 | — | 2 | 2 | no | — |
| 194 | C2×C16 | — | 4 | 4 | no | — |
| 195 | C2×C16 | — | 8 | 4 | no | — |
| 196 | C2×C16 | — | 8 | 2 | no | — |
| 197 | C2×C16 | C4×D8 | 1 | 2 | yes | — |
| 198 | C2×C16 | #440 | 1 | 2 | yes | — |
| 199 | C2×C16 | #474 | 1 | 2 | yes | — |
| 200 | C2×C16 | #478 | 1 | 2 | yes | — |
| 201 | C2×C16 | C2×SD32 | 1 | 2 | yes | — |
| 202 | C2×C16 | #508 | 1 | 2 | yes | — |
| 203 | C2×C16 | #509 | 1 | 2 | yes | — |
| 204 | C2×C16 | #509 | 1 | 2 | yes | — |
| 205 | #108 | — | 2 | 4 | no | — |
| 206 | #108 | — | 4 | 2 | no | — |
| 207 | #108 | — | 16 | 2 | no | — |
| 208 | #108 | — | 16 | 2 | no | — |
| 209 | #107 | — | 4 | 2 | no | — |
| 210 | #107 | — | 8 | 4 | no | — |
| 211 | #106 | — | 4 | 2 | no | — |
| 212 | C4⋊C8 | — | 16 | 2 | no | — |
| 213 | #104 | — | 16 | 8 | no | — |
| 214 | #102 | — | 16 | 4 | no | — |
| 215 | #99 | — | 32 | 2 | no | — |
| 216 | #97 | — | 8 | 2 | no | — |
| 217 | C4×C8 | — | 2 | 2 | no | — |
| 218 | C4×C8 | — | 2 | 1 | no | — |
| 219 | C4×C8 | — | 2 | 1 | no | — |
| 220 | C4×C8 | — | 8 | 2 | no | — |
| 221 | C4×C8 | #385 | 1 | 1 | yes | characteristic |
| 222 | C4×C8 | #452 | 1 | 2 | yes | — |
| 223 | C4×C8 | #455 | 1 | 1 | yes | characteristic |
| 224 | C32 | — | 8 | 4 | no | — |
| 225 | #41 | — | 16 | 4 | no | — |
| 226 | #41 | — | 16 | 4 | no | — |
| 227 | C2×Q8 | — | 8 | 2 | no | — |
| 228 | C2×D4 | — | 8 | 2 | no | — |
| 229 | C2×D4 | — | 16 | 2 | no | — |
| 230 | C2×D4 | — | 16 | 2 | no | — |
| 231 | C22×C4 | — | 2 | 1 | no | — |
| 232 | C22×C4 | — | 16 | 2 | no | — |
| 233 | C22×C4 | #1619 | 1 | 1 | yes | characteristic |
| 234 | Q16 | — | 8 | 4 | no | — |
| 235 | Q16 | — | 8 | 4 | no | — |
| 236 | SD16 | — | 16 | 4 | no | — |
| 237 | SD16 | — | 16 | 4 | no | — |
| 238 | D8 | — | 8 | 4 | no | — |
| 239 | D8 | — | 8 | 4 | no | — |
| 240 | D8 | — | 16 | 8 | no | — |
| 241 | M16 | — | 4 | 4 | no | — |
| 242 | M16 | — | 8 | 2 | no | — |
| 243 | M16 | — | 32 | 4 | no | — |
| 244 | M16 | — | 32 | 2 | no | — |
| 245 | C2×C8 | — | 2 | 2 | no | — |
| 246 | C2×C8 | — | 2 | 2 | no | — |
| 247 | C2×C8 | — | 4 | 4 | no | — |
| 248 | C2×C8 | — | 8 | 2 | no | — |
| 249 | C2×C8 | — | 32 | 4 | no | — |
| 250 | C2×C8 | — | 32 | 1 | no | — |
| 251 | C2×C8 | #1641 | 1 | 2 | yes | — |
| 252 | C2×C8 | #1862 | 1 | 2 | yes | — |
| 253 | C2×C8 | #2186 | 1 | 2 | yes | — |
| 254 | C2×C8 | #2187 | 1 | 2 | yes | — |
| 255 | C2×C8 | #2192 | 1 | 2 | yes | — |
| 256 | C2×C8 | #2194 | 1 | 2 | yes | — |
| 257 | C4⋊C4 | — | 8 | 2 | no | — |
| 258 | C4⋊C4 | — | 16 | 2 | no | — |
| 259 | C22⋊C4 | — | 8 | 2 | no | — |
| 260 | C22⋊C4 | — | 32 | 2 | no | — |
| 261 | C42 | — | 2 | 1 | no | — |
| 262 | C42 | — | 8 | 2 | no | — |
| 263 | C42 | #1630 | 1 | 1 | yes | characteristic |
| 264 | C42 | #1856 | 1 | 1 | yes | characteristic |
| 265 | C16 | — | 2 | 2 | no | — |
| 266 | C16 | — | 2 | 2 | no | — |
| 267 | C16 | — | 8 | 4 | no | — |
| 268 | C16 | — | 16 | 2 | no | — |
| 269 | C16 | C4×SD32 | 1 | 2 | yes | — |
| 270 | C16 | #2177 | 1 | 2 | yes | — |
| 271 | C16 | #2226 | 1 | 2 | yes | — |
| 272 | C16 | #2228 | 1 | 2 | yes | — |
| 273 | C23 | — | 16 | 2 | no | — |
| 274 | C23 | #12312 | 1 | 1 | yes | characteristic |
| 275 | Q8 | — | 16 | 4 | no | — |
| 276 | D4 | — | 16 | 4 | no | — |
| 277 | D4 | — | 32 | 4 | no | — |
| 278 | D4 | — | 32 | 4 | no | — |
| 279 | C2×C4 | — | 2 | 1 | no | — |
| 280 | C2×C4 | — | 2 | 1 | no | — |
| 281 | C2×C4 | — | 4 | 2 | no | — |
| 282 | C2×C4 | — | 16 | 2 | no | — |
| 283 | C2×C4 | — | 16 | 2 | no | — |
| 284 | C2×C4 | — | 32 | 2 | no | — |
| 285 | C2×C4 | — | 32 | 2 | no | — |
| 286 | C2×C4 | #8920 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #12310 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | #12317 | 1 | 2 | yes | — |
| 289 | C2×C4 | #12320 | 1 | 2 | yes | — |
| 290 | C8 | — | 2 | 4 | no | — |
| 291 | C8 | — | 4 | 2 | no | — |
| 292 | C8 | — | 8 | 4 | no | — |
| 293 | C8 | — | 64 | 1 | no | — |
| 294 | C8 | #12381 | 1 | 2 | yes | — |
| 295 | C8 | #12384 | 1 | 2 | yes | — |
| 296 | C8 | #12937 | 1 | 2 | yes | — |
| 297 | C8 | #12940 | 1 | 2 | yes | — |
| 298 | C22 | — | 4 | 1 | no | — |
| 299 | C22 | — | 32 | 2 | no | — |
| 300 | C22 | — | 64 | 2 | no | — |
| 301 | C22 | #123298 | 1 | 2 | yes | — |
| 302 | C22 | #136488 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 2 | 2 | no | — |
| 304 | C4 | — | 4 | 1 | no | — |
| 305 | C4 | — | 32 | 2 | no | — |
| 306 | C4 | — | 32 | 2 | no | — |
| 307 | C4 | #123276 | 1 | 1 | yes | characteristic |
| 308 | C4 | #123285 | 1 | 1 | yes | characteristic |
| 309 | C4 | #136495 | 1 | 2 | yes | — |
| 310 | C2 | — | 2 | 1 | no | — |
| 311 | C2 | — | 2 | 2 | no | — |
| 312 | C2 | — | 64 | 2 | no | — |
| 313 | C2 | #72225202 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #123285 |
| commutator | C8×C16 | C22×C4 |
| Frattini, cosocle | C2×C8×C16 | C23 |
| socle | C2 | #72225202 |
| prime radical | #491465 | C22 |
| LCS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ G | |
| LECS | G ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #72225342 | C2 | semidirect |
| 4. #72216535 | C2 | semidirect |
| 17. #93867 | C4 | semidirect |
| 18. #93866 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72225579 | 2 |
| #72225406 | 2 |
| #72225342 | 2 |
| #72216535 | 2 |
| #72216251 | 2 |