Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
1080 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 180 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 | #72216560 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#93866 | C2 | 1 | 2 | yes | maximal |
| 3 | #72216243 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #72216165 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #496104 | — | 2 | 2 | no | — |
| 6 | C2×#12515 | C22 | 1 | 1 | yes | characteristic |
| 7 | C4×#1712 | C4 | 1 | 2 | yes | — |
| 8 | #151981 | — | 2 | 2 | no | — |
| 9 | C2×#7459 | C22 | 1 | 4 | yes | — |
| 10 | #141681 | — | 2 | 2 | no | — |
| 11 | #141665 | C4 | 1 | 2 | yes | — |
| 12 | #135778 | C22 | 1 | 1 | yes | characteristic |
| 13 | C8×#346 | C22 | 1 | 1 | yes | characteristic |
| 14 | #94019 | — | 2 | 2 | no | — |
| 15 | #93866 | C4 | 1 | 8 | yes | complemented |
| 16 | C2×#2246 | — | 2 | 2 | no | — |
| 17 | C2×#2177 | — | 4 | 2 | no | — |
| 18 | C2×#2177 | — | 4 | 2 | no | — |
| 19 | C4×#493 | — | 2 | 2 | no | — |
| 20 | C4×#493 | — | 4 | 2 | no | — |
| 21 | C2×#1712 | C2×C4 | 1 | 2 | yes | — |
| 22 | C2×#1422 | — | 2 | 4 | no | — |
| 23 | C2×#1398 | — | 4 | 2 | no | — |
| 24 | #13239 | — | 4 | 2 | no | — |
| 25 | #13239 | — | 4 | 2 | no | — |
| 26 | #13202 | — | 2 | 4 | no | — |
| 27 | #12906 | — | 2 | 2 | no | — |
| 28 | C4×#346 | D4 | 1 | 2 | yes | — |
| 29 | #12515 | — | 2 | 2 | no | — |
| 30 | #12515 | C2×C4 | 1 | 4 | yes | — |
| 31 | C2×#1311 | D4 | 1 | 2 | yes | — |
| 32 | C2×C8×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×C8×C16 | C23 | 1 | 1 | yes | characteristic |
| 34 | #9824 | — | 2 | 2 | no | — |
| 35 | #9824 | — | 4 | 2 | no | — |
| 36 | #8818 | C2×C4 | 1 | 2 | yes | — |
| 37 | C4×C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | #7468 | — | 4 | 2 | no | — |
| 39 | #7459 | — | 2 | 4 | no | — |
| 40 | #7459 | C2×C4 | 1 | 8 | yes | complemented |
| 41 | C22×D16 | — | 4 | 2 | no | — |
| 42 | C22×D16 | — | 4 | 2 | no | — |
| 43 | C2×#493 | — | 2 | 2 | no | — |
| 44 | C2×#493 | — | 4 | 2 | no | — |
| 45 | C2×C4×D8 | — | 4 | 4 | no | — |
| 46 | C2×C4×D8 | — | 8 | 4 | no | — |
| 47 | C4×#127 | — | 4 | 2 | no | — |
| 48 | C2×M64 | — | 4 | 4 | no | — |
| 49 | #2246 | — | 2 | 8 | no | — |
| 50 | #2246 | — | 4 | 4 | no | — |
| 51 | #2177 | — | 8 | 8 | no | — |
| 52 | #2177 | — | 8 | 8 | no | — |
| 53 | C2×#367 | — | 4 | 2 | no | — |
| 54 | C2×#367 | — | 4 | 2 | no | — |
| 55 | C2×#357 | — | 4 | 4 | no | — |
| 56 | C2×#357 | — | 4 | 4 | no | — |
| 57 | C2×#346 | D8 | 1 | 2 | yes | — |
| 58 | C2×#346 | D8 | 1 | 2 | yes | — |
| 59 | C2×#346 | SD16 | 1 | 2 | yes | — |
| 60 | C2×#346 | SD16 | 1 | 2 | yes | — |
| 61 | C2×C4×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 62 | C2×C4×C16 | C2×D4 | 1 | 2 | yes | — |
| 63 | #1970 | — | 4 | 4 | no | — |
| 64 | #1970 | — | 8 | 4 | no | — |
| 65 | #1957 | — | 4 | 4 | no | — |
| 66 | #1957 | — | 4 | 2 | no | — |
| 67 | #1845 | — | 2 | 2 | no | — |
| 68 | #1845 | — | 4 | 2 | no | — |
| 69 | C42×C8 | — | 2 | 1 | no | — |
| 70 | C42×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | #1712 | — | 2 | 2 | no | — |
| 72 | #1712 | — | 4 | 2 | no | — |
| 73 | #1712 | C42 | 1 | 4 | yes | — |
| 74 | C2×C82 | — | 2 | 1 | no | — |
| 75 | C2×C82 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C2×C82 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | #1422 | — | 2 | 8 | no | — |
| 78 | #1422 | — | 4 | 4 | no | — |
| 79 | #1398 | — | 8 | 4 | no | — |
| 80 | #1311 | — | 2 | 4 | no | — |
| 81 | C8×C16 | — | 4 | 1 | no | — |
| 82 | C8×C16 | C22×C4 | 1 | 2 | yes | — |
| 83 | C8×C16 | #41 | 1 | 2 | yes | — |
| 84 | C22×D8 | — | 4 | 4 | no | — |
| 85 | C22×D8 | — | 8 | 4 | no | — |
| 86 | C2×#127 | — | 4 | 2 | no | — |
| 87 | C2×C4×D4 | — | 8 | 4 | no | — |
| 88 | C2×C4×D4 | — | 8 | 2 | no | — |
| 89 | C2×D16 | — | 4 | 8 | no | — |
| 90 | C2×D16 | — | 4 | 8 | no | — |
| 91 | C2×D16 | — | 8 | 4 | no | — |
| 92 | C2×D16 | — | 8 | 4 | no | — |
| 93 | C2×D16 | — | 8 | 2 | no | — |
| 94 | C2×D16 | — | 8 | 4 | no | — |
| 95 | C2×D16 | — | 8 | 2 | no | — |
| 96 | C22×C16 | #102 | 1 | 2 | yes | — |
| 97 | C22×C16 | #102 | 1 | 2 | yes | — |
| 98 | C22×C16 | C2×D8 | 1 | 2 | yes | — |
| 99 | C22×C16 | C2×SD16 | 1 | 2 | yes | — |
| 100 | #493 | — | 2 | 4 | no | — |
| 101 | #493 | — | 4 | 4 | no | — |
| 102 | #493 | — | 4 | 4 | no | — |
| 103 | #493 | — | 4 | 2 | no | — |
| 104 | #493 | — | 4 | 4 | no | — |
| 105 | #493 | — | 8 | 4 | no | — |
| 106 | C4×D8 | — | 8 | 8 | no | — |
| 107 | C4×D8 | — | 8 | 8 | no | — |
| 108 | C4×D8 | — | 16 | 16 | no | — |
| 109 | C2×#107 | — | 4 | 4 | no | — |
| 110 | C2×#107 | — | 8 | 4 | no | — |
| 111 | C2×#102 | — | 8 | 8 | no | — |
| 112 | C2×#102 | — | 8 | 4 | no | — |
| 113 | C2×C4×C8 | — | 2 | 1 | no | — |
| 114 | C2×C4×C8 | — | 2 | 4 | no | — |
| 115 | C2×C4×C8 | — | 2 | 1 | no | — |
| 116 | C2×C4×C8 | #100 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C8 | #102 | 1 | 2 | yes | — |
| 118 | C2×C4×C8 | #102 | 1 | 2 | yes | — |
| 119 | C2×C4×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | #390 | — | 8 | 4 | no | — |
| 121 | #390 | — | 8 | 2 | no | — |
| 122 | #384 | — | 4 | 2 | no | — |
| 123 | C43 | #100 | 1 | 1 | yes | characteristic |
| 124 | M64 | — | 4 | 8 | no | — |
| 125 | M64 | — | 8 | 4 | no | — |
| 126 | C2×C32 | — | 8 | 4 | no | — |
| 127 | #367 | — | 8 | 4 | no | — |
| 128 | #367 | — | 8 | 4 | no | — |
| 129 | #357 | — | 8 | 8 | no | — |
| 130 | #357 | — | 8 | 8 | no | — |
| 131 | #346 | — | 2 | 4 | no | — |
| 132 | #346 | — | 2 | 2 | no | — |
| 133 | #346 | — | 2 | 4 | no | — |
| 134 | #346 | — | 2 | 2 | no | — |
| 135 | #346 | — | 4 | 2 | no | — |
| 136 | C4×C16 | — | 2 | 2 | no | — |
| 137 | C4×C16 | — | 2 | 2 | no | — |
| 138 | C4×C16 | C4×D4 | 1 | 4 | yes | — |
| 139 | C4×C16 | #135 | 1 | 4 | yes | — |
| 140 | C82 | — | 2 | 1 | no | — |
| 141 | C82 | — | 2 | 2 | no | — |
| 142 | C82 | — | 4 | 2 | no | — |
| 143 | C82 | C2×C42 | 1 | 1 | yes | characteristic |
| 144 | C82 | #117 | 1 | 1 | yes | characteristic |
| 145 | C22×D4 | — | 8 | 4 | no | — |
| 146 | C22×D4 | — | 8 | 2 | no | — |
| 147 | C23×C4 | — | 16 | 2 | no | — |
| 148 | C2×D8 | — | 4 | 8 | no | — |
| 149 | C2×D8 | — | 8 | 8 | no | — |
| 150 | C2×D8 | — | 8 | 8 | no | — |
| 151 | C2×D8 | — | 8 | 4 | no | — |
| 152 | C2×D8 | — | 8 | 4 | no | — |
| 153 | C2×D8 | — | 8 | 4 | no | — |
| 154 | C2×D8 | — | 8 | 4 | no | — |
| 155 | C2×D8 | — | 8 | 4 | no | — |
| 156 | C2×D8 | — | 16 | 8 | no | — |
| 157 | C2×D8 | — | 16 | 8 | no | — |
| 158 | C22×C8 | — | 2 | 2 | no | — |
| 159 | C22×C8 | — | 2 | 2 | no | — |
| 160 | C22×C8 | #331 | 1 | 2 | yes | — |
| 161 | C22×C8 | C2×#102 | 1 | 2 | yes | — |
| 162 | #127 | — | 4 | 4 | no | — |
| 163 | #127 | — | 8 | 4 | no | — |
| 164 | #127 | — | 8 | 2 | no | — |
| 165 | C4×D4 | — | 16 | 8 | no | — |
| 166 | C4×D4 | — | 16 | 16 | no | — |
| 167 | C2×C4⋊C4 | — | 8 | 4 | no | — |
| 168 | C2×C4⋊C4 | — | 8 | 2 | no | — |
| 169 | C2×C22⋊C4 | — | 16 | 4 | no | — |
| 170 | C2×C22⋊C4 | — | 16 | 2 | no | — |
| 171 | C2×C42 | — | 2 | 2 | no | — |
| 172 | C2×C42 | #331 | 1 | 1 | yes | characteristic |
| 173 | C2×C42 | #331 | 1 | 1 | yes | characteristic |
| 174 | C2×C42 | C2×#100 | 1 | 1 | yes | characteristic |
| 175 | D16 | — | 8 | 16 | no | — |
| 176 | D16 | — | 8 | 8 | no | — |
| 177 | D16 | — | 8 | 8 | no | — |
| 178 | C2×C16 | — | 2 | 2 | no | — |
| 179 | C2×C16 | — | 2 | 2 | no | — |
| 180 | C2×C16 | — | 2 | 2 | no | — |
| 181 | C2×C16 | — | 2 | 2 | no | — |
| 182 | C2×C16 | — | 2 | 2 | no | — |
| 183 | C2×C16 | — | 2 | 2 | no | — |
| 184 | C2×C16 | — | 2 | 2 | no | — |
| 185 | C2×C16 | #361 | 1 | 4 | yes | — |
| 186 | C2×C16 | C4×D8 | 1 | 4 | yes | — |
| 187 | C2×C16 | C4×SD16 | 1 | 4 | yes | — |
| 188 | C2×C16 | #509 | 1 | 4 | yes | — |
| 189 | C2×C16 | #509 | 1 | 2 | yes | — |
| 190 | C2×C16 | #510 | 1 | 2 | yes | — |
| 191 | #107 | — | 8 | 4 | no | — |
| 192 | #107 | — | 8 | 4 | no | — |
| 193 | #107 | — | 16 | 8 | no | — |
| 194 | #102 | — | 16 | 8 | no | — |
| 195 | #102 | — | 16 | 16 | no | — |
| 196 | C4×C8 | — | 2 | 2 | no | — |
| 197 | C4×C8 | — | 2 | 2 | no | — |
| 198 | C4×C8 | — | 2 | 2 | no | — |
| 199 | C4×C8 | — | 2 | 2 | no | — |
| 200 | C4×C8 | — | 2 | 2 | no | — |
| 201 | C4×C8 | — | 2 | 1 | no | — |
| 202 | C4×C8 | — | 2 | 1 | no | — |
| 203 | C4×C8 | — | 4 | 1 | no | — |
| 204 | C4×C8 | — | 4 | 1 | no | — |
| 205 | C4×C8 | — | 4 | 8 | no | — |
| 206 | C4×C8 | — | 4 | 1 | no | — |
| 207 | C4×C8 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 208 | C4×C8 | #413 | 1 | 1 | yes | characteristic |
| 209 | C4×C8 | #416 | 1 | 2 | yes | — |
| 210 | C32 | — | 8 | 8 | no | — |
| 211 | C24 | — | 16 | 2 | no | — |
| 212 | C2×D4 | — | 8 | 8 | no | — |
| 213 | C2×D4 | — | 8 | 4 | no | — |
| 214 | C2×D4 | — | 16 | 8 | no | — |
| 215 | C2×D4 | — | 16 | 8 | no | — |
| 216 | C2×D4 | — | 16 | 4 | no | — |
| 217 | C2×D4 | — | 16 | 2 | no | — |
| 218 | C2×D4 | — | 16 | 4 | no | — |
| 219 | C2×D4 | — | 16 | 4 | no | — |
| 220 | C22×C4 | — | 2 | 1 | no | — |
| 221 | C22×C4 | — | 2 | 1 | no | — |
| 222 | C22×C4 | — | 16 | 2 | no | — |
| 223 | C22×C4 | — | 32 | 4 | no | — |
| 224 | C22×C4 | — | 32 | 8 | no | — |
| 225 | C22×C4 | #1357 | 1 | 2 | yes | — |
| 226 | C22×C4 | C2×#331 | 1 | 1 | yes | characteristic |
| 227 | D8 | — | 8 | 8 | no | — |
| 228 | D8 | — | 8 | 8 | no | — |
| 229 | D8 | — | 16 | 16 | no | — |
| 230 | D8 | — | 16 | 16 | no | — |
| 231 | D8 | — | 16 | 4 | no | — |
| 232 | D8 | — | 16 | 4 | no | — |
| 233 | C2×C8 | — | 2 | 2 | no | — |
| 234 | C2×C8 | — | 2 | 2 | no | — |
| 235 | C2×C8 | — | 2 | 8 | no | — |
| 236 | C2×C8 | — | 2 | 2 | no | — |
| 237 | C2×C8 | — | 2 | 2 | no | — |
| 238 | C2×C8 | — | 2 | 2 | no | — |
| 239 | C2×C8 | — | 4 | 4 | no | — |
| 240 | C2×C8 | — | 4 | 2 | no | — |
| 241 | C2×C8 | #1540 | 1 | 2 | yes | — |
| 242 | C2×C8 | C4×#102 | 1 | 2 | yes | — |
| 243 | C2×C8 | #2139 | 1 | 2 | yes | — |
| 244 | C2×C8 | #2140 | 1 | 2 | yes | — |
| 245 | C2×C8 | C2×#361 | 1 | 2 | yes | — |
| 246 | C2×C8 | #2148 | 1 | 2 | yes | — |
| 247 | C4⋊C4 | — | 16 | 8 | no | — |
| 248 | C4⋊C4 | — | 16 | 4 | no | — |
| 249 | C22⋊C4 | — | 32 | 4 | no | — |
| 250 | C22⋊C4 | — | 32 | 8 | no | — |
| 251 | C42 | — | 2 | 1 | no | — |
| 252 | C42 | — | 2 | 2 | no | — |
| 253 | C42 | — | 2 | 2 | no | — |
| 254 | C42 | — | 4 | 4 | no | — |
| 255 | C42 | #1543 | 1 | 1 | yes | characteristic |
| 256 | C42 | C4×#100 | 1 | 1 | yes | characteristic |
| 257 | C16 | — | 2 | 4 | no | — |
| 258 | C16 | — | 2 | 4 | no | — |
| 259 | C16 | — | 4 | 2 | no | — |
| 260 | C16 | #2175 | 1 | 4 | yes | — |
| 261 | C16 | #2177 | 1 | 4 | yes | — |
| 262 | C23 | — | 16 | 4 | no | — |
| 263 | C23 | — | 32 | 4 | no | — |
| 264 | C23 | — | 32 | 4 | no | — |
| 265 | C23 | — | 32 | 2 | no | — |
| 266 | C23 | — | 32 | 2 | no | — |
| 267 | C23 | C2×#1357 | 1 | 1 | yes | characteristic |
| 268 | D4 | — | 16 | 16 | no | — |
| 269 | D4 | — | 16 | 8 | no | — |
| 270 | D4 | — | 32 | 8 | no | — |
| 271 | D4 | — | 32 | 4 | no | — |
| 272 | C2×C4 | — | 2 | 4 | no | — |
| 273 | C2×C4 | — | 2 | 1 | no | — |
| 274 | C2×C4 | — | 2 | 1 | no | — |
| 275 | C2×C4 | — | 2 | 2 | no | — |
| 276 | C2×C4 | — | 4 | 2 | no | — |
| 277 | C2×C4 | — | 4 | 2 | no | — |
| 278 | C2×C4 | — | 4 | 1 | no | — |
| 279 | C2×C4 | — | 32 | 4 | no | — |
| 280 | C2×C4 | — | 32 | 2 | no | — |
| 281 | C2×C4 | — | 64 | 8 | no | — |
| 282 | C2×C4 | C4×#331 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #11920 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #11930 | 1 | 2 | yes | — |
| 285 | C2×C4 | #11931 | 1 | 2 | yes | — |
| 286 | C8 | — | 2 | 4 | no | — |
| 287 | C8 | — | 2 | 4 | no | — |
| 288 | C8 | — | 4 | 2 | no | — |
| 289 | C8 | — | 8 | 2 | no | — |
| 290 | C8 | #11951 | 1 | 2 | yes | — |
| 291 | C8 | #11952 | 1 | 2 | yes | — |
| 292 | C8 | #12637 | 1 | 2 | yes | — |
| 293 | C8 | C4×#361 | 1 | 2 | yes | — |
| 294 | C22 | — | 2 | 2 | no | — |
| 295 | C22 | — | 32 | 8 | no | — |
| 296 | C22 | — | 32 | 4 | no | — |
| 297 | C22 | — | 64 | 4 | no | — |
| 298 | C22 | — | 64 | 2 | no | — |
| 299 | C22 | — | 64 | 2 | no | — |
| 300 | C22 | #122887 | 1 | 1 | yes | characteristic |
| 301 | C22 | #122892 | 1 | 1 | yes | characteristic |
| 302 | C22 | C4×#1357 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 2 | 2 | no | — |
| 304 | C4 | — | 2 | 2 | no | — |
| 305 | C4 | — | 4 | 2 | no | — |
| 306 | C4 | — | 8 | 1 | no | — |
| 307 | C4 | — | 64 | 2 | no | — |
| 308 | C4 | #122889 | 1 | 2 | yes | — |
| 309 | C4 | #128900 | 1 | 2 | yes | — |
| 310 | C2 | — | 2 | 1 | no | — |
| 311 | C2 | — | 2 | 1 | no | — |
| 312 | C2 | — | 64 | 4 | no | — |
| 313 | C2 | #72214160 | 1 | 2 | yes | — |
| 314 | C2 | #72216562 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #122887 |
| commutator | C82 | C2×C42 |
| Frattini, cosocle | C2×C8×C16 | C23 |
| prime radical | C2×#12515 | C22 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C42×C8 ◁ C8×#346 ◁ 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 |
|---|---|---|
| 1. #72216560 | C2 | semidirect |
| 3. #72216243 | C2 | semidirect |
| 15. #93866 | C4 | semidirect |
| 40. #7459 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72216560 | 2 |
| C2×#93866 | 2 |
| #72216243 | 2 |
| #72216165 | 2 |