Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
1880 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 288 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 | #72226853 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #72226848 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #72225365 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #72225198 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#123403 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #498562 | — | 2 | 4 | no | — |
| 7 | C2×#12515 | C22 | 1 | 4 | yes | — |
| 8 | C2×#12515 | C22 | 1 | 2 | yes | — |
| 9 | #231561 | C22 | 1 | 4 | yes | — |
| 10 | #141258 | — | 2 | 4 | no | — |
| 11 | #141247 | C22 | 1 | 8 | yes | — |
| 12 | #137536 | — | 2 | 8 | no | — |
| 13 | C8×#367 | C22 | 1 | 2 | yes | — |
| 14 | #129140 | C22 | 1 | 4 | yes | — |
| 15 | #123403 | C4 | 1 | 8 | yes | complemented |
| 16 | C2×#7101 | C22 | 1 | 2 | yes | — |
| 17 | C2×#7096 | C22 | 1 | 4 | yes | — |
| 18 | C2×C162 | C22 | 1 | 1 | yes | characteristic |
| 19 | #92931 | C22 | 1 | 4 | yes | — |
| 20 | C2×#2246 | — | 2 | 4 | no | — |
| 21 | C2×#2246 | — | 2 | 4 | no | — |
| 22 | C2×#2246 | — | 4 | 2 | no | — |
| 23 | C2×#2177 | — | 4 | 4 | no | — |
| 24 | #23185 | — | 2 | 8 | no | — |
| 25 | C2×#1712 | C23 | 1 | 4 | yes | — |
| 26 | #13252 | — | 2 | 8 | no | — |
| 27 | #13239 | — | 4 | 4 | no | — |
| 28 | #13012 | — | 4 | 8 | no | — |
| 29 | C4×#367 | D4 | 1 | 2 | yes | — |
| 30 | #12662 | — | 2 | 4 | no | — |
| 31 | #12515 | — | 2 | 8 | no | — |
| 32 | #12515 | — | 4 | 2 | no | — |
| 33 | #12515 | C2×C4 | 1 | 16 | yes | complemented |
| 34 | #12515 | C2×C4 | 1 | 8 | yes | complemented |
| 35 | C2×#1372 | D4 | 1 | 2 | yes | — |
| 36 | C2×#1370 | C23 | 1 | 4 | yes | — |
| 37 | C2×#1361 | — | 2 | 8 | no | — |
| 38 | C2×C8×C16 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×C8×C16 | C23 | 1 | 2 | yes | — |
| 40 | #9734 | — | 2 | 8 | no | — |
| 41 | #9061 | — | 4 | 4 | no | — |
| 42 | C8×#107 | C23 | 1 | 4 | yes | — |
| 43 | #7171 | D4 | 1 | 4 | yes | — |
| 44 | #7167 | — | 2 | 4 | no | — |
| 45 | #7101 | — | 2 | 4 | no | — |
| 46 | #7096 | — | 4 | 4 | no | — |
| 47 | C162 | C2×C4 | 1 | 4 | yes | complemented |
| 48 | C22×D16 | — | 4 | 4 | no | — |
| 49 | C22×D16 | — | 4 | 4 | no | — |
| 50 | C22×D16 | — | 8 | 4 | no | — |
| 51 | C22×D16 | — | 8 | 4 | no | — |
| 52 | C2×#493 | — | 2 | 8 | no | — |
| 53 | C2×#493 | — | 2 | 4 | no | — |
| 54 | C2×C4×D8 | — | 4 | 8 | no | — |
| 55 | #2366 | — | 4 | 4 | no | — |
| 56 | #2246 | — | 2 | 16 | no | — |
| 57 | #2246 | — | 2 | 16 | no | — |
| 58 | #2246 | — | 4 | 8 | no | — |
| 59 | #2246 | — | 4 | 8 | no | — |
| 60 | #2246 | — | 8 | 4 | no | — |
| 61 | #2246 | — | 8 | 4 | no | — |
| 62 | #2177 | — | 8 | 16 | no | — |
| 63 | C2×#367 | D8 | 1 | 4 | yes | — |
| 64 | C2×C4⋊C16 | — | 2 | 4 | no | — |
| 65 | C2×#357 | — | 4 | 8 | no | — |
| 66 | C2×#357 | — | 4 | 8 | no | — |
| 67 | C2×#346 | — | 4 | 4 | no | — |
| 68 | C2×C4×C16 | — | 2 | 1 | no | — |
| 69 | C2×C4×C16 | C2×D4 | 1 | 2 | yes | — |
| 70 | C2×C4×C16 | C2×D4 | 1 | 2 | yes | — |
| 71 | #1982 | — | 4 | 8 | no | — |
| 72 | #1970 | — | 4 | 8 | no | — |
| 73 | #1879 | — | 8 | 8 | no | — |
| 74 | C4×#107 | C2×D4 | 1 | 4 | yes | — |
| 75 | C8×C4⋊C4 | — | 2 | 4 | no | — |
| 76 | #1712 | — | 2 | 8 | no | — |
| 77 | #1712 | — | 2 | 4 | no | — |
| 78 | #1712 | C22×C4 | 1 | 8 | yes | — |
| 79 | C2×#335 | C2×D4 | 1 | 4 | yes | — |
| 80 | C2×#331 | — | 4 | 4 | no | — |
| 81 | C2×C82 | C24 | 1 | 1 | yes | characteristic |
| 82 | #1380 | D8 | 1 | 4 | yes | — |
| 83 | #1374 | — | 4 | 4 | no | — |
| 84 | #1372 | — | 2 | 4 | no | — |
| 85 | #1370 | — | 2 | 4 | no | — |
| 86 | #1370 | — | 2 | 4 | no | — |
| 87 | #1361 | — | 8 | 8 | no | — |
| 88 | C8×C16 | — | 2 | 1 | no | — |
| 89 | C8×C16 | C22×C4 | 1 | 2 | yes | — |
| 90 | C8×C16 | C22×C4 | 1 | 4 | yes | — |
| 91 | C8×C16 | #41 | 1 | 4 | yes | — |
| 92 | #1279 | — | 2 | 4 | no | — |
| 93 | C22×D8 | — | 4 | 8 | no | — |
| 94 | C22×D8 | — | 4 | 8 | no | — |
| 95 | C22×D8 | — | 8 | 4 | no | — |
| 96 | C2×#127 | — | 4 | 4 | no | — |
| 97 | C2×C4×D4 | — | 8 | 8 | no | — |
| 98 | C2×D16 | — | 4 | 16 | no | — |
| 99 | C2×D16 | — | 4 | 16 | no | — |
| 100 | C2×D16 | — | 8 | 16 | no | — |
| 101 | C2×D16 | — | 8 | 16 | no | — |
| 102 | C2×D16 | — | 8 | 8 | no | — |
| 103 | C2×D16 | — | 8 | 8 | no | — |
| 104 | C2×D16 | — | 16 | 4 | no | — |
| 105 | C2×D16 | — | 16 | 4 | no | — |
| 106 | C2×D16 | — | 16 | 8 | no | — |
| 107 | C2×D16 | — | 16 | 8 | no | — |
| 108 | C2×D16 | — | 16 | 8 | no | — |
| 109 | C2×D16 | — | 16 | 8 | no | — |
| 110 | C22×C16 | — | 2 | 2 | no | — |
| 111 | C22×C16 | — | 2 | 2 | no | — |
| 112 | C22×C16 | — | 4 | 4 | no | — |
| 113 | C22×C16 | C2×D8 | 1 | 2 | yes | — |
| 114 | C22×C16 | C2×D8 | 1 | 2 | yes | — |
| 115 | #493 | — | 2 | 16 | no | — |
| 116 | #493 | — | 2 | 8 | no | — |
| 117 | #493 | — | 4 | 8 | no | — |
| 118 | #493 | — | 4 | 4 | no | — |
| 119 | #493 | — | 4 | 8 | no | — |
| 120 | #493 | — | 4 | 8 | no | — |
| 121 | #493 | — | 8 | 4 | no | — |
| 122 | C4×D8 | — | 8 | 16 | no | — |
| 123 | C4×D8 | — | 16 | 8 | no | — |
| 124 | C2×#107 | C2×D8 | 1 | 8 | yes | — |
| 125 | C2×C4⋊C8 | — | 2 | 4 | no | — |
| 126 | C2×#102 | — | 8 | 8 | no | — |
| 127 | C2×#102 | — | 8 | 8 | no | — |
| 128 | C2×C4×C8 | — | 4 | 4 | no | — |
| 129 | C2×C4×C8 | C22×D4 | 1 | 2 | yes | — |
| 130 | C2×C4×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 131 | #390 | — | 8 | 8 | no | — |
| 132 | #386 | — | 16 | 4 | no | — |
| 133 | C4×C4⋊C4 | — | 2 | 4 | no | — |
| 134 | #367 | — | 2 | 8 | no | — |
| 135 | C4⋊C16 | — | 4 | 4 | no | — |
| 136 | C4⋊C16 | — | 4 | 4 | no | — |
| 137 | #357 | — | 8 | 16 | no | — |
| 138 | #357 | — | 16 | 8 | no | — |
| 139 | #346 | — | 8 | 8 | no | — |
| 140 | C4×C16 | — | 2 | 2 | no | — |
| 141 | C4×C16 | — | 2 | 2 | no | — |
| 142 | C4×C16 | — | 2 | 2 | no | — |
| 143 | C4×C16 | C4×D4 | 1 | 4 | yes | — |
| 144 | C4×C16 | C4×D4 | 1 | 4 | yes | — |
| 145 | C4×C16 | #135 | 1 | 4 | yes | — |
| 146 | #340 | — | 2 | 4 | no | — |
| 147 | #336 | — | 4 | 4 | no | — |
| 148 | #335 | — | 2 | 4 | no | — |
| 149 | #335 | — | 2 | 4 | no | — |
| 150 | #331 | — | 16 | 4 | no | — |
| 151 | C82 | C23×C4 | 1 | 1 | yes | characteristic |
| 152 | C82 | C2×#41 | 1 | 2 | yes | — |
| 153 | C82 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 154 | C22×D4 | — | 8 | 8 | no | — |
| 155 | C22×D4 | — | 8 | 4 | no | — |
| 156 | C23×C4 | — | 16 | 4 | no | — |
| 157 | C2×D8 | — | 4 | 16 | no | — |
| 158 | C2×D8 | — | 4 | 16 | no | — |
| 159 | C2×D8 | — | 8 | 8 | no | — |
| 160 | C2×D8 | — | 8 | 8 | no | — |
| 161 | C2×D8 | — | 8 | 8 | no | — |
| 162 | C2×D8 | — | 8 | 8 | no | — |
| 163 | C2×D8 | — | 8 | 8 | no | — |
| 164 | C2×D8 | — | 8 | 8 | no | — |
| 165 | C2×D8 | — | 8 | 8 | no | — |
| 166 | C2×D8 | — | 16 | 8 | no | — |
| 167 | C2×D8 | — | 16 | 8 | no | — |
| 168 | C2×D8 | — | 16 | 8 | no | — |
| 169 | C2×D8 | — | 16 | 8 | no | — |
| 170 | C2×D8 | — | 16 | 4 | no | — |
| 171 | C22×C8 | — | 2 | 1 | no | — |
| 172 | C22×C8 | — | 4 | 4 | no | — |
| 173 | C22×C8 | C22×D8 | 1 | 2 | yes | — |
| 174 | C22×C8 | #576 | 1 | 2 | yes | — |
| 175 | #127 | — | 4 | 8 | no | — |
| 176 | #127 | — | 8 | 8 | no | — |
| 177 | #127 | — | 8 | 4 | no | — |
| 178 | C4×D4 | — | 16 | 16 | no | — |
| 179 | C4×D4 | — | 32 | 8 | no | — |
| 180 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 181 | C2×C22⋊C4 | — | 16 | 8 | no | — |
| 182 | C2×C22⋊C4 | — | 16 | 4 | no | — |
| 183 | C2×C42 | — | 4 | 4 | no | — |
| 184 | C2×C42 | D42 | 1 | 1 | yes | characteristic |
| 185 | D16 | — | 8 | 32 | no | — |
| 186 | D16 | — | 16 | 16 | no | — |
| 187 | D16 | — | 16 | 32 | no | — |
| 188 | D16 | — | 16 | 32 | no | — |
| 189 | C2×C16 | — | 2 | 4 | no | — |
| 190 | C2×C16 | — | 2 | 8 | no | — |
| 191 | C2×C16 | — | 2 | 8 | no | — |
| 192 | C2×C16 | — | 2 | 4 | no | — |
| 193 | C2×C16 | — | 2 | 4 | no | — |
| 194 | C2×C16 | — | 8 | 4 | no | — |
| 195 | C2×C16 | — | 8 | 4 | no | — |
| 196 | C2×C16 | — | 8 | 4 | no | — |
| 197 | C2×C16 | C4×D8 | 1 | 4 | yes | — |
| 198 | C2×C16 | C4×D8 | 1 | 4 | yes | — |
| 199 | C2×C16 | #509 | 1 | 8 | yes | — |
| 200 | #107 | — | 2 | 8 | no | — |
| 201 | #107 | — | 2 | 8 | no | — |
| 202 | C4⋊C8 | — | 4 | 4 | no | — |
| 203 | C4⋊C8 | — | 4 | 4 | no | — |
| 204 | #102 | — | 16 | 16 | no | — |
| 205 | #102 | — | 32 | 8 | no | — |
| 206 | C4×C8 | — | 2 | 1 | no | — |
| 207 | C4×C8 | — | 8 | 4 | no | — |
| 208 | C4×C8 | — | 8 | 4 | no | — |
| 209 | C4×C8 | C2×C4×D4 | 1 | 2 | yes | — |
| 210 | C4×C8 | #518 | 1 | 1 | yes | characteristic |
| 211 | C4×C8 | #550 | 1 | 2 | yes | — |
| 212 | C4×C8 | #559 | 1 | 1 | yes | characteristic |
| 213 | C4×C8 | C2×#135 | 1 | 2 | yes | — |
| 214 | C4×C8 | #576 | 1 | 2 | yes | — |
| 215 | #95 | — | 4 | 4 | no | — |
| 216 | C24 | — | 16 | 4 | no | — |
| 217 | C2×D4 | — | 8 | 16 | no | — |
| 218 | C2×D4 | — | 8 | 8 | no | — |
| 219 | C2×D4 | — | 16 | 8 | no | — |
| 220 | C2×D4 | — | 16 | 8 | no | — |
| 221 | C2×D4 | — | 16 | 8 | no | — |
| 222 | C2×D4 | — | 16 | 4 | no | — |
| 223 | C2×D4 | — | 16 | 8 | no | — |
| 224 | C2×D4 | — | 16 | 8 | no | — |
| 225 | C2×D4 | — | 32 | 4 | no | — |
| 226 | C2×D4 | — | 32 | 4 | no | — |
| 227 | C22×C4 | — | 4 | 8 | no | — |
| 228 | C22×C4 | — | 32 | 16 | no | — |
| 229 | C22×C4 | — | 64 | 4 | no | — |
| 230 | C22×C4 | D4×D8 | 1 | 2 | yes | — |
| 231 | C22×C4 | #3292 | 1 | 1 | yes | characteristic |
| 232 | D8 | — | 8 | 16 | no | — |
| 233 | D8 | — | 8 | 16 | no | — |
| 234 | D8 | — | 16 | 16 | no | — |
| 235 | D8 | — | 16 | 16 | no | — |
| 236 | D8 | — | 16 | 8 | no | — |
| 237 | D8 | — | 16 | 8 | no | — |
| 238 | D8 | — | 32 | 8 | no | — |
| 239 | D8 | — | 32 | 8 | no | — |
| 240 | C2×C8 | — | 2 | 4 | no | — |
| 241 | C2×C8 | — | 2 | 4 | no | — |
| 242 | C2×C8 | — | 2 | 1 | no | — |
| 243 | C2×C8 | — | 2 | 1 | no | — |
| 244 | C2×C8 | — | 8 | 4 | no | — |
| 245 | C2×C8 | — | 8 | 4 | no | — |
| 246 | C2×C8 | — | 8 | 4 | no | — |
| 247 | C2×C8 | C2×C4×D8 | 1 | 2 | yes | — |
| 248 | C2×C8 | #2948 | 1 | 2 | yes | — |
| 249 | C2×C8 | #3391 | 1 | 2 | yes | — |
| 250 | C2×C8 | #3396 | 1 | 2 | yes | — |
| 251 | C2×C8 | C2×#509 | 1 | 4 | yes | — |
| 252 | C2×C8 | #3415 | 1 | 4 | yes | — |
| 253 | C4⋊C4 | — | 4 | 8 | no | — |
| 254 | C4⋊C4 | — | 4 | 8 | no | — |
| 255 | C22⋊C4 | — | 32 | 16 | no | — |
| 256 | C22⋊C4 | — | 64 | 4 | no | — |
| 257 | C42 | — | 8 | 4 | no | — |
| 258 | C42 | — | 8 | 4 | no | — |
| 259 | C42 | #2366 | 1 | 1 | yes | characteristic |
| 260 | C42 | #3287 | 1 | 2 | yes | — |
| 261 | C42 | #3292 | 1 | 1 | yes | characteristic |
| 262 | C16 | — | 2 | 8 | no | — |
| 263 | C16 | — | 2 | 8 | no | — |
| 264 | C16 | — | 2 | 4 | no | — |
| 265 | C16 | — | 16 | 4 | no | — |
| 266 | C16 | #2177 | 1 | 8 | yes | complemented |
| 267 | C23 | — | 16 | 8 | no | — |
| 268 | C23 | — | 32 | 8 | no | — |
| 269 | C23 | — | 32 | 8 | no | — |
| 270 | C23 | — | 32 | 4 | no | — |
| 271 | C23 | — | 32 | 4 | no | — |
| 272 | C23 | D82 | 1 | 1 | yes | characteristic |
| 273 | D4 | — | 16 | 16 | no | — |
| 274 | D4 | — | 16 | 16 | no | — |
| 275 | D4 | — | 32 | 8 | no | — |
| 276 | D4 | — | 32 | 8 | no | — |
| 277 | D4 | — | 32 | 8 | no | — |
| 278 | D4 | — | 64 | 4 | no | — |
| 279 | C2×C4 | — | 2 | 2 | no | — |
| 280 | C2×C4 | — | 8 | 8 | no | — |
| 281 | C2×C4 | — | 8 | 8 | no | — |
| 282 | C2×C4 | — | 8 | 8 | no | — |
| 283 | C2×C4 | — | 64 | 16 | no | — |
| 284 | C2×C4 | #19975 | 1 | 2 | yes | — |
| 285 | C2×C4 | #20021 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #23185 | 1 | 2 | yes | — |
| 287 | C2×C4 | #23201 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | #33840 | 1 | 4 | yes | — |
| 289 | C2×C4 | #33858 | 1 | 4 | yes | — |
| 290 | C8 | — | 2 | 2 | no | — |
| 291 | C8 | — | 2 | 2 | no | — |
| 292 | C8 | — | 2 | 2 | no | — |
| 293 | C8 | — | 2 | 2 | no | — |
| 294 | C8 | — | 16 | 4 | no | — |
| 295 | C8 | C2×#2177 | 1 | 2 | yes | — |
| 296 | C8 | #33456 | 1 | 2 | yes | — |
| 297 | C8 | #33966 | 1 | 4 | yes | — |
| 298 | C22 | — | 32 | 16 | no | — |
| 299 | C22 | — | 32 | 8 | no | — |
| 300 | C22 | — | 64 | 4 | no | — |
| 301 | C22 | — | 64 | 8 | no | — |
| 302 | C22 | — | 64 | 4 | no | — |
| 303 | C22 | #231561 | 1 | 1 | yes | characteristic |
| 304 | C22 | #492514 | 1 | 4 | yes | — |
| 305 | C22 | #492532 | 1 | 2 | yes | — |
| 306 | C4 | — | 2 | 1 | no | — |
| 307 | C4 | — | 2 | 1 | no | — |
| 308 | C4 | — | 16 | 8 | no | — |
| 309 | C4 | #492817 | 1 | 4 | yes | — |
| 310 | C4 | #498562 | 1 | 2 | yes | — |
| 311 | C4 | #498572 | 1 | 2 | yes | — |
| 312 | C2 | — | 64 | 8 | no | — |
| 313 | C2 | #72216416 | 1 | 1 | yes | characteristic |
| 314 | C2 | #72226848 | 1 | 2 | yes | — |
| 315 | C2 | #72226856 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D82 |
| commutator | C82 | C23×C4 |
| Frattini, cosocle | C2×C82 | C24 |
| prime radical | C2×#123403 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72226853 | C2 | semidirect |
| 2. #72226848 | C2 | semidirect |
| 3. #72225365 | C2 | semidirect |
| 4. #72225198 | C2 | semidirect |
| 15. #123403 | C4 | semidirect |
| 33. #12515 | C2×C4 | semidirect |
| 34. #12515 | C2×C4 | semidirect |
| 47. C162 | C2×C4 | semidirect |
| 266. C16 | #2177 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72226853 | 2 |
| #72226848 | 2 |
| #72225365 | 2 |
| #72225198 | 2 |
| C2×#123403 | 2 |