Finite Group DatabasebetaFinite Group DatabasebetaOrder 4096.
1365 conjugacy classes of subgroups; only the 295 normal subgroups are shown.
This group has 1365 conjugacy classes of subgroups — too many to draw. Only its 295 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 | #72225076 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72225071 | — | 2 | 2 | no | — |
| 3 | #72216382 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #72216379 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72216298 | — | 2 | 2 | no | — |
| 6 | #72216291 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #72216290 | — | 2 | 4 | no | — |
| 8 | #72216283 | — | 2 | 4 | no | — |
| 9 | #72216278 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #72216272 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #72216268 | C4 | 1 | 2 | yes | complemented |
| 12 | #72216265 | — | 2 | 2 | no | — |
| 13 | #72216263 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #72216261 | — | 2 | 2 | no | — |
| 15 | #72216239 | C22 | 1 | 1 | yes | characteristic |
| 16 | #72216236 | C22 | 1 | 2 | yes | — |
| 17 | #72216220 | C22 | 1 | 2 | yes | — |
| 18 | C16×M64 | C22 | 1 | 1 | yes | characteristic |
| 19 | #72216197 | C22 | 1 | 1 | yes | characteristic |
| 20 | #513082 | — | 4 | 4 | no | — |
| 21 | #491485 | — | 2 | 2 | no | — |
| 22 | #491485 | — | 2 | 2 | no | — |
| 23 | #491465 | — | 2 | 2 | no | — |
| 24 | #152896 | — | 4 | 8 | no | — |
| 25 | #152407 | — | 4 | 4 | no | — |
| 26 | #151732 | — | 2 | 2 | no | — |
| 27 | #151724 | — | 2 | 2 | no | — |
| 28 | #151718 | — | 2 | 4 | no | — |
| 29 | #151715 | — | 4 | 4 | no | — |
| 30 | #150160 | — | 4 | 8 | no | — |
| 31 | C8×M64 | D4 | 1 | 4 | yes | — |
| 32 | C8×M64 | D4 | 1 | 2 | yes | — |
| 33 | #135796 | — | 2 | 2 | no | — |
| 34 | #135781 | — | 2 | 1 | no | — |
| 35 | #135778 | — | 2 | 2 | no | — |
| 36 | #135778 | — | 2 | 1 | no | — |
| 37 | #123406 | — | 2 | 2 | no | — |
| 38 | #123405 | — | 2 | 2 | no | — |
| 39 | #123405 | — | 2 | 2 | no | — |
| 40 | #123404 | C2×C4 | 1 | 2 | yes | — |
| 41 | #123403 | C2×C4 | 1 | 2 | yes | — |
| 42 | C2×C162 | C23 | 1 | 1 | yes | characteristic |
| 43 | #93665 | — | 2 | 2 | no | — |
| 44 | #93665 | — | 2 | 2 | no | — |
| 45 | #93664 | D4 | 1 | 2 | yes | — |
| 46 | C16×C32 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C16×C32 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | #33733 | — | 4 | 4 | no | — |
| 49 | #33733 | — | 4 | 4 | no | — |
| 50 | #33713 | — | 4 | 4 | no | — |
| 51 | #19603 | — | 4 | 2 | no | — |
| 52 | #13733 | — | 4 | 4 | no | — |
| 53 | #13725 | — | 4 | 4 | no | — |
| 54 | #13719 | — | 4 | 8 | no | — |
| 55 | #13697 | — | 8 | 8 | no | — |
| 56 | #13654 | — | 8 | 8 | no | — |
| 57 | C2×#1426 | — | 8 | 8 | no | — |
| 58 | C2×#1418 | — | 8 | 16 | no | — |
| 59 | C4×M64 | D8 | 1 | 8 | yes | — |
| 60 | C4×M64 | D8 | 1 | 4 | yes | — |
| 61 | #12909 | — | 4 | 2 | no | — |
| 62 | #12906 | — | 4 | 4 | no | — |
| 63 | #12906 | — | 4 | 2 | no | — |
| 64 | #12521 | — | 2 | 2 | no | — |
| 65 | #12521 | — | 2 | 2 | no | — |
| 66 | #12518 | — | 4 | 4 | no | — |
| 67 | #12517 | — | 4 | 2 | no | — |
| 68 | #12516 | — | 2 | 2 | no | — |
| 69 | #12515 | — | 2 | 2 | no | — |
| 70 | #12515 | — | 2 | 2 | no | — |
| 71 | #12515 | — | 2 | 2 | no | — |
| 72 | #12509 | — | 4 | 2 | no | — |
| 73 | #12509 | — | 4 | 2 | no | — |
| 74 | #12503 | — | 4 | 4 | no | — |
| 75 | #12503 | — | 4 | 4 | no | — |
| 76 | #12023 | — | 4 | 4 | no | — |
| 77 | C2×C8×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C2×C8×C16 | C2×D4 | 1 | 2 | yes | — |
| 79 | #8823 | — | 4 | 1 | no | — |
| 80 | #8818 | — | 4 | 1 | no | — |
| 81 | #7385 | — | 8 | 16 | no | — |
| 82 | #7329 | — | 2 | 4 | no | — |
| 83 | #7329 | SD16 | 1 | 4 | yes | — |
| 84 | C8×C32 | — | 2 | 4 | no | — |
| 85 | C8×C32 | C22⋊C4 | 1 | 2 | yes | — |
| 86 | C8×C32 | C22⋊C4 | 1 | 2 | yes | — |
| 87 | C162 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C162 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C162 | #41 | 1 | 2 | yes | — |
| 90 | C2×#508 | — | 8 | 8 | no | — |
| 91 | #3169 | — | 8 | 2 | no | — |
| 92 | #3169 | — | 8 | 2 | no | — |
| 93 | #3147 | — | 8 | 2 | no | — |
| 94 | C2×D32 | — | 8 | 8 | no | — |
| 95 | C2×M64 | — | 2 | 8 | no | — |
| 96 | C2×M64 | D16 | 1 | 8 | yes | — |
| 97 | #2252 | — | 4 | 4 | no | — |
| 98 | #2252 | — | 4 | 4 | no | — |
| 99 | #2249 | — | 8 | 8 | no | — |
| 100 | #2248 | — | 8 | 4 | no | — |
| 101 | #2247 | — | 4 | 4 | no | — |
| 102 | #2246 | — | 4 | 4 | no | — |
| 103 | #2246 | — | 4 | 4 | no | — |
| 104 | #2246 | — | 4 | 4 | no | — |
| 105 | #2240 | — | 8 | 4 | no | — |
| 106 | #2240 | — | 8 | 4 | no | — |
| 107 | #2215 | — | 8 | 8 | no | — |
| 108 | #2215 | — | 8 | 8 | no | — |
| 109 | C4×D16 | — | 8 | 8 | no | — |
| 110 | C4×D16 | — | 8 | 8 | no | — |
| 111 | C2×#368 | — | 8 | 4 | no | — |
| 112 | C2×#367 | — | 8 | 4 | no | — |
| 113 | C2×#367 | — | 8 | 8 | no | — |
| 114 | C2×C4×C16 | C2×D8 | 1 | 2 | yes | — |
| 115 | C2×C4×C16 | C2×D8 | 1 | 4 | yes | — |
| 116 | #1848 | — | 8 | 1 | no | — |
| 117 | #1845 | — | 8 | 1 | no | — |
| 118 | #1845 | — | 8 | 2 | no | — |
| 119 | #1713 | — | 4 | 2 | no | — |
| 120 | #1712 | — | 4 | 2 | no | — |
| 121 | #1709 | — | 8 | 2 | no | — |
| 122 | #1706 | — | 8 | 2 | no | — |
| 123 | #1706 | — | 8 | 2 | no | — |
| 124 | #1567 | — | 8 | 2 | no | — |
| 125 | C2×C82 | #127 | 1 | 1 | yes | characteristic |
| 126 | #1426 | — | 16 | 16 | no | — |
| 127 | #1424 | — | 8 | 8 | no | — |
| 128 | #1418 | — | 16 | 32 | no | — |
| 129 | #1415 | — | 8 | 16 | no | — |
| 130 | #1397 | — | 2 | 8 | no | — |
| 131 | #1397 | SD32 | 1 | 8 | yes | — |
| 132 | C4×C32 | — | 2 | 8 | no | — |
| 133 | C4×C32 | #102 | 1 | 4 | yes | — |
| 134 | C4×C32 | #102 | 1 | 4 | yes | — |
| 135 | #1372 | — | 8 | 8 | no | — |
| 136 | C8×C16 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 137 | C8×C16 | C4×D4 | 1 | 2 | yes | — |
| 138 | C8×C16 | #121 | 1 | 2 | yes | — |
| 139 | C8×C16 | #124 | 1 | 1 | yes | characteristic |
| 140 | C8×C16 | C2×SD16 | 1 | 2 | yes | — |
| 141 | C8×C16 | #135 | 1 | 4 | yes | — |
| 142 | C2×#135 | — | 16 | 4 | no | — |
| 143 | #532 | — | 16 | 2 | no | — |
| 144 | #508 | — | 16 | 16 | no | — |
| 145 | #508 | — | 16 | 16 | no | — |
| 146 | C2×Q32 | — | 8 | 8 | no | — |
| 147 | C2×SD32 | — | 16 | 8 | no | — |
| 148 | C2×D16 | — | 8 | 8 | no | — |
| 149 | C2×D16 | — | 8 | 8 | no | — |
| 150 | C2×D16 | — | 8 | 8 | no | — |
| 151 | C22×C16 | C2×D16 | 1 | 4 | yes | — |
| 152 | C22×C16 | #509 | 1 | 8 | yes | — |
| 153 | #500 | — | 8 | 2 | no | — |
| 154 | #500 | — | 8 | 2 | no | — |
| 155 | #495 | — | 16 | 4 | no | — |
| 156 | #494 | — | 8 | 2 | no | — |
| 157 | #493 | — | 8 | 2 | no | — |
| 158 | #493 | — | 8 | 2 | no | — |
| 159 | #493 | — | 8 | 2 | no | — |
| 160 | #492 | — | 16 | 2 | no | — |
| 161 | #486 | — | 16 | 2 | no | — |
| 162 | #486 | — | 16 | 2 | no | — |
| 163 | #466 | — | 16 | 4 | no | — |
| 164 | #466 | — | 16 | 4 | no | — |
| 165 | C4×D8 | — | 16 | 4 | no | — |
| 166 | C4×D8 | — | 16 | 4 | no | — |
| 167 | C2×#108 | — | 16 | 2 | no | — |
| 168 | C2×#107 | — | 16 | 2 | no | — |
| 169 | C2×#107 | — | 16 | 4 | no | — |
| 170 | #423 | — | 16 | 1 | no | — |
| 171 | C2×C4×C8 | #493 | 1 | 1 | yes | characteristic |
| 172 | C2×C4×C8 | #493 | 1 | 2 | yes | — |
| 173 | #384 | — | 16 | 1 | no | — |
| 174 | D32 | — | 16 | 16 | no | — |
| 175 | M64 | — | 2 | 16 | no | — |
| 176 | M64 | — | 4 | 16 | no | — |
| 177 | C2×C32 | — | 4 | 8 | no | — |
| 178 | C2×C32 | #357 | 1 | 8 | yes | — |
| 179 | C2×C32 | #357 | 1 | 8 | yes | — |
| 180 | #368 | — | 16 | 8 | no | — |
| 181 | #367 | — | 16 | 16 | no | — |
| 182 | #367 | — | 16 | 8 | no | — |
| 183 | #366 | — | 8 | 8 | no | — |
| 184 | #366 | — | 8 | 8 | no | — |
| 185 | #357 | — | 16 | 8 | no | — |
| 186 | #357 | — | 16 | 8 | no | — |
| 187 | C4×C16 | C2×#102 | 1 | 2 | yes | — |
| 188 | C4×C16 | C4×D8 | 1 | 4 | yes | — |
| 189 | C4×C16 | #466 | 1 | 4 | yes | — |
| 190 | C4×C16 | #486 | 1 | 2 | yes | — |
| 191 | C4×C16 | C2×SD32 | 1 | 4 | yes | — |
| 192 | C4×C16 | #509 | 1 | 8 | yes | — |
| 193 | #335 | — | 16 | 4 | no | — |
| 194 | C82 | #390 | 1 | 1 | yes | characteristic |
| 195 | C82 | #492 | 1 | 1 | yes | characteristic |
| 196 | C82 | #495 | 1 | 2 | yes | — |
| 197 | C2×#41 | — | 32 | 2 | no | — |
| 198 | #135 | — | 32 | 8 | no | — |
| 199 | #135 | — | 32 | 8 | no | — |
| 200 | C2×Q16 | — | 16 | 4 | no | — |
| 201 | C2×SD16 | — | 32 | 4 | no | — |
| 202 | C2×D8 | — | 16 | 4 | no | — |
| 203 | C2×D8 | — | 16 | 4 | no | — |
| 204 | C2×D8 | — | 16 | 4 | no | — |
| 205 | C2×M16 | — | 32 | 1 | no | — |
| 206 | C22×C8 | #2246 | 1 | 2 | yes | — |
| 207 | C22×C8 | #2250 | 1 | 4 | yes | — |
| 208 | C4⋊Q8 | — | 16 | 2 | no | — |
| 209 | #127 | — | 16 | 2 | no | — |
| 210 | #124 | — | 32 | 2 | no | — |
| 211 | #121 | — | 32 | 2 | no | — |
| 212 | #121 | — | 32 | 2 | no | — |
| 213 | C4×D4 | — | 32 | 2 | no | — |
| 214 | C4×D4 | — | 32 | 2 | no | — |
| 215 | C2×C4⋊C4 | — | 32 | 1 | no | — |
| 216 | C2×C4⋊C4 | — | 32 | 2 | no | — |
| 217 | C2×C42 | #1712 | 1 | 1 | yes | characteristic |
| 218 | Q32 | — | 16 | 16 | no | — |
| 219 | SD32 | — | 32 | 16 | no | — |
| 220 | D16 | — | 16 | 16 | no | — |
| 221 | D16 | — | 16 | 16 | no | — |
| 222 | D16 | — | 16 | 16 | no | — |
| 223 | C2×C16 | — | 2 | 8 | no | — |
| 224 | C2×C16 | — | 2 | 16 | no | — |
| 225 | C2×C16 | C2×#357 | 1 | 4 | yes | — |
| 226 | C2×C16 | #2177 | 1 | 8 | yes | — |
| 227 | C2×C16 | #2218 | 1 | 8 | yes | — |
| 228 | C2×C16 | #2240 | 1 | 4 | yes | — |
| 229 | #108 | — | 32 | 4 | no | — |
| 230 | #107 | — | 16 | 4 | no | — |
| 231 | #107 | — | 16 | 4 | no | — |
| 232 | #107 | — | 32 | 4 | no | — |
| 233 | #107 | — | 32 | 8 | no | — |
| 234 | C4⋊C8 | — | 32 | 2 | no | — |
| 235 | #102 | — | 32 | 4 | no | — |
| 236 | #102 | — | 32 | 4 | no | — |
| 237 | C4×C8 | #1706 | 1 | 2 | yes | — |
| 238 | C4×C8 | #1709 | 1 | 1 | yes | characteristic |
| 239 | C4×C8 | #1957 | 1 | 1 | yes | characteristic |
| 240 | C4×C8 | #1970 | 1 | 2 | yes | — |
| 241 | C4×C8 | #2248 | 1 | 2 | yes | — |
| 242 | C4×C8 | #2250 | 1 | 4 | yes | — |
| 243 | C32 | — | 2 | 8 | no | — |
| 244 | C32 | — | 8 | 8 | no | — |
| 245 | C32 | #1418 | 1 | 16 | yes | complemented |
| 246 | #41 | — | 64 | 4 | no | — |
| 247 | #41 | — | 64 | 4 | no | — |
| 248 | C2×Q8 | — | 32 | 2 | no | — |
| 249 | C2×D4 | — | 32 | 2 | no | — |
| 250 | C2×D4 | — | 32 | 2 | no | — |
| 251 | C2×D4 | — | 32 | 2 | no | — |
| 252 | C2×D4 | — | 64 | 2 | no | — |
| 253 | C22×C4 | — | 64 | 2 | no | — |
| 254 | C22×C4 | — | 64 | 1 | no | — |
| 255 | C22×C4 | #12515 | 1 | 1 | yes | characteristic |
| 256 | C22×C4 | #12519 | 1 | 2 | yes | — |
| 257 | Q16 | — | 32 | 8 | no | — |
| 258 | SD16 | — | 64 | 8 | no | — |
| 259 | D8 | — | 32 | 8 | no | — |
| 260 | D8 | — | 32 | 8 | no | — |
| 261 | D8 | — | 32 | 8 | no | — |
| 262 | M16 | — | 64 | 2 | no | — |
| 263 | C2×C8 | — | 2 | 4 | no | — |
| 264 | C2×C8 | — | 2 | 8 | no | — |
| 265 | C2×C8 | — | 64 | 1 | no | — |
| 266 | C2×C8 | #12506 | 1 | 4 | yes | — |
| 267 | C2×C8 | #12509 | 1 | 2 | yes | — |
| 268 | C2×C8 | #13202 | 1 | 2 | yes | — |
| 269 | C2×C8 | #13239 | 1 | 4 | yes | — |
| 270 | C4⋊C4 | — | 32 | 2 | no | — |
| 271 | C4⋊C4 | — | 32 | 2 | no | — |
| 272 | C4⋊C4 | — | 64 | 4 | no | — |
| 273 | C4⋊C4 | — | 64 | 2 | no | — |
| 274 | C22⋊C4 | — | 64 | 2 | no | — |
| 275 | C22⋊C4 | — | 64 | 2 | no | — |
| 276 | C42 | #9824 | 1 | 1 | yes | characteristic |
| 277 | C42 | #12517 | 1 | 1 | yes | characteristic |
| 278 | C42 | #12519 | 1 | 2 | yes | — |
| 279 | C16 | — | 2 | 4 | no | — |
| 280 | C16 | — | 2 | 8 | no | — |
| 281 | C16 | C2×#1418 | 1 | 8 | yes | — |
| 282 | C16 | #13658 | 1 | 16 | yes | — |
| 283 | C23 | — | 64 | 2 | no | — |
| 284 | C23 | #123412 | 1 | 1 | yes | characteristic |
| 285 | Q8 | — | 64 | 4 | no | — |
| 286 | D4 | — | 64 | 4 | no | — |
| 287 | D4 | — | 64 | 4 | no | — |
| 288 | D4 | — | 64 | 4 | no | — |
| 289 | D4 | — | 128 | 4 | no | — |
| 290 | C2×C4 | — | 2 | 4 | no | — |
| 291 | C2×C4 | — | 2 | 2 | no | — |
| 292 | C2×C4 | — | 64 | 2 | no | — |
| 293 | C2×C4 | — | 128 | 1 | no | — |
| 294 | C2×C4 | — | 128 | 2 | no | — |
| 295 | C2×C4 | — | 128 | 2 | no | — |
| 296 | C2×C4 | — | 128 | 2 | no | — |
| 297 | C2×C4 | #123412 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #123419 | 1 | 2 | yes | — |
| 299 | C2×C4 | #141665 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #141681 | 1 | 2 | yes | — |
| 301 | C8 | — | 2 | 4 | no | — |
| 302 | C8 | — | 2 | 2 | no | — |
| 303 | C8 | — | 128 | 1 | no | — |
| 304 | C8 | #152896 | 1 | 4 | yes | — |
| 305 | C8 | #152984 | 1 | 8 | yes | — |
| 306 | C22 | — | 2 | 1 | no | — |
| 307 | C22 | — | 2 | 2 | no | — |
| 308 | C22 | — | 128 | 2 | no | — |
| 309 | C22 | — | 128 | 2 | no | — |
| 310 | C22 | — | 128 | 2 | no | — |
| 311 | C22 | #72216196 | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 256 | 1 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 2 | 2 | no | — |
| 315 | C4 | — | 128 | 2 | no | — |
| 316 | C4 | #72216290 | 1 | 2 | yes | — |
| 317 | C4 | #72216324 | 1 | 4 | yes | — |
| 318 | C2 | — | 256 | 2 | no | — |
| 319 | C2 | — | 4 | 1 | no | — |
| 320 | C2 | #72216382 | 1 | 1 | yes | characteristic |
| 321 | C2 | #72216385 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #72216196 |
| commutator | C162 | C22×C4 |
| Frattini, cosocle | C2×C162 | C23 |
| prime radical | #72216278 | C2 |
| LCS | G ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G | |
| LECS | G ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72225076 | C2 | semidirect |
| 3. #72216382 | C2 | semidirect |
| 4. #72216379 | C2 | semidirect |
| 6. #72216291 | C2 | semidirect |
| 10. #72216272 | C4 | semidirect |
| 11. #72216268 | C4 | semidirect |
| 13. #72216263 | C4 | semidirect |
| 245. C32 | #1418 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72225076 | 2 |
| #72216382 | 2 |
| #72216379 | 2 |
| #72216291 | 2 |
| #72216278 | 2 |