Finite Group DatabasebetaFinite Group DatabasebetaOrder 4096.
592 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 87 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 | #72227191 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #72225395 | — | 2 | 2 | no | — |
| 3 | #72216382 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #72216290 | — | 2 | 2 | no | — |
| 5 | #72216236 | C4 | 1 | 1 | yes | characteristic |
| 6 | #72216220 | C22 | 1 | 1 | yes | characteristic |
| 7 | #72216197 | C4 | 1 | 1 | yes | characteristic |
| 8 | #491485 | — | 4 | 1 | no | — |
| 9 | #491465 | — | 2 | 1 | no | — |
| 10 | #153582 | — | 4 | 2 | no | — |
| 11 | #152896 | — | 4 | 2 | no | — |
| 12 | #152896 | — | 4 | 2 | no | — |
| 13 | C8×M64 | D4 | 1 | 2 | yes | — |
| 14 | #135778 | — | 2 | 1 | no | — |
| 15 | #135778 | — | 4 | 1 | no | — |
| 16 | #123406 | — | 2 | 1 | no | — |
| 17 | #123405 | — | 4 | 1 | no | — |
| 18 | #123404 | C8 | 1 | 1 | yes | characteristic |
| 19 | #123403 | C8 | 1 | 1 | yes | characteristic |
| 20 | C2×C162 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | #94748 | — | 4 | 2 | no | — |
| 22 | #94746 | — | 2 | 4 | no | — |
| 23 | #93665 | — | 4 | 1 | no | — |
| 24 | #33733 | — | 8 | 2 | no | — |
| 25 | #33713 | — | 4 | 2 | no | — |
| 26 | #19603 | — | 4 | 1 | no | — |
| 27 | C2×#1418 | — | 8 | 4 | no | — |
| 28 | C2×#1418 | — | 8 | 2 | no | — |
| 29 | C2×#1418 | — | 8 | 2 | no | — |
| 30 | C4×M64 | D8 | 1 | 2 | yes | — |
| 31 | C4×M64 | SD16 | 1 | 2 | yes | — |
| 32 | #12906 | — | 4 | 2 | no | — |
| 33 | #12906 | — | 8 | 2 | no | — |
| 34 | #12521 | — | 4 | 1 | no | — |
| 35 | #12518 | — | 4 | 2 | no | — |
| 36 | #12517 | — | 4 | 1 | no | — |
| 37 | #12516 | — | 2 | 1 | no | — |
| 38 | #12515 | — | 2 | 1 | no | — |
| 39 | #12515 | — | 4 | 1 | no | — |
| 40 | #12509 | — | 4 | 2 | no | — |
| 41 | #12503 | — | 8 | 2 | no | — |
| 42 | C2×C8×C16 | — | 2 | 1 | no | — |
| 43 | C2×C8×C16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 44 | #8818 | — | 4 | 1 | no | — |
| 45 | #7544 | — | 4 | 4 | no | — |
| 46 | #7513 | — | 8 | 2 | no | — |
| 47 | #7511 | — | 8 | 2 | no | — |
| 48 | #7329 | — | 2 | 2 | no | — |
| 49 | C8×C32 | — | 2 | 2 | no | — |
| 50 | C162 | — | 2 | 1 | no | — |
| 51 | C162 | C2×C8 | 1 | 1 | yes | characteristic |
| 52 | C162 | M16 | 1 | 1 | yes | characteristic |
| 53 | C2×#508 | — | 8 | 2 | no | — |
| 54 | C2×#508 | — | 8 | 2 | no | — |
| 55 | #3169 | — | 8 | 2 | no | — |
| 56 | #3147 | — | 8 | 1 | no | — |
| 57 | C2×M64 | — | 2 | 2 | no | — |
| 58 | C2×M64 | D16 | 1 | 2 | yes | — |
| 59 | C2×M64 | SD32 | 1 | 2 | yes | — |
| 60 | #2252 | — | 8 | 2 | no | — |
| 61 | #2249 | — | 8 | 2 | no | — |
| 62 | #2249 | — | 8 | 2 | no | — |
| 63 | #2248 | — | 8 | 2 | no | — |
| 64 | #2247 | — | 4 | 2 | no | — |
| 65 | #2246 | — | 4 | 2 | no | — |
| 66 | #2246 | — | 8 | 2 | no | — |
| 67 | #2240 | — | 8 | 4 | no | — |
| 68 | #2215 | — | 16 | 4 | no | — |
| 69 | C4×D16 | — | 16 | 4 | no | — |
| 70 | C2×#367 | — | 8 | 2 | no | — |
| 71 | C2×#367 | — | 8 | 2 | no | — |
| 72 | C2×#367 | — | 16 | 4 | no | — |
| 73 | C2×C4×C16 | — | 2 | 2 | no | — |
| 74 | C2×C4×C16 | #102 | 1 | 2 | yes | — |
| 75 | #1845 | — | 8 | 2 | no | — |
| 76 | #1845 | — | 8 | 1 | no | — |
| 77 | #1713 | — | 4 | 1 | no | — |
| 78 | #1712 | — | 4 | 1 | no | — |
| 79 | #1709 | — | 8 | 1 | no | — |
| 80 | #1706 | — | 8 | 2 | no | — |
| 81 | C2×C82 | #100 | 1 | 1 | yes | characteristic |
| 82 | M128 | — | 8 | 4 | no | — |
| 83 | #1418 | — | 16 | 4 | no | — |
| 84 | #1418 | — | 16 | 4 | no | — |
| 85 | #1418 | — | 16 | 8 | no | — |
| 86 | #1397 | — | 2 | 2 | no | — |
| 87 | #1397 | — | 2 | 2 | no | — |
| 88 | C4×C32 | — | 2 | 2 | no | — |
| 89 | C4×C32 | — | 2 | 2 | no | — |
| 90 | C8×C16 | — | 2 | 2 | no | — |
| 91 | C8×C16 | — | 2 | 1 | no | — |
| 92 | C8×C16 | — | 2 | 1 | no | — |
| 93 | C8×C16 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 94 | C8×C16 | #99 | 1 | 1 | yes | characteristic |
| 95 | C8×C16 | #104 | 1 | 2 | yes | — |
| 96 | C2×#135 | — | 16 | 2 | no | — |
| 97 | #532 | — | 16 | 1 | no | — |
| 98 | #508 | — | 16 | 2 | no | — |
| 99 | #508 | — | 16 | 4 | no | — |
| 100 | #508 | — | 16 | 2 | no | — |
| 101 | #508 | — | 16 | 2 | no | — |
| 102 | #508 | — | 16 | 4 | no | — |
| 103 | #508 | — | 16 | 2 | no | — |
| 104 | C2×Q32 | — | 8 | 2 | no | — |
| 105 | C2×Q32 | — | 8 | 2 | no | — |
| 106 | C2×SD32 | — | 16 | 2 | no | — |
| 107 | C2×SD32 | — | 16 | 2 | no | — |
| 108 | C2×D16 | — | 8 | 2 | no | — |
| 109 | C2×D16 | — | 8 | 2 | no | — |
| 110 | C2×D16 | — | 16 | 4 | no | — |
| 111 | C22×C16 | — | 2 | 4 | no | — |
| 112 | C22×C16 | #357 | 1 | 2 | yes | — |
| 113 | C22×C16 | #361 | 1 | 2 | yes | — |
| 114 | #500 | — | 8 | 2 | no | — |
| 115 | #495 | — | 16 | 2 | no | — |
| 116 | #494 | — | 8 | 1 | no | — |
| 117 | #493 | — | 8 | 2 | no | — |
| 118 | #493 | — | 8 | 1 | no | — |
| 119 | #492 | — | 16 | 1 | no | — |
| 120 | #486 | — | 16 | 2 | no | — |
| 121 | #466 | — | 16 | 4 | no | — |
| 122 | C4×D8 | — | 16 | 4 | no | — |
| 123 | C2×#107 | — | 16 | 4 | no | — |
| 124 | C2×#107 | — | 16 | 2 | no | — |
| 125 | C2×C4×C8 | — | 2 | 1 | no | — |
| 126 | C2×C4×C8 | #331 | 1 | 1 | yes | characteristic |
| 127 | #384 | — | 16 | 1 | no | — |
| 128 | M64 | — | 4 | 2 | no | — |
| 129 | M64 | — | 4 | 2 | no | — |
| 130 | M64 | — | 4 | 4 | no | — |
| 131 | C2×C32 | — | 2 | 2 | no | — |
| 132 | C2×C32 | — | 2 | 2 | no | — |
| 133 | C2×C32 | — | 4 | 2 | no | — |
| 134 | #367 | — | 16 | 2 | no | — |
| 135 | #367 | — | 16 | 4 | no | — |
| 136 | #367 | — | 16 | 2 | no | — |
| 137 | #367 | — | 32 | 8 | no | — |
| 138 | #366 | — | 16 | 4 | no | — |
| 139 | #357 | — | 16 | 8 | no | — |
| 140 | C4×C16 | — | 2 | 2 | no | — |
| 141 | C4×C16 | — | 2 | 2 | no | — |
| 142 | C4×C16 | — | 2 | 4 | no | — |
| 143 | C4×C16 | D4⋊C8 | 1 | 2 | yes | — |
| 144 | C4×C16 | #327 | 1 | 2 | yes | — |
| 145 | C4×C16 | #359 | 1 | 2 | yes | — |
| 146 | C4×C16 | #360 | 1 | 2 | yes | — |
| 147 | C82 | — | 2 | 1 | no | — |
| 148 | C82 | #323 | 1 | 1 | yes | characteristic |
| 149 | C82 | #329 | 1 | 1 | yes | characteristic |
| 150 | C64 | — | 16 | 4 | no | — |
| 151 | C2×#41 | — | 32 | 1 | no | — |
| 152 | #135 | — | 32 | 2 | no | — |
| 153 | #135 | — | 32 | 2 | no | — |
| 154 | #135 | — | 32 | 2 | no | — |
| 155 | #135 | — | 32 | 2 | no | — |
| 156 | C2×Q16 | — | 16 | 2 | no | — |
| 157 | C2×SD16 | — | 32 | 2 | no | — |
| 158 | C2×D8 | — | 16 | 4 | no | — |
| 159 | C2×D8 | — | 16 | 2 | no | — |
| 160 | C22×C8 | — | 2 | 2 | no | — |
| 161 | C22×C8 | #1361 | 1 | 2 | yes | — |
| 162 | C4⋊Q8 | — | 16 | 1 | no | — |
| 163 | #127 | — | 16 | 1 | no | — |
| 164 | #124 | — | 32 | 1 | no | — |
| 165 | #121 | — | 32 | 2 | no | — |
| 166 | C4×D4 | — | 32 | 2 | no | — |
| 167 | C2×C4⋊C4 | — | 32 | 2 | no | — |
| 168 | C2×C4⋊C4 | — | 32 | 1 | no | — |
| 169 | C2×C42 | #1357 | 1 | 1 | yes | characteristic |
| 170 | Q32 | — | 16 | 2 | no | — |
| 171 | Q32 | — | 16 | 2 | no | — |
| 172 | Q32 | — | 16 | 4 | no | — |
| 173 | SD32 | — | 32 | 4 | no | — |
| 174 | SD32 | — | 32 | 2 | no | — |
| 175 | SD32 | — | 32 | 2 | no | — |
| 176 | D16 | — | 16 | 2 | no | — |
| 177 | D16 | — | 16 | 2 | no | — |
| 178 | D16 | — | 16 | 4 | no | — |
| 179 | D16 | — | 32 | 8 | no | — |
| 180 | C2×C16 | — | 2 | 4 | no | — |
| 181 | C2×C16 | — | 2 | 2 | no | — |
| 182 | C2×C16 | — | 2 | 4 | no | — |
| 183 | C2×C16 | — | 2 | 2 | no | — |
| 184 | C2×C16 | — | 2 | 4 | no | — |
| 185 | C2×C16 | — | 4 | 8 | no | — |
| 186 | C2×C16 | #1331 | 1 | 2 | yes | — |
| 187 | C2×C16 | #1333 | 1 | 2 | yes | — |
| 188 | C2×C16 | #1347 | 1 | 2 | yes | — |
| 189 | C2×C16 | #1349 | 1 | 2 | yes | — |
| 190 | #107 | — | 16 | 4 | no | — |
| 191 | #107 | — | 32 | 2 | no | — |
| 192 | #107 | — | 32 | 2 | no | — |
| 193 | #107 | — | 32 | 8 | no | — |
| 194 | #102 | — | 32 | 4 | no | — |
| 195 | C4×C8 | — | 2 | 2 | no | — |
| 196 | C4×C8 | — | 2 | 1 | no | — |
| 197 | C4×C8 | — | 2 | 1 | no | — |
| 198 | C4×C8 | #1270 | 1 | 1 | yes | characteristic |
| 199 | C4×C8 | #1359 | 1 | 1 | yes | characteristic |
| 200 | C4×C8 | #1360 | 1 | 2 | yes | — |
| 201 | C32 | — | 4 | 2 | no | — |
| 202 | C32 | — | 4 | 2 | no | — |
| 203 | C32 | — | 8 | 2 | no | — |
| 204 | #41 | — | 64 | 2 | no | — |
| 205 | #41 | — | 64 | 2 | no | — |
| 206 | C2×Q8 | — | 32 | 1 | no | — |
| 207 | C2×D4 | — | 32 | 2 | no | — |
| 208 | C2×D4 | — | 32 | 1 | no | — |
| 209 | C2×D4 | — | 64 | 1 | no | — |
| 210 | C22×C4 | — | 2 | 1 | no | — |
| 211 | C22×C4 | — | 64 | 1 | no | — |
| 212 | C22×C4 | — | 64 | 1 | no | — |
| 213 | C22×C4 | #7096 | 1 | 1 | yes | characteristic |
| 214 | Q16 | — | 32 | 2 | no | — |
| 215 | Q16 | — | 32 | 2 | no | — |
| 216 | SD16 | — | 64 | 2 | no | — |
| 217 | SD16 | — | 64 | 2 | no | — |
| 218 | D8 | — | 32 | 8 | no | — |
| 219 | D8 | — | 32 | 2 | no | — |
| 220 | D8 | — | 32 | 2 | no | — |
| 221 | C2×C8 | — | 2 | 2 | no | — |
| 222 | C2×C8 | — | 2 | 2 | no | — |
| 223 | C2×C8 | — | 2 | 2 | no | — |
| 224 | C2×C8 | — | 2 | 2 | no | — |
| 225 | C2×C8 | — | 4 | 4 | no | — |
| 226 | C2×C8 | #7094 | 1 | 2 | yes | — |
| 227 | C2×C8 | #7112 | 1 | 2 | yes | — |
| 228 | C4⋊C4 | — | 32 | 2 | no | — |
| 229 | C4⋊C4 | — | 64 | 2 | no | — |
| 230 | C4⋊C4 | — | 64 | 4 | no | — |
| 231 | C22⋊C4 | — | 64 | 2 | no | — |
| 232 | C42 | — | 2 | 1 | no | — |
| 233 | C42 | #7092 | 1 | 1 | yes | characteristic |
| 234 | C42 | #7107 | 1 | 1 | yes | characteristic |
| 235 | C16 | — | 2 | 8 | no | — |
| 236 | C16 | — | 2 | 2 | no | — |
| 237 | C16 | — | 2 | 2 | no | — |
| 238 | C16 | — | 4 | 4 | no | — |
| 239 | C16 | #7390 | 1 | 4 | yes | — |
| 240 | C16 | #7422 | 1 | 2 | yes | — |
| 241 | C16 | #7424 | 1 | 2 | yes | — |
| 242 | C23 | — | 64 | 1 | no | — |
| 243 | C23 | #93866 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 64 | 2 | no | — |
| 245 | D4 | — | 64 | 4 | no | — |
| 246 | D4 | — | 64 | 2 | no | — |
| 247 | D4 | — | 128 | 2 | no | — |
| 248 | C2×C4 | — | 2 | 1 | no | — |
| 249 | C2×C4 | — | 2 | 2 | no | — |
| 250 | C2×C4 | — | 2 | 1 | no | — |
| 251 | C2×C4 | — | 4 | 2 | no | — |
| 252 | C2×C4 | — | 64 | 1 | no | — |
| 253 | C2×C4 | — | 128 | 1 | no | — |
| 254 | C2×C4 | — | 128 | 2 | no | — |
| 255 | C2×C4 | — | 128 | 1 | no | — |
| 256 | C2×C4 | — | 128 | 1 | no | — |
| 257 | C2×C4 | #92866 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | #93868 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 2 | 4 | no | — |
| 260 | C8 | — | 2 | 2 | no | — |
| 261 | C8 | — | 4 | 2 | no | — |
| 262 | C8 | #93922 | 1 | 2 | yes | — |
| 263 | C8 | #93924 | 1 | 2 | yes | — |
| 264 | C22 | — | 2 | 1 | no | — |
| 265 | C22 | — | 4 | 1 | no | — |
| 266 | C22 | — | 128 | 2 | no | — |
| 267 | C22 | — | 128 | 1 | no | — |
| 268 | C22 | #72216565 | 1 | 1 | yes | characteristic |
| 269 | C4 | — | 256 | 1 | no | — |
| 270 | C4 | — | 2 | 2 | no | — |
| 271 | C4 | — | 2 | 1 | no | — |
| 272 | C4 | — | 4 | 1 | no | — |
| 273 | C4 | — | 128 | 1 | no | — |
| 274 | C4 | #72227269 | 1 | 2 | yes | — |
| 275 | C2 | — | 256 | 1 | no | — |
| 276 | C2 | — | 2 | 1 | no | — |
| 277 | C2 | — | 4 | 1 | no | — |
| 278 | C2 | #72227271 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72227271 |
| commutator | C162 | C2×C8 |
| Frattini, cosocle | #72216220 | C22 |
| prime radical | #72216236 | C4 |
| LCS | G ▷ C162 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ #72216220 ◁ G | |
| LECS | G ▷ #72216220 ▷ 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. #72227191 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72227191 | 2 |
| #72216382 | 2 |