Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
746 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 746 conjugacy classes of subgroups — too many to draw. Only its 146 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 | C2×AGL(1,ℤ/16) | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #21709 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #21697 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13535 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13534 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13341 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13331 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12596 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#509 | C22 | 1 | 4 | yes | — |
| 10 | #3150 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#444 | — | 2 | 1 | no | — |
| 13 | C2×#444 | C22 | 1 | 2 | yes | — |
| 14 | C2×#442 | C22 | 1 | 2 | yes | — |
| 15 | C2×C4×D8 | — | 2 | 1 | no | — |
| 16 | #2931 | — | 2 | 1 | no | — |
| 17 | #2328 | — | 2 | 1 | no | — |
| 18 | AGL(1,ℤ/16) | — | 2 | 16 | no | — |
| 19 | C2×#365 | C22 | 1 | 4 | yes | — |
| 20 | C2×#361 | C22 | 1 | 4 | yes | — |
| 21 | C2×#360 | C22 | 1 | 4 | yes | — |
| 22 | C2×#347 | C22 | 1 | 2 | yes | — |
| 23 | #1974 | — | 2 | 1 | no | — |
| 24 | #1974 | C22 | 1 | 2 | yes | — |
| 25 | #1972 | C22 | 1 | 2 | yes | — |
| 26 | #1970 | — | 2 | 1 | no | — |
| 27 | #1963 | — | 2 | 1 | no | — |
| 28 | #1961 | — | 2 | 1 | no | — |
| 29 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1846 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1377 | C22 | 1 | 2 | yes | — |
| 33 | #1376 | C22 | 1 | 2 | yes | — |
| 34 | C2×#135 | — | 2 | 1 | no | — |
| 35 | C2×#135 | C23 | 1 | 2 | yes | — |
| 36 | C22×D8 | — | 2 | 1 | no | — |
| 37 | C22×D8 | C23 | 1 | 2 | yes | — |
| 38 | C2×#131 | — | 2 | 2 | no | — |
| 39 | #532 | — | 2 | 1 | no | — |
| 40 | C2×#127 | — | 2 | 1 | no | — |
| 41 | C2×C4×D4 | — | 2 | 2 | no | — |
| 42 | #509 | — | 2 | 8 | no | — |
| 43 | #509 | — | 2 | 8 | no | — |
| 44 | C2×SD32 | — | 2 | 4 | no | — |
| 45 | C2×D16 | — | 2 | 4 | no | — |
| 46 | C2×M32 | C23 | 1 | 4 | yes | — |
| 47 | #497 | — | 2 | 2 | no | — |
| 48 | #496 | — | 2 | 2 | no | — |
| 49 | #493 | — | 2 | 4 | no | — |
| 50 | #444 | — | 2 | 8 | no | — |
| 51 | #444 | — | 4 | 4 | no | — |
| 52 | #442 | — | 2 | 8 | no | — |
| 53 | C4×D8 | — | 4 | 4 | no | — |
| 54 | #436 | — | 4 | 2 | no | — |
| 55 | C8×D4 | — | 4 | 2 | no | — |
| 56 | C2×#108 | — | 2 | 1 | no | — |
| 57 | C2×#108 | C23 | 1 | 2 | yes | — |
| 58 | C2×#107 | — | 2 | 1 | no | — |
| 59 | C2×#106 | C23 | 1 | 2 | yes | — |
| 60 | #423 | — | 2 | 1 | no | — |
| 61 | C2×#104 | — | 2 | 2 | no | — |
| 62 | C2×#104 | — | 2 | 2 | no | — |
| 63 | C2×#102 | — | 2 | 2 | no | — |
| 64 | C2×#102 | — | 2 | 2 | no | — |
| 65 | #408 | — | 2 | 2 | no | — |
| 66 | C2×#97 | D4 | 1 | 1 | yes | characteristic |
| 67 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 68 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 69 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 70 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 72 | #390 | — | 2 | 2 | no | — |
| 73 | #386 | — | 4 | 4 | no | — |
| 74 | #382 | — | 2 | 1 | no | — |
| 75 | #376 | D4 | 1 | 1 | yes | characteristic |
| 76 | #365 | — | 2 | 8 | no | — |
| 77 | #361 | — | 2 | 8 | no | — |
| 78 | #360 | — | 2 | 8 | no | — |
| 79 | #347 | D4 | 1 | 8 | yes | complemented |
| 80 | C2×#41 | — | 2 | 2 | no | — |
| 81 | C22×D4 | — | 2 | 2 | no | — |
| 82 | C22×D4 | — | 4 | 2 | no | — |
| 83 | C23×C4 | — | 4 | 2 | no | — |
| 84 | #135 | — | 2 | 4 | no | — |
| 85 | #135 | — | 2 | 4 | no | — |
| 86 | #135 | — | 4 | 2 | no | — |
| 87 | #135 | — | 4 | 2 | no | — |
| 88 | C2×Q16 | — | 2 | 1 | no | — |
| 89 | C2×Q16 | #41 | 1 | 2 | yes | — |
| 90 | C2×SD16 | — | 2 | 2 | no | — |
| 91 | C2×SD16 | — | 2 | 2 | no | — |
| 92 | C2×D8 | — | 2 | 4 | no | — |
| 93 | C2×D8 | — | 2 | 1 | no | — |
| 94 | C2×D8 | — | 2 | 4 | no | — |
| 95 | C2×D8 | — | 2 | 4 | no | — |
| 96 | C2×D8 | — | 2 | 2 | no | — |
| 97 | C2×D8 | — | 4 | 2 | no | — |
| 98 | C2×D8 | — | 4 | 2 | no | — |
| 99 | C2×D8 | #41 | 1 | 4 | yes | — |
| 100 | C2×D8 | #41 | 1 | 2 | yes | — |
| 101 | #131 | — | 4 | 4 | no | — |
| 102 | #131 | — | 4 | 4 | no | — |
| 103 | C2×M16 | — | 2 | 2 | no | — |
| 104 | C2×M16 | — | 4 | 2 | no | — |
| 105 | C22×C8 | — | 4 | 2 | no | — |
| 106 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 110 | C4⋊Q8 | — | 2 | 1 | no | — |
| 111 | #127 | — | 2 | 4 | no | — |
| 112 | #127 | — | 2 | 1 | no | — |
| 113 | #127 | — | 4 | 2 | no | — |
| 114 | #124 | — | 2 | 2 | no | — |
| 115 | #121 | — | 4 | 2 | no | — |
| 116 | C4×D4 | — | 4 | 2 | no | — |
| 117 | C4×D4 | — | 4 | 4 | no | — |
| 118 | C4×D4 | — | 4 | 4 | no | — |
| 119 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 120 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 121 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 122 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 125 | SD32 | — | 2 | 16 | no | — |
| 126 | D16 | — | 2 | 16 | no | — |
| 127 | M32 | — | 2 | 4 | no | — |
| 128 | M32 | C2×D4 | 1 | 8 | yes | — |
| 129 | C2×C16 | — | 2 | 4 | no | — |
| 130 | #108 | — | 2 | 4 | no | — |
| 131 | #108 | — | 4 | 2 | no | — |
| 132 | #107 | — | 4 | 2 | no | — |
| 133 | #106 | — | 2 | 4 | no | — |
| 134 | C4⋊C8 | — | 4 | 2 | no | — |
| 135 | #104 | — | 4 | 4 | no | — |
| 136 | #104 | — | 4 | 4 | no | — |
| 137 | #102 | — | 4 | 4 | no | — |
| 138 | #102 | — | 4 | 4 | no | — |
| 139 | C22⋊C8 | — | 4 | 4 | no | — |
| 140 | #97 | — | 2 | 2 | no | — |
| 141 | #97 | — | 2 | 2 | no | — |
| 142 | #97 | C2×D4 | 1 | 4 | yes | — |
| 143 | C4×C8 | — | 2 | 2 | no | — |
| 144 | C4×C8 | — | 2 | 2 | no | — |
| 145 | C4×C8 | — | 2 | 2 | no | — |
| 146 | #95 | — | 2 | 2 | no | — |
| 147 | #95 | — | 4 | 2 | no | — |
| 148 | C24 | — | 4 | 2 | no | — |
| 149 | #41 | — | 4 | 2 | no | — |
| 150 | #41 | — | 4 | 2 | no | — |
| 151 | #41 | — | 4 | 2 | no | — |
| 152 | #41 | — | 4 | 2 | no | — |
| 153 | C2×Q8 | — | 2 | 2 | no | — |
| 154 | C2×D4 | — | 2 | 2 | no | — |
| 155 | C2×D4 | — | 2 | 4 | no | — |
| 156 | C2×D4 | — | 4 | 2 | no | — |
| 157 | C2×D4 | — | 4 | 8 | no | — |
| 158 | C2×D4 | — | 4 | 2 | no | — |
| 159 | C2×D4 | — | 4 | 2 | no | — |
| 160 | C2×D4 | — | 4 | 2 | no | — |
| 161 | C2×D4 | — | 4 | 2 | no | — |
| 162 | C2×D4 | — | 4 | 2 | no | — |
| 163 | C2×D4 | — | 4 | 2 | no | — |
| 164 | C2×D4 | — | 16 | 4 | no | — |
| 165 | C22×C4 | — | 2 | 1 | no | — |
| 166 | C22×C4 | — | 2 | 1 | no | — |
| 167 | C22×C4 | — | 4 | 2 | no | — |
| 168 | C22×C4 | — | 4 | 2 | no | — |
| 169 | C22×C4 | — | 8 | 4 | no | — |
| 170 | C22×C4 | — | 16 | 4 | no | — |
| 171 | C22×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 172 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 173 | Q16 | — | 2 | 4 | no | — |
| 174 | Q16 | — | 4 | 2 | no | — |
| 175 | SD16 | — | 4 | 4 | no | — |
| 176 | SD16 | — | 4 | 4 | no | — |
| 177 | D8 | — | 2 | 4 | no | — |
| 178 | D8 | — | 2 | 8 | no | — |
| 179 | D8 | — | 4 | 8 | no | — |
| 180 | D8 | — | 4 | 2 | no | — |
| 181 | D8 | — | 4 | 4 | no | — |
| 182 | M16 | — | 4 | 2 | no | — |
| 183 | M16 | — | 4 | 2 | no | — |
| 184 | M16 | — | 8 | 4 | no | — |
| 185 | C2×C8 | — | 2 | 1 | no | — |
| 186 | C2×C8 | — | 2 | 1 | no | — |
| 187 | C2×C8 | — | 2 | 1 | no | — |
| 188 | C2×C8 | — | 2 | 1 | no | — |
| 189 | C2×C8 | — | 2 | 1 | no | — |
| 190 | C2×C8 | — | 2 | 1 | no | — |
| 191 | C2×C8 | — | 4 | 2 | no | — |
| 192 | C2×C8 | — | 8 | 4 | no | — |
| 193 | C2×C8 | #127 | 1 | 2 | yes | — |
| 194 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 195 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 196 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 197 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 198 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 199 | C4⋊C4 | — | 4 | 2 | no | — |
| 200 | C4⋊C4 | — | 4 | 2 | no | — |
| 201 | C4⋊C4 | — | 4 | 2 | no | — |
| 202 | C22⋊C4 | — | 4 | 4 | no | — |
| 203 | C22⋊C4 | — | 8 | 4 | no | — |
| 204 | C22⋊C4 | — | 16 | 4 | no | — |
| 205 | C42 | — | 2 | 1 | no | — |
| 206 | C42 | — | 2 | 1 | no | — |
| 207 | C42 | — | 2 | 1 | no | — |
| 208 | C42 | — | 2 | 1 | no | — |
| 209 | C42 | #127 | 1 | 2 | yes | — |
| 210 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 211 | C16 | — | 2 | 8 | no | — |
| 212 | C23 | — | 4 | 4 | no | — |
| 213 | C23 | — | 4 | 2 | no | — |
| 214 | C23 | — | 8 | 4 | no | — |
| 215 | C23 | — | 8 | 4 | no | — |
| 216 | C23 | — | 8 | 2 | no | — |
| 217 | C23 | — | 8 | 2 | no | — |
| 218 | C23 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 219 | Q8 | — | 4 | 2 | no | — |
| 220 | Q8 | — | 4 | 2 | no | — |
| 221 | D4 | — | 4 | 4 | no | — |
| 222 | D4 | — | 4 | 2 | no | — |
| 223 | D4 | — | 4 | 4 | no | — |
| 224 | D4 | — | 4 | 2 | no | — |
| 225 | D4 | — | 8 | 4 | no | — |
| 226 | D4 | — | 8 | 2 | no | — |
| 227 | D4 | — | 8 | 2 | no | — |
| 228 | D4 | — | 8 | 2 | no | — |
| 229 | D4 | — | 8 | 2 | no | — |
| 230 | D4 | — | 16 | 8 | no | — |
| 231 | C2×C4 | — | 2 | 2 | no | — |
| 232 | C2×C4 | — | 2 | 2 | no | — |
| 233 | C2×C4 | — | 2 | 2 | no | — |
| 234 | C2×C4 | — | 4 | 2 | no | — |
| 235 | C2×C4 | — | 4 | 2 | no | — |
| 236 | C2×C4 | — | 4 | 2 | no | — |
| 237 | C2×C4 | — | 4 | 2 | no | — |
| 238 | C2×C4 | — | 8 | 2 | no | — |
| 239 | C2×C4 | — | 8 | 2 | no | — |
| 240 | C2×C4 | — | 8 | 4 | no | — |
| 241 | C2×C4 | — | 8 | 2 | no | — |
| 242 | C2×C4 | — | 16 | 8 | no | — |
| 243 | C2×C4 | #525 | 1 | 1 | yes | characteristic |
| 244 | C2×C4 | #525 | 1 | 1 | yes | characteristic |
| 245 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 246 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 247 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 248 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 249 | C8 | — | 2 | 2 | no | — |
| 250 | C8 | — | 2 | 2 | no | — |
| 251 | C8 | — | 2 | 2 | no | — |
| 252 | C8 | — | 8 | 2 | no | — |
| 253 | C8 | #525 | 1 | 4 | yes | — |
| 254 | C22 | — | 2 | 2 | no | — |
| 255 | C22 | — | 8 | 8 | no | — |
| 256 | C22 | — | 8 | 4 | no | — |
| 257 | C22 | — | 8 | 2 | no | — |
| 258 | C22 | — | 8 | 4 | no | — |
| 259 | C22 | — | 16 | 2 | no | — |
| 260 | C22 | — | 16 | 2 | no | — |
| 261 | C22 | — | 16 | 4 | no | — |
| 262 | C22 | #3109 | 1 | 1 | yes | characteristic |
| 263 | C22 | #3109 | 1 | 1 | yes | characteristic |
| 264 | C22 | #3151 | 1 | 1 | yes | characteristic |
| 265 | C4 | — | 4 | 2 | no | — |
| 266 | C4 | — | 4 | 2 | no | — |
| 267 | C4 | — | 4 | 2 | no | — |
| 268 | C4 | — | 8 | 2 | no | — |
| 269 | C4 | — | 16 | 2 | no | — |
| 270 | C4 | #2397 | 1 | 1 | yes | characteristic |
| 271 | C4 | #3109 | 1 | 1 | yes | characteristic |
| 272 | C4 | #3109 | 1 | 1 | yes | characteristic |
| 273 | C4 | #3151 | 1 | 1 | yes | characteristic |
| 274 | C2 | — | 2 | 1 | no | — |
| 275 | C2 | — | 2 | 1 | no | — |
| 276 | C2 | — | 16 | 2 | no | — |
| 277 | C2 | — | 16 | 4 | no | — |
| 278 | C2 | #23802 | 1 | 1 | yes | characteristic |
| 279 | C2 | #33692 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×AGL(1,ℤ/16) | C2 | semidirect |
| 2. #21709 | C2 | semidirect |
| 3. #21697 | C2 | semidirect |
| 4. #13535 | C2 | semidirect |
| 5. #13534 | C2 | semidirect |
| 6. #13341 | C2 | semidirect |
| 7. #13331 | C2 | semidirect |
| 8. #12596 | C2 | semidirect |
| 79. #347 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AGL(1,ℤ/16) | 2 |
| #21709 | 2 |
| #21697 | 2 |
| #13535 | 2 |
| #13534 | 2 |
| #13341 | 2 |
| #13331 | 2 |
| #12596 | 2 |