Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
540 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 126 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 | #33179 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20829 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10631 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10631 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #9780 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #9198 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #8795 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #8157 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#469 | — | 2 | 1 | no | — |
| 11 | C2×#466 | — | 2 | 1 | no | — |
| 12 | #2921 | C22 | 1 | 2 | yes | — |
| 13 | C2×#405 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2619 | C22 | 1 | 2 | yes | — |
| 15 | #1972 | — | 2 | 1 | no | — |
| 16 | #1970 | — | 2 | 1 | no | — |
| 17 | #1961 | C22 | 1 | 2 | yes | — |
| 18 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 19 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1846 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1837 | C22 | 1 | 2 | yes | — |
| 22 | #1811 | — | 2 | 1 | no | — |
| 23 | #1810 | — | 2 | 1 | no | — |
| 24 | #1778 | — | 2 | 1 | no | — |
| 25 | C4×#108 | — | 2 | 1 | no | — |
| 26 | #1545 | C22 | 1 | 4 | yes | — |
| 27 | #1545 | C22 | 1 | 4 | yes | — |
| 28 | #1458 | C22 | 1 | 2 | yes | — |
| 29 | #1452 | C22 | 1 | 2 | yes | — |
| 30 | #1286 | C22 | 1 | 4 | yes | — |
| 31 | #1286 | C22 | 1 | 4 | yes | — |
| 32 | #1281 | C22 | 1 | 2 | yes | — |
| 33 | C22×D8 | — | 2 | 1 | no | — |
| 34 | C22×D8 | — | 2 | 1 | no | — |
| 35 | C22×M16 | D4 | 1 | 1 | yes | characteristic |
| 36 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×#127 | C23 | 1 | 2 | yes | — |
| 38 | C2×#121 | — | 2 | 2 | no | — |
| 39 | C2×#117 | C23 | 1 | 1 | yes | characteristic |
| 40 | #493 | — | 2 | 4 | no | — |
| 41 | #469 | — | 4 | 4 | no | — |
| 42 | #466 | — | 4 | 4 | no | — |
| 43 | #433 | — | 4 | 4 | no | — |
| 44 | #431 | C23 | 1 | 4 | yes | — |
| 45 | C2×#108 | — | 2 | 1 | no | — |
| 46 | C2×#108 | — | 2 | 1 | no | — |
| 47 | C2×#108 | — | 2 | 1 | no | — |
| 48 | C2×#108 | — | 2 | 1 | no | — |
| 49 | C2×#107 | — | 2 | 1 | no | — |
| 50 | C2×#106 | — | 2 | 1 | no | — |
| 51 | #423 | C23 | 1 | 2 | yes | — |
| 52 | #423 | C23 | 1 | 2 | yes | — |
| 53 | C2×#102 | — | 2 | 2 | no | — |
| 54 | C2×#102 | — | 2 | 2 | no | — |
| 55 | #407 | — | 2 | 2 | no | — |
| 56 | #405 | — | 2 | 4 | no | — |
| 57 | C4×M16 | — | 2 | 2 | no | — |
| 58 | C2×#97 | D4 | 1 | 1 | yes | characteristic |
| 59 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 62 | #394 | — | 2 | 4 | no | — |
| 63 | #392 | — | 2 | 4 | no | — |
| 64 | #390 | — | 2 | 2 | no | — |
| 65 | #382 | C23 | 1 | 2 | yes | — |
| 66 | #341 | — | 2 | 2 | no | — |
| 67 | #341 | — | 2 | 2 | no | — |
| 68 | #331 | — | 4 | 4 | no | — |
| 69 | #331 | — | 4 | 4 | no | — |
| 70 | #324 | — | 2 | 2 | no | — |
| 71 | #323 | — | 2 | 2 | no | — |
| 72 | #322 | — | 2 | 4 | no | — |
| 73 | C22×D4 | — | 2 | 2 | no | — |
| 74 | C22×D4 | — | 2 | 4 | no | — |
| 75 | C22×D4 | — | 4 | 2 | no | — |
| 76 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C2×D8 | — | 4 | 2 | no | — |
| 78 | C2×D8 | — | 4 | 2 | no | — |
| 79 | C2×D8 | — | 4 | 2 | no | — |
| 80 | C2×D8 | — | 4 | 2 | no | — |
| 81 | C2×D8 | — | 4 | 2 | no | — |
| 82 | C2×D8 | — | 4 | 2 | no | — |
| 83 | C2×M16 | — | 2 | 2 | no | — |
| 84 | C2×M16 | — | 2 | 2 | no | — |
| 85 | C2×M16 | — | 2 | 2 | no | — |
| 86 | C2×M16 | — | 2 | 2 | no | — |
| 87 | C2×M16 | — | 4 | 1 | no | — |
| 88 | C22×C8 | — | 2 | 2 | no | — |
| 89 | C22×C8 | — | 2 | 2 | no | — |
| 90 | C22×C8 | — | 4 | 1 | no | — |
| 91 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | #127 | — | 2 | 4 | no | — |
| 96 | #127 | — | 2 | 2 | no | — |
| 97 | #127 | — | 2 | 2 | no | — |
| 98 | #121 | — | 4 | 4 | no | — |
| 99 | #121 | — | 4 | 4 | no | — |
| 100 | #117 | — | 2 | 2 | no | — |
| 101 | #117 | — | 2 | 2 | no | — |
| 102 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 103 | C2×C4⋊C4 | #41 | 1 | 2 | yes | — |
| 104 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 105 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 106 | C2×C22⋊C4 | #41 | 1 | 2 | yes | — |
| 107 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 109 | #108 | — | 4 | 2 | no | — |
| 110 | #108 | — | 4 | 2 | no | — |
| 111 | #108 | — | 4 | 2 | no | — |
| 112 | #108 | — | 4 | 2 | no | — |
| 113 | #107 | — | 4 | 2 | no | — |
| 114 | #106 | — | 4 | 2 | no | — |
| 115 | C4⋊C8 | — | 2 | 4 | no | — |
| 116 | C4⋊C8 | — | 2 | 4 | no | — |
| 117 | C4⋊C8 | — | 2 | 4 | no | — |
| 118 | #102 | — | 4 | 4 | no | — |
| 119 | #102 | — | 4 | 4 | no | — |
| 120 | C22⋊C8 | — | 2 | 4 | no | — |
| 121 | C22⋊C8 | — | 4 | 4 | no | — |
| 122 | #97 | — | 2 | 2 | no | — |
| 123 | #97 | — | 2 | 2 | no | — |
| 124 | #97 | — | 4 | 2 | no | — |
| 125 | C4×C8 | — | 2 | 2 | no | — |
| 126 | C4×C8 | — | 2 | 4 | no | — |
| 127 | C4×C8 | — | 4 | 2 | no | — |
| 128 | C4×C8 | #41 | 1 | 4 | yes | — |
| 129 | #95 | — | 2 | 4 | no | — |
| 130 | C24 | — | 4 | 2 | no | — |
| 131 | C24 | #124 | 1 | 1 | yes | characteristic |
| 132 | C2×D4 | — | 4 | 2 | no | — |
| 133 | C2×D4 | — | 4 | 2 | no | — |
| 134 | C2×D4 | — | 4 | 4 | no | — |
| 135 | C2×D4 | — | 4 | 4 | no | — |
| 136 | C2×D4 | — | 4 | 2 | no | — |
| 137 | C2×D4 | — | 4 | 2 | no | — |
| 138 | C2×D4 | — | 4 | 2 | no | — |
| 139 | C2×D4 | — | 4 | 2 | no | — |
| 140 | C2×D4 | — | 8 | 4 | no | — |
| 141 | C2×D4 | — | 16 | 4 | no | — |
| 142 | C2×D4 | — | 16 | 4 | no | — |
| 143 | C22×C4 | — | 2 | 1 | no | — |
| 144 | C22×C4 | — | 2 | 1 | no | — |
| 145 | C22×C4 | — | 2 | 1 | no | — |
| 146 | C22×C4 | — | 2 | 1 | no | — |
| 147 | C22×C4 | — | 2 | 1 | no | — |
| 148 | C22×C4 | — | 2 | 1 | no | — |
| 149 | C22×C4 | — | 4 | 2 | no | — |
| 150 | C22×C4 | #124 | 1 | 1 | yes | characteristic |
| 151 | C22×C4 | #124 | 1 | 2 | yes | — |
| 152 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 153 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 154 | D8 | — | 8 | 4 | no | — |
| 155 | D8 | — | 8 | 4 | no | — |
| 156 | M16 | — | 4 | 4 | no | — |
| 157 | M16 | — | 4 | 2 | no | — |
| 158 | M16 | — | 4 | 2 | no | — |
| 159 | M16 | — | 4 | 2 | no | — |
| 160 | M16 | — | 4 | 2 | no | — |
| 161 | C2×C8 | — | 2 | 1 | no | — |
| 162 | C2×C8 | — | 2 | 1 | no | — |
| 163 | C2×C8 | — | 2 | 1 | no | — |
| 164 | C2×C8 | — | 2 | 1 | no | — |
| 165 | C2×C8 | — | 2 | 1 | no | — |
| 166 | C2×C8 | — | 2 | 1 | no | — |
| 167 | C2×C8 | — | 2 | 1 | no | — |
| 168 | C2×C8 | — | 2 | 1 | no | — |
| 169 | C2×C8 | — | 4 | 4 | no | — |
| 170 | C2×C8 | — | 4 | 2 | no | — |
| 171 | C2×C8 | — | 4 | 2 | no | — |
| 172 | C2×C8 | — | 4 | 4 | no | — |
| 173 | C2×C8 | #123 | 1 | 2 | yes | — |
| 174 | C2×C8 | #123 | 1 | 2 | yes | — |
| 175 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 176 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 177 | C4⋊C4 | — | 2 | 4 | no | — |
| 178 | C4⋊C4 | — | 4 | 2 | no | — |
| 179 | C4⋊C4 | — | 4 | 2 | no | — |
| 180 | C22⋊C4 | — | 2 | 4 | no | — |
| 181 | C22⋊C4 | — | 8 | 4 | no | — |
| 182 | C22⋊C4 | — | 16 | 4 | no | — |
| 183 | C42 | — | 2 | 1 | no | — |
| 184 | C42 | — | 2 | 1 | no | — |
| 185 | C42 | C2×#41 | 1 | 2 | yes | — |
| 186 | C42 | H(5,𝔽2) | 1 | 2 | yes | — |
| 187 | C23 | — | 2 | 1 | no | — |
| 188 | C23 | — | 2 | 1 | no | — |
| 189 | C23 | — | 2 | 1 | no | — |
| 190 | C23 | — | 4 | 2 | no | — |
| 191 | C23 | — | 8 | 4 | no | — |
| 192 | C23 | — | 8 | 4 | no | — |
| 193 | C23 | — | 8 | 2 | no | — |
| 194 | C23 | — | 8 | 2 | no | — |
| 195 | C23 | — | 8 | 2 | no | — |
| 196 | C23 | C2×#124 | 1 | 1 | yes | characteristic |
| 197 | D4 | — | 8 | 2 | no | — |
| 198 | D4 | — | 8 | 2 | no | — |
| 199 | D4 | — | 8 | 2 | no | — |
| 200 | D4 | — | 8 | 2 | no | — |
| 201 | D4 | — | 16 | 4 | no | — |
| 202 | D4 | — | 16 | 4 | no | — |
| 203 | D4 | — | 16 | 8 | no | — |
| 204 | C2×C4 | — | 2 | 1 | no | — |
| 205 | C2×C4 | — | 2 | 1 | no | — |
| 206 | C2×C4 | — | 2 | 2 | no | — |
| 207 | C2×C4 | — | 2 | 1 | no | — |
| 208 | C2×C4 | — | 4 | 2 | no | — |
| 209 | C2×C4 | — | 4 | 1 | no | — |
| 210 | C2×C4 | — | 4 | 1 | no | — |
| 211 | C2×C4 | — | 4 | 1 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 4 | 2 | no | — |
| 214 | C2×C4 | — | 8 | 4 | no | — |
| 215 | C2×C4 | — | 8 | 2 | no | — |
| 216 | C2×C4 | C2×#123 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | #553 | 1 | 4 | yes | — |
| 219 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 223 | C8 | — | 2 | 2 | no | — |
| 224 | C8 | — | 2 | 2 | no | — |
| 225 | C8 | — | 2 | 2 | no | — |
| 226 | C8 | — | 2 | 2 | no | — |
| 227 | C8 | — | 8 | 4 | no | — |
| 228 | C8 | — | 8 | 2 | no | — |
| 229 | C8 | — | 8 | 2 | no | — |
| 230 | C22 | — | 2 | 2 | no | — |
| 231 | C22 | — | 4 | 2 | no | — |
| 232 | C22 | — | 4 | 1 | no | — |
| 233 | C22 | — | 4 | 2 | no | — |
| 234 | C22 | — | 8 | 1 | no | — |
| 235 | C22 | — | 16 | 2 | no | — |
| 236 | C22 | — | 16 | 2 | no | — |
| 237 | C22 | — | 16 | 4 | no | — |
| 238 | C22 | — | 16 | 2 | no | — |
| 239 | C22 | — | 16 | 4 | no | — |
| 240 | C22 | #2504 | 1 | 1 | yes | characteristic |
| 241 | C22 | #3135 | 1 | 1 | yes | characteristic |
| 242 | C22 | #3138 | 1 | 1 | yes | characteristic |
| 243 | C4 | — | 2 | 4 | no | — |
| 244 | C4 | — | 4 | 1 | no | — |
| 245 | C4 | — | 4 | 2 | no | — |
| 246 | C4 | — | 16 | 2 | no | — |
| 247 | C4 | #3094 | 1 | 1 | yes | characteristic |
| 248 | C4 | #3094 | 1 | 1 | yes | characteristic |
| 249 | C4 | #3214 | 1 | 1 | yes | characteristic |
| 250 | C4 | #3214 | 1 | 1 | yes | characteristic |
| 251 | C2 | — | 2 | 1 | no | — |
| 252 | C2 | — | 2 | 1 | no | — |
| 253 | C2 | — | 8 | 1 | no | — |
| 254 | C2 | — | 32 | 2 | no | — |
| 255 | C2 | #27981 | 1 | 1 | yes | characteristic |
| 256 | C2 | #27989 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #33179 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20829 | C2 | semidirect |
| 3. #10631 | C2 | semidirect |
| 4. #10631 | C2 | semidirect |
| 5. #9780 | C2 | semidirect |
| 6. #9198 | C2 | semidirect |
| 7. #8795 | C2 | semidirect |
| 8. #8157 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33179 | 2 |
| #20829 | 2 |
| #10631 | 2 |
| #9780 | 2 |
| #9198 | 2 |
| #8795 | 2 |
| #8157 | 2 |