Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
504 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 72 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×#2238 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #30247 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#1339 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11367 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7193 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#469 | — | 2 | 1 | no | — |
| 7 | C2×#469 | C22 | 1 | 2 | yes | — |
| 8 | C2×#430 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2856 | — | 2 | 1 | no | — |
| 10 | #2238 | — | 2 | 4 | no | — |
| 11 | #2238 | — | 2 | 4 | no | — |
| 12 | C2×#365 | — | 2 | 2 | no | — |
| 13 | C2×#361 | — | 2 | 2 | no | — |
| 14 | C2×#349 | C22 | 1 | 2 | yes | — |
| 15 | #2094 | — | 2 | 2 | no | — |
| 16 | #2011 | — | 2 | 2 | no | — |
| 17 | C2×#344 | — | 2 | 1 | no | — |
| 18 | C2×#339 | — | 2 | 1 | no | — |
| 19 | C2×H(3,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 20 | #1728 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1339 | — | 2 | 4 | no | — |
| 22 | C22×D8 | — | 2 | 1 | no | — |
| 23 | C22×D8 | D4 | 1 | 2 | yes | — |
| 24 | C22×M16 | — | 2 | 1 | no | — |
| 25 | C22×M16 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×#127 | — | 2 | 1 | no | — |
| 28 | C2×#121 | — | 2 | 2 | no | — |
| 29 | C2×#121 | — | 4 | 2 | no | — |
| 30 | C2×#117 | — | 2 | 1 | no | — |
| 31 | C22×C42 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×M32 | — | 2 | 2 | no | — |
| 33 | #469 | — | 2 | 2 | no | — |
| 34 | #469 | — | 2 | 2 | no | — |
| 35 | #469 | — | 2 | 2 | no | — |
| 36 | #469 | — | 2 | 2 | no | — |
| 37 | #469 | — | 4 | 2 | no | — |
| 38 | #469 | — | 4 | 2 | no | — |
| 39 | #430 | — | 2 | 2 | no | — |
| 40 | #430 | — | 2 | 1 | no | — |
| 41 | #430 | D4 | 1 | 2 | yes | complemented |
| 42 | C2×#108 | — | 2 | 2 | no | — |
| 43 | C2×#106 | — | 2 | 1 | no | — |
| 44 | C2×#106 | D4 | 1 | 2 | yes | — |
| 45 | C2×#102 | — | 2 | 2 | no | — |
| 46 | C2×#102 | — | 2 | 2 | no | — |
| 47 | #390 | — | 2 | 2 | no | — |
| 48 | #390 | — | 4 | 2 | no | — |
| 49 | #384 | — | 2 | 1 | no | — |
| 50 | #365 | — | 2 | 8 | no | — |
| 51 | #361 | — | 2 | 8 | no | — |
| 52 | #349 | — | 2 | 2 | no | — |
| 53 | #349 | D4 | 1 | 4 | yes | complemented |
| 54 | #344 | — | 4 | 2 | no | — |
| 55 | #339 | — | 4 | 2 | no | — |
| 56 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 57 | #336 | — | 2 | 2 | no | — |
| 58 | C22×D4 | — | 2 | 2 | no | — |
| 59 | C22×D4 | — | 4 | 2 | no | — |
| 60 | C22×D4 | — | 4 | 2 | no | — |
| 61 | C23×C4 | — | 2 | 1 | no | — |
| 62 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×D8 | — | 2 | 2 | no | — |
| 64 | C2×D8 | — | 2 | 2 | no | — |
| 65 | C2×D8 | — | 4 | 2 | no | — |
| 66 | C2×D8 | — | 4 | 2 | no | — |
| 67 | C2×D8 | — | 4 | 2 | no | — |
| 68 | C2×D8 | — | 8 | 1 | no | — |
| 69 | C2×D8 | — | 8 | 1 | no | — |
| 70 | C2×M16 | — | 2 | 1 | no | — |
| 71 | C2×M16 | — | 2 | 2 | no | — |
| 72 | C2×M16 | — | 4 | 1 | no | — |
| 73 | C2×M16 | — | 4 | 1 | no | — |
| 74 | C2×M16 | — | 4 | 1 | no | — |
| 75 | C2×M16 | — | 4 | 1 | no | — |
| 76 | C2×M16 | — | 4 | 1 | no | — |
| 77 | C2×M16 | — | 4 | 1 | no | — |
| 78 | C2×M16 | — | 4 | 1 | no | — |
| 79 | C2×M16 | — | 4 | 1 | no | — |
| 80 | C2×M16 | — | 4 | 1 | no | — |
| 81 | C2×M16 | SD16 | 1 | 2 | yes | — |
| 82 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 84 | C2×M16 | #41 | 1 | 1 | yes | characteristic |
| 85 | C22×C8 | — | 2 | 1 | no | — |
| 86 | C22×C8 | — | 4 | 1 | no | — |
| 87 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 88 | #127 | — | 8 | 1 | no | — |
| 89 | #127 | — | 8 | 1 | no | — |
| 90 | #121 | — | 4 | 2 | no | — |
| 91 | #121 | — | 4 | 2 | no | — |
| 92 | #121 | — | 4 | 2 | no | — |
| 93 | #121 | — | 4 | 2 | no | — |
| 94 | #121 | — | 8 | 8 | no | — |
| 95 | #117 | — | 2 | 2 | no | — |
| 96 | #117 | — | 4 | 1 | no | — |
| 97 | #117 | — | 4 | 1 | no | — |
| 98 | #117 | — | 4 | 1 | no | — |
| 99 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 100 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 101 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 102 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 103 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 104 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 105 | C2×C42 | — | 2 | 1 | no | — |
| 106 | C2×C42 | — | 4 | 1 | no | — |
| 107 | C2×C42 | — | 4 | 1 | no | — |
| 108 | C2×C42 | — | 4 | 1 | no | — |
| 109 | C2×C42 | SD16 | 1 | 2 | yes | — |
| 110 | M32 | — | 2 | 4 | no | — |
| 111 | M32 | — | 4 | 2 | no | — |
| 112 | C2×C16 | — | 4 | 2 | no | — |
| 113 | #108 | — | 2 | 4 | no | — |
| 114 | #108 | — | 4 | 2 | no | — |
| 115 | #106 | — | 2 | 2 | no | — |
| 116 | #106 | — | 2 | 2 | no | — |
| 117 | #106 | — | 4 | 2 | no | — |
| 118 | #102 | — | 4 | 4 | no | — |
| 119 | #102 | — | 4 | 2 | no | — |
| 120 | #102 | — | 4 | 2 | no | — |
| 121 | C24 | — | 4 | 2 | no | — |
| 122 | C24 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 123 | C2×D4 | — | 4 | 2 | no | — |
| 124 | C2×D4 | — | 4 | 2 | no | — |
| 125 | C2×D4 | — | 4 | 2 | no | — |
| 126 | C2×D4 | — | 4 | 2 | no | — |
| 127 | C2×D4 | — | 4 | 2 | no | — |
| 128 | C2×D4 | — | 4 | 2 | no | — |
| 129 | C2×D4 | — | 8 | 4 | no | — |
| 130 | C2×D4 | — | 8 | 4 | no | — |
| 131 | C2×D4 | — | 8 | 4 | no | — |
| 132 | C2×D4 | — | 8 | 8 | no | — |
| 133 | C2×D4 | — | 8 | 2 | no | — |
| 134 | C2×D4 | — | 8 | 2 | no | — |
| 135 | C22×C4 | — | 2 | 1 | no | — |
| 136 | C22×C4 | — | 2 | 1 | no | — |
| 137 | C22×C4 | — | 2 | 1 | no | — |
| 138 | C22×C4 | — | 2 | 1 | no | — |
| 139 | C22×C4 | — | 2 | 2 | no | — |
| 140 | C22×C4 | — | 4 | 4 | no | — |
| 141 | C22×C4 | — | 4 | 2 | no | — |
| 142 | C22×C4 | — | 8 | 1 | no | — |
| 143 | C22×C4 | — | 8 | 1 | no | — |
| 144 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 145 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 146 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 147 | C22×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 148 | C22×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 149 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 150 | D8 | — | 4 | 4 | no | — |
| 151 | D8 | — | 8 | 2 | no | — |
| 152 | D8 | — | 8 | 4 | no | — |
| 153 | M16 | — | 4 | 1 | no | — |
| 154 | M16 | — | 4 | 2 | no | — |
| 155 | M16 | — | 4 | 1 | no | — |
| 156 | M16 | — | 4 | 2 | no | — |
| 157 | M16 | — | 4 | 2 | no | — |
| 158 | M16 | — | 4 | 2 | no | — |
| 159 | M16 | — | 8 | 1 | no | — |
| 160 | M16 | — | 8 | 1 | no | — |
| 161 | M16 | — | 8 | 1 | no | — |
| 162 | C2×C8 | — | 2 | 2 | no | — |
| 163 | C2×C8 | — | 2 | 2 | no | — |
| 164 | C2×C8 | — | 2 | 2 | no | — |
| 165 | C2×C8 | — | 4 | 1 | no | — |
| 166 | C2×C8 | — | 4 | 1 | no | — |
| 167 | C2×C8 | — | 4 | 2 | no | — |
| 168 | C2×C8 | — | 8 | 1 | no | — |
| 169 | C2×C8 | — | 8 | 1 | no | — |
| 170 | C2×C8 | #120 | 1 | 2 | yes | — |
| 171 | C2×C8 | #121 | 1 | 2 | yes | — |
| 172 | C4⋊C4 | — | 4 | 2 | no | — |
| 173 | C4⋊C4 | — | 4 | 2 | no | — |
| 174 | C4⋊C4 | — | 4 | 2 | no | — |
| 175 | C4⋊C4 | — | 8 | 1 | no | — |
| 176 | C4⋊C4 | — | 8 | 4 | no | — |
| 177 | C22⋊C4 | — | 4 | 2 | no | — |
| 178 | C22⋊C4 | — | 8 | 8 | no | — |
| 179 | C22⋊C4 | — | 8 | 1 | no | — |
| 180 | C22⋊C4 | — | 8 | 2 | no | — |
| 181 | C22⋊C4 | — | 8 | 2 | no | — |
| 182 | C42 | — | 4 | 2 | no | — |
| 183 | C42 | — | 4 | 2 | no | — |
| 184 | C42 | — | 4 | 1 | no | — |
| 185 | C42 | — | 4 | 1 | no | — |
| 186 | C42 | — | 8 | 1 | no | — |
| 187 | C16 | — | 4 | 4 | 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 | #450 | 1 | 1 | yes | characteristic |
| 197 | C23 | #460 | 1 | 1 | yes | characteristic |
| 198 | C23 | #496 | 1 | 1 | yes | characteristic |
| 199 | D4 | — | 8 | 2 | no | — |
| 200 | D4 | — | 8 | 2 | no | — |
| 201 | D4 | — | 8 | 2 | no | — |
| 202 | D4 | — | 8 | 2 | no | — |
| 203 | D4 | — | 16 | 8 | no | — |
| 204 | D4 | — | 16 | 4 | no | — |
| 205 | D4 | — | 16 | 4 | no | — |
| 206 | C2×C4 | — | 2 | 1 | no | — |
| 207 | C2×C4 | — | 2 | 1 | no | — |
| 208 | C2×C4 | — | 2 | 1 | no | — |
| 209 | C2×C4 | — | 2 | 1 | no | — |
| 210 | C2×C4 | — | 2 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 2 | 1 | no | — |
| 213 | C2×C4 | — | 2 | 1 | no | — |
| 214 | C2×C4 | — | 2 | 1 | no | — |
| 215 | C2×C4 | — | 2 | 1 | no | — |
| 216 | C2×C4 | — | 2 | 4 | no | — |
| 217 | C2×C4 | — | 4 | 2 | no | — |
| 218 | C2×C4 | — | 4 | 2 | no | — |
| 219 | C2×C4 | — | 8 | 4 | no | — |
| 220 | C2×C4 | — | 8 | 2 | no | — |
| 221 | C2×C4 | — | 8 | 4 | no | — |
| 222 | C2×C4 | — | 8 | 2 | no | — |
| 223 | C2×C4 | — | 16 | 2 | no | — |
| 224 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | #450 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 227 | C2×C4 | #454 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 230 | C2×C4 | #460 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | #492 | 1 | 1 | yes | characteristic |
| 232 | C8 | — | 2 | 2 | no | — |
| 233 | C8 | — | 2 | 2 | no | — |
| 234 | C8 | — | 4 | 2 | no | — |
| 235 | C8 | — | 8 | 2 | no | — |
| 236 | C22 | — | 2 | 2 | no | — |
| 237 | C22 | — | 2 | 1 | no | — |
| 238 | C22 | — | 2 | 1 | no | — |
| 239 | C22 | — | 4 | 1 | no | — |
| 240 | C22 | — | 4 | 1 | no | — |
| 241 | C22 | — | 4 | 2 | no | — |
| 242 | C22 | — | 8 | 1 | no | — |
| 243 | C22 | — | 16 | 4 | no | — |
| 244 | C22 | — | 16 | 2 | no | — |
| 245 | C22 | — | 16 | 2 | no | — |
| 246 | C22 | — | 16 | 4 | no | — |
| 247 | C22 | — | 16 | 2 | no | — |
| 248 | C22 | #1613 | 1 | 1 | yes | characteristic |
| 249 | C22 | #1616 | 1 | 1 | yes | characteristic |
| 250 | C22 | #2014 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 2 | 1 | no | — |
| 252 | C4 | — | 2 | 1 | no | — |
| 253 | C4 | — | 4 | 4 | no | — |
| 254 | C4 | — | 16 | 2 | no | — |
| 255 | C4 | — | 16 | 2 | no | — |
| 256 | C4 | #1613 | 1 | 1 | yes | characteristic |
| 257 | C4 | #1616 | 1 | 1 | yes | characteristic |
| 258 | C4 | #2006 | 1 | 1 | yes | characteristic |
| 259 | C4 | #2022 | 1 | 1 | yes | characteristic |
| 260 | C2 | — | 2 | 1 | no | — |
| 261 | C2 | — | 2 | 1 | no | — |
| 262 | C2 | — | 4 | 1 | no | — |
| 263 | C2 | — | 4 | 1 | no | — |
| 264 | C2 | — | 32 | 2 | no | — |
| 265 | C2 | #9968 | 1 | 1 | yes | characteristic |
| 266 | C2 | #12307 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| prime radical | #30247 | C2 |
| LCS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2238 | C2 | semidirect |
| 3. C2×#1339 | C2 | semidirect |
| 4. #11367 | C2 | semidirect |
| 5. #7193 | C2 | semidirect |
| 41. #430 | D4 | semidirect |
| 53. #349 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2238 | 2 |
| #30247 | 2 |
| C2×#1339 | 2 |
| #11367 | 2 |
| #7193 | 2 |