Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1670 conjugacy classes of subgroups.
The subgroup lattice has 287 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60666 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #60647 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33708 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #33704 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #33702 | C2 | 1 | 12 | yes | maximal, complemented |
| 6 | #33701 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×#2244 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #33381 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #33326 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C2×#576 | — | 2 | 1 | no | — |
| 11 | C2×#576 | C22 | 1 | 2 | yes | — |
| 12 | #3504 | — | 2 | 1 | no | — |
| 13 | #3498 | — | 2 | 1 | no | — |
| 14 | C2×#504 | C22 | 1 | 2 | yes | — |
| 15 | #3166 | — | 2 | 3 | no | — |
| 16 | #3164 | C22 | 1 | 6 | yes | complemented |
| 17 | #3163 | — | 2 | 3 | no | — |
| 18 | #3160 | C22 | 1 | 3 | yes | complemented |
| 19 | C2×#499 | C22 | 1 | 3 | yes | complemented |
| 20 | #3155 | — | 2 | 3 | no | — |
| 21 | #3155 | C22 | 1 | 6 | yes | — |
| 22 | #3151 | — | 2 | 3 | no | — |
| 23 | #3147 | C22 | 1 | 3 | yes | — |
| 24 | C2×#493 | C22 | 1 | 3 | yes | — |
| 25 | #2913 | C22 | 1 | 2 | yes | — |
| 26 | #2912 | — | 2 | 1 | no | — |
| 27 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2245 | C22 | 1 | 1 | yes | characteristic, complemented |
| 29 | #2245 | C22 | 1 | 3 | yes | complemented |
| 30 | #2244 | C22 | 1 | 3 | yes | complemented |
| 31 | #2244 | C22 | 1 | 12 | yes | complemented |
| 32 | #2244 | C22 | 1 | 6 | yes | complemented |
| 33 | #2244 | C22 | 1 | 3 | yes | complemented |
| 34 | #2243 | C22 | 1 | 12 | yes | complemented |
| 35 | #2242 | C22 | 1 | 12 | yes | complemented |
| 36 | #2240 | C22 | 1 | 12 | yes | complemented |
| 37 | #2140 | C22 | 1 | 12 | yes | complemented |
| 38 | #2139 | C22 | 1 | 12 | yes | complemented |
| 39 | #2138 | C22 | 1 | 12 | yes | complemented |
| 40 | C2×#357 | C22 | 1 | 12 | yes | complemented |
| 41 | #2108 | C22 | 1 | 6 | yes | complemented |
| 42 | C4×M32 | C22 | 1 | 3 | yes | complemented |
| 43 | C2×#346 | C22 | 1 | 3 | yes | complemented |
| 44 | C2×H(5,𝔽2) | — | 2 | 2 | no | — |
| 45 | #576 | — | 2 | 16 | no | — |
| 46 | #576 | — | 4 | 8 | no | — |
| 47 | C2×AGL(1,ℤ/8) | — | 2 | 6 | no | — |
| 48 | C2×AGL(1,ℤ/8) | — | 2 | 6 | no | — |
| 49 | C2×#135 | — | 2 | 3 | no | — |
| 50 | C2×#135 | C23 | 1 | 6 | yes | — |
| 51 | C22×D8 | — | 2 | 3 | no | — |
| 52 | C22×D8 | C23 | 1 | 6 | yes | — |
| 53 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 54 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 55 | #557 | — | 4 | 2 | no | — |
| 56 | #554 | — | 4 | 6 | no | — |
| 57 | #550 | — | 4 | 2 | no | — |
| 58 | D42 | — | 4 | 6 | no | — |
| 59 | #544 | — | 2 | 6 | no | — |
| 60 | #535 | — | 2 | 6 | no | — |
| 61 | #533 | — | 2 | 3 | no | — |
| 62 | #532 | — | 2 | 3 | no | — |
| 63 | C2×#127 | — | 2 | 3 | no | — |
| 64 | C2×#125 | — | 2 | 3 | no | — |
| 65 | #520 | — | 2 | 2 | no | — |
| 66 | C4×#41 | D4 | 1 | 1 | yes | characteristic |
| 67 | #504 | — | 2 | 8 | no | — |
| 68 | C2×M32 | C23 | 1 | 6 | yes | — |
| 69 | C22×C16 | C23 | 1 | 6 | yes | — |
| 70 | #501 | — | 2 | 12 | no | — |
| 71 | #500 | C23 | 1 | 6 | yes | — |
| 72 | #499 | — | 2 | 3 | no | — |
| 73 | #499 | — | 2 | 3 | no | — |
| 74 | #499 | — | 2 | 3 | no | — |
| 75 | #499 | C23 | 1 | 3 | yes | — |
| 76 | #499 | C23 | 1 | 6 | yes | — |
| 77 | #499 | C23 | 1 | 1 | yes | characteristic |
| 78 | #498 | — | 2 | 3 | no | — |
| 79 | #496 | — | 2 | 12 | no | — |
| 80 | #495 | — | 2 | 3 | no | — |
| 81 | #495 | C23 | 1 | 6 | yes | — |
| 82 | #494 | C23 | 1 | 1 | yes | characteristic |
| 83 | #493 | — | 2 | 3 | no | — |
| 84 | #493 | — | 2 | 3 | no | — |
| 85 | #493 | C23 | 1 | 3 | yes | — |
| 86 | #493 | C23 | 1 | 6 | yes | — |
| 87 | #492 | — | 2 | 3 | no | — |
| 88 | #428 | — | 2 | 6 | no | — |
| 89 | #428 | — | 2 | 6 | no | — |
| 90 | #427 | — | 2 | 3 | no | — |
| 91 | #427 | C23 | 1 | 6 | yes | — |
| 92 | C2×#107 | C23 | 1 | 6 | yes | — |
| 93 | C2×#106 | — | 2 | 3 | no | — |
| 94 | #405 | D4 | 1 | 3 | yes | — |
| 95 | #405 | C23 | 1 | 3 | yes | — |
| 96 | C4×M16 | D4 | 1 | 3 | yes | — |
| 97 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 98 | #358 | C23 | 1 | 4 | yes | — |
| 99 | #358 | C23 | 1 | 12 | yes | — |
| 100 | #357 | C23 | 1 | 12 | yes | — |
| 101 | #357 | C23 | 1 | 24 | yes | — |
| 102 | #357 | C23 | 1 | 12 | yes | — |
| 103 | #346 | C23 | 1 | 1 | yes | characteristic |
| 104 | #346 | C23 | 1 | 3 | yes | — |
| 105 | #346 | C23 | 1 | 6 | yes | — |
| 106 | C4×C16 | C23 | 1 | 6 | yes | — |
| 107 | H(5,𝔽2) | — | 4 | 8 | no | — |
| 108 | H(5,𝔽2) | — | 4 | 8 | no | — |
| 109 | C2×#41 | — | 2 | 6 | no | — |
| 110 | C2×#41 | — | 4 | 2 | no | — |
| 111 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C22×D4 | — | 2 | 6 | no | — |
| 113 | C22×D4 | — | 4 | 6 | no | — |
| 114 | C22×D4 | — | 4 | 6 | no | — |
| 115 | AGL(1,ℤ/8) | — | 4 | 24 | no | — |
| 116 | AGL(1,ℤ/8) | — | 4 | 24 | no | — |
| 117 | #135 | — | 2 | 24 | no | — |
| 118 | #135 | — | 4 | 12 | no | — |
| 119 | C2×Q16 | — | 2 | 1 | no | — |
| 120 | C2×Q16 | C24 | 1 | 2 | yes | — |
| 121 | C2×SD16 | — | 2 | 6 | no | — |
| 122 | C2×SD16 | — | 2 | 6 | no | — |
| 123 | C2×D8 | — | 2 | 3 | no | — |
| 124 | C2×D8 | — | 2 | 6 | no | — |
| 125 | C2×D8 | — | 2 | 6 | no | — |
| 126 | C2×D8 | — | 2 | 24 | no | — |
| 127 | C2×D8 | — | 4 | 12 | no | — |
| 128 | C2×D8 | C24 | 1 | 6 | yes | — |
| 129 | #131 | SD16 | 1 | 8 | yes | — |
| 130 | #131 | #41 | 1 | 8 | yes | — |
| 131 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 132 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 133 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 134 | C22×C8 | C24 | 1 | 3 | yes | — |
| 135 | C4⋊Q8 | — | 2 | 6 | no | — |
| 136 | C4⋊Q8 | — | 2 | 1 | no | — |
| 137 | C4⋊Q8 | — | 4 | 6 | no | — |
| 138 | #127 | — | 2 | 3 | no | — |
| 139 | #127 | — | 2 | 6 | no | — |
| 140 | #127 | — | 4 | 6 | no | — |
| 141 | #125 | — | 2 | 3 | no | — |
| 142 | #125 | — | 2 | 6 | no | — |
| 143 | #125 | — | 2 | 1 | no | — |
| 144 | #125 | — | 4 | 6 | no | — |
| 145 | #125 | — | 4 | 6 | no | — |
| 146 | #124 | — | 2 | 6 | no | — |
| 147 | C22⋊Q8 | — | 4 | 6 | no | — |
| 148 | C22⋊Q8 | — | 4 | 12 | no | — |
| 149 | #121 | — | 4 | 6 | no | — |
| 150 | #121 | — | 4 | 12 | no | — |
| 151 | #120 | — | 4 | 12 | no | — |
| 152 | C4×Q8 | — | 4 | 2 | no | — |
| 153 | C4×Q8 | — | 4 | 6 | no | — |
| 154 | C4×Q8 | D8 | 1 | 2 | yes | — |
| 155 | C4×D4 | — | 4 | 6 | no | — |
| 156 | C4×D4 | — | 4 | 6 | no | — |
| 157 | C4×D4 | D8 | 1 | 6 | yes | — |
| 158 | #117 | — | 2 | 6 | no | — |
| 159 | #117 | C2×D4 | 1 | 3 | yes | — |
| 160 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 161 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 162 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 163 | M32 | — | 2 | 12 | no | — |
| 164 | C2×C16 | — | 2 | 12 | no | — |
| 165 | C2×C16 | C24 | 1 | 6 | yes | — |
| 166 | C2×C16 | C24 | 1 | 2 | yes | — |
| 167 | #107 | — | 2 | 6 | no | — |
| 168 | #107 | — | 2 | 6 | no | — |
| 169 | #107 | C24 | 1 | 6 | yes | — |
| 170 | #107 | C24 | 1 | 2 | yes | — |
| 171 | #106 | — | 2 | 3 | no | — |
| 172 | #106 | — | 2 | 6 | no | — |
| 173 | #106 | — | 2 | 1 | no | — |
| 174 | #106 | — | 2 | 6 | no | — |
| 175 | #97 | C2×D4 | 1 | 6 | yes | — |
| 176 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 177 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 178 | C4×C8 | C24 | 1 | 3 | yes | — |
| 179 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 180 | C24 | — | 4 | 6 | no | — |
| 181 | #41 | — | 4 | 12 | no | — |
| 182 | #41 | — | 4 | 12 | no | — |
| 183 | #41 | — | 8 | 8 | no | — |
| 184 | #41 | #124 | 1 | 4 | yes | — |
| 185 | #41 | C2×SD16 | 1 | 4 | yes | — |
| 186 | C2×Q8 | — | 2 | 2 | no | — |
| 187 | C2×Q8 | — | 4 | 6 | no | — |
| 188 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 189 | C2×D4 | — | 2 | 6 | no | — |
| 190 | C2×D4 | — | 4 | 6 | no | — |
| 191 | C2×D4 | — | 4 | 12 | no | — |
| 192 | C2×D4 | — | 4 | 6 | no | — |
| 193 | C2×D4 | — | 4 | 6 | no | — |
| 194 | C2×D4 | — | 4 | 12 | no | — |
| 195 | C2×D4 | — | 4 | 12 | no | — |
| 196 | C2×D4 | — | 8 | 24 | no | — |
| 197 | C2×D4 | — | 16 | 12 | no | — |
| 198 | C2×D4 | C2×D8 | 1 | 3 | yes | — |
| 199 | C22×C4 | — | 4 | 6 | no | — |
| 200 | C22×C4 | — | 4 | 6 | no | — |
| 201 | C22×C4 | C2×D8 | 1 | 6 | yes | — |
| 202 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 203 | Q16 | — | 2 | 4 | no | — |
| 204 | Q16 | — | 4 | 2 | no | — |
| 205 | SD16 | — | 4 | 12 | no | — |
| 206 | SD16 | — | 4 | 12 | no | — |
| 207 | D8 | — | 2 | 12 | no | — |
| 208 | D8 | — | 4 | 12 | no | — |
| 209 | D8 | — | 4 | 6 | no | — |
| 210 | D8 | — | 4 | 12 | no | — |
| 211 | M16 | #124 | 1 | 12 | yes | — |
| 212 | M16 | C2×SD16 | 1 | 12 | yes | — |
| 213 | C2×C8 | C2×SD16 | 1 | 12 | yes | — |
| 214 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 215 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 216 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 217 | C2×C8 | C2×#41 | 1 | 12 | yes | — |
| 218 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 219 | C4⋊C4 | — | 2 | 6 | no | — |
| 220 | C4⋊C4 | — | 2 | 2 | no | — |
| 221 | C4⋊C4 | — | 4 | 6 | no | — |
| 222 | C4⋊C4 | — | 4 | 6 | no | — |
| 223 | C4⋊C4 | — | 4 | 12 | no | — |
| 224 | C4⋊C4 | — | 4 | 2 | no | — |
| 225 | C4⋊C4 | — | 4 | 2 | no | — |
| 226 | C4⋊C4 | — | 4 | 6 | no | — |
| 227 | C4⋊C4 | C2×D8 | 1 | 6 | yes | — |
| 228 | C22⋊C4 | — | 4 | 6 | no | — |
| 229 | C22⋊C4 | — | 4 | 12 | no | — |
| 230 | C22⋊C4 | C2×D8 | 1 | 6 | yes | — |
| 231 | C42 | — | 4 | 6 | no | — |
| 232 | C42 | C2×D8 | 1 | 6 | yes | — |
| 233 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 234 | C42 | C22×D4 | 1 | 3 | yes | — |
| 235 | C16 | — | 2 | 2 | no | — |
| 236 | C16 | — | 2 | 6 | no | — |
| 237 | C23 | — | 4 | 6 | no | — |
| 238 | C23 | — | 8 | 12 | no | — |
| 239 | C23 | — | 16 | 12 | no | — |
| 240 | C23 | C22×D8 | 1 | 3 | yes | — |
| 241 | Q8 | — | 4 | 2 | no | — |
| 242 | Q8 | — | 4 | 2 | no | — |
| 243 | Q8 | — | 16 | 6 | no | — |
| 244 | Q8 | #486 | 1 | 4 | yes | — |
| 245 | D4 | — | 4 | 6 | no | — |
| 246 | D4 | — | 4 | 6 | no | — |
| 247 | D4 | — | 8 | 12 | no | — |
| 248 | D4 | — | 8 | 12 | no | — |
| 249 | D4 | — | 8 | 12 | no | — |
| 250 | D4 | — | 16 | 12 | no | — |
| 251 | D4 | #486 | 1 | 12 | yes | — |
| 252 | C2×C4 | — | 4 | 6 | no | — |
| 253 | C2×C4 | — | 4 | 2 | no | — |
| 254 | C2×C4 | — | 4 | 6 | no | — |
| 255 | C2×C4 | — | 4 | 2 | no | — |
| 256 | C2×C4 | — | 8 | 12 | no | — |
| 257 | C2×C4 | — | 16 | 6 | no | — |
| 258 | C2×C4 | C2×#124 | 1 | 6 | yes | — |
| 259 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 260 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 261 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 262 | C2×C4 | C22×SD16 | 1 | 6 | yes | — |
| 263 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 264 | C8 | C2×#124 | 1 | 6 | yes | — |
| 265 | C8 | C22×SD16 | 1 | 2 | yes | — |
| 266 | C8 | C22×SD16 | 1 | 6 | yes | — |
| 267 | C8 | C22×#41 | 1 | 2 | yes | — |
| 268 | C22 | — | 2 | 6 | no | — |
| 269 | C22 | — | 8 | 12 | no | — |
| 270 | C22 | — | 8 | 6 | no | — |
| 271 | C22 | — | 32 | 12 | no | — |
| 272 | C22 | C2×#486 | 1 | 6 | yes | — |
| 273 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 274 | C4 | — | 2 | 2 | no | — |
| 275 | C4 | — | 2 | 6 | no | — |
| 276 | C4 | — | 8 | 6 | no | — |
| 277 | C4 | — | 8 | 2 | no | — |
| 278 | C4 | — | 8 | 2 | no | — |
| 279 | C4 | C2×#486 | 1 | 6 | yes | — |
| 280 | C4 | C22×#124 | 1 | 1 | yes | characteristic |
| 281 | C4 | C23×SD16 | 1 | 1 | yes | characteristic |
| 282 | C2 | — | 2 | 6 | no | — |
| 283 | C2 | — | 16 | 6 | no | — |
| 284 | C2 | C22×#486 | 1 | 1 | yes | characteristic |
| 285 | C2 | #62636 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #60647 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #60666 | C2 | semidirect |
| 3. #33708 | C2 | semidirect |
| 4. #33704 | C2 | semidirect |
| 5. #33702 | C2 | semidirect |
| 6. #33701 | C2 | semidirect |
| 7. C2×#2244 | C2 | semidirect |
| 8. #33381 | C2 | semidirect |
| 9. #33326 | C2 | semidirect |
| 16. #3164 | C22 | semidirect |
| 18. #3160 | C22 | semidirect |
| 19. C2×#499 | C22 | semidirect |
| 28. #2245 | C22 | semidirect |
| 29. #2245 | C22 | semidirect |
| 30. #2244 | C22 | semidirect |
| 31. #2244 | C22 | semidirect |
| 32. #2244 | C22 | semidirect |
| 33. #2244 | C22 | semidirect |
| 34. #2243 | C22 | semidirect |
| 35. #2242 | C22 | semidirect |
| 36. #2240 | C22 | semidirect |
| 37. #2140 | C22 | semidirect |
| 38. #2139 | C22 | semidirect |
| 39. #2138 | C22 | semidirect |
| 40. C2×#357 | C22 | semidirect |
| 41. #2108 | C22 | semidirect |
| 42. C4×M32 | C22 | semidirect |
| 43. C2×#346 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60666 | 2 |
| #60647 | 2 |
| #33708 | 2 |
| #33704 | 2 |
| #33702 | 2 |
| #33701 | 2 |
| C2×#2244 | 2 |
| #33381 | 2 |
| #33326 | 2 |