Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4316 conjugacy classes of subgroups.
The subgroup lattice has 293 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3498 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2991 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #17782 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 4 | C2×#1560 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#1545 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×#1543 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C22×H(5,𝔽2) | C4 | 1 | 2 | yes | — |
| 8 | #3498 | — | 2 | 8 | no | — |
| 9 | #3498 | C4 | 1 | 8 | yes | — |
| 10 | #3498 | C22 | 1 | 8 | yes | — |
| 11 | C2×#550 | — | 2 | 2 | no | — |
| 12 | C2×D42 | — | 2 | 6 | no | — |
| 13 | C2×#535 | C4 | 1 | 6 | yes | — |
| 14 | C2×#532 | C22 | 1 | 3 | yes | — |
| 15 | C22×#127 | C22 | 1 | 3 | yes | — |
| 16 | C2×C4×#41 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2991 | — | 2 | 8 | no | — |
| 18 | #2991 | C22 | 1 | 16 | yes | complemented |
| 19 | C2×C8×Q8 | — | 2 | 2 | no | — |
| 20 | C2×#436 | — | 2 | 6 | no | — |
| 21 | C2×#431 | C22 | 1 | 6 | yes | complemented |
| 22 | C2×#423 | C22 | 1 | 6 | yes | complemented |
| 23 | #1560 | C22 | 1 | 24 | yes | complemented |
| 24 | #1545 | C22 | 1 | 48 | yes | complemented |
| 25 | #1543 | C22 | 1 | 24 | yes | complemented |
| 26 | C2×#333 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | C2×#333 | C22 | 1 | 3 | yes | complemented |
| 28 | C2×#331 | C22 | 1 | 6 | yes | complemented |
| 29 | C2×#331 | C22 | 1 | 6 | yes | complemented |
| 30 | C2×H(5,𝔽2) | — | 2 | 8 | no | — |
| 31 | C2×H(5,𝔽2) | — | 4 | 16 | no | — |
| 32 | C2×H(5,𝔽2) | C2×C4 | 1 | 16 | yes | — |
| 33 | C22×#41 | — | 2 | 2 | no | — |
| 34 | C22×#41 | C2×C4 | 1 | 6 | yes | — |
| 35 | C22×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C23×D4 | — | 2 | 6 | no | — |
| 37 | C23×D4 | — | 2 | 6 | no | — |
| 38 | C23×D4 | C2×C4 | 1 | 6 | yes | — |
| 39 | C22×M16 | — | 2 | 6 | no | — |
| 40 | #550 | — | 2 | 16 | no | — |
| 41 | #550 | — | 2 | 16 | no | — |
| 42 | D42 | — | 2 | 48 | no | — |
| 43 | D42 | — | 2 | 48 | no | — |
| 44 | #535 | — | 2 | 24 | no | — |
| 45 | #535 | C2×C4 | 1 | 48 | yes | — |
| 46 | #532 | — | 2 | 12 | no | — |
| 47 | #532 | C2×C4 | 1 | 12 | yes | — |
| 48 | #532 | C23 | 1 | 12 | yes | — |
| 49 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×#127 | — | 2 | 6 | no | — |
| 51 | C2×#127 | — | 2 | 12 | no | — |
| 52 | C2×#127 | C2×C4 | 1 | 6 | yes | — |
| 53 | C2×#127 | C2×C4 | 1 | 12 | yes | — |
| 54 | C2×#127 | C23 | 1 | 3 | yes | — |
| 55 | C2×#127 | C23 | 1 | 12 | yes | — |
| 56 | C2×#124 | C2×C4 | 1 | 6 | yes | — |
| 57 | C2×#121 | — | 2 | 6 | no | — |
| 58 | C2×#121 | — | 2 | 12 | no | — |
| 59 | C2×#120 | — | 2 | 12 | no | — |
| 60 | C4×#41 | C2×C4 | 1 | 4 | yes | — |
| 61 | C4×#41 | D4 | 1 | 8 | yes | — |
| 62 | C4×#41 | C23 | 1 | 4 | yes | — |
| 63 | C2×C4×Q8 | D4 | 1 | 2 | yes | — |
| 64 | C2×C4×D4 | — | 2 | 6 | no | — |
| 65 | C2×C4×D4 | D4 | 1 | 6 | yes | — |
| 66 | C2×#117 | C2×C4 | 1 | 3 | yes | — |
| 67 | C22×C42 | C23 | 1 | 3 | yes | — |
| 68 | C8×Q8 | — | 2 | 16 | no | — |
| 69 | #436 | — | 2 | 48 | no | — |
| 70 | #431 | — | 2 | 12 | no | — |
| 71 | #431 | C23 | 1 | 24 | yes | complemented |
| 72 | #423 | — | 2 | 12 | no | — |
| 73 | #423 | C23 | 1 | 24 | yes | complemented |
| 74 | C2×C4⋊C8 | — | 2 | 6 | no | — |
| 75 | C2×C4⋊C8 | C23 | 1 | 6 | yes | — |
| 76 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 77 | C2×C22⋊C8 | — | 2 | 6 | no | — |
| 78 | C2×C4×C8 | — | 2 | 6 | no | — |
| 79 | #333 | C23 | 1 | 4 | yes | complemented |
| 80 | #333 | C23 | 1 | 12 | yes | complemented |
| 81 | #331 | C23 | 1 | 24 | yes | complemented |
| 82 | #331 | C23 | 1 | 24 | yes | complemented |
| 83 | C25 | — | 2 | 6 | no | — |
| 84 | H(5,𝔽2) | — | 4 | 64 | no | — |
| 85 | H(5,𝔽2) | — | 4 | 64 | no | — |
| 86 | C2×#41 | — | 2 | 8 | no | — |
| 87 | C2×#41 | — | 2 | 16 | no | — |
| 88 | C2×#41 | — | 2 | 12 | no | — |
| 89 | C2×#41 | — | 4 | 24 | no | — |
| 90 | C2×#41 | — | 4 | 16 | no | — |
| 91 | C2×#41 | C22⋊C4 | 1 | 4 | yes | — |
| 92 | C2×#41 | C22×C4 | 1 | 24 | yes | — |
| 93 | C2×#41 | C22×C4 | 1 | 4 | yes | — |
| 94 | C22×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 95 | C22×Q8 | C22×C4 | 1 | 2 | yes | — |
| 96 | C22×D4 | — | 2 | 6 | no | — |
| 97 | C22×D4 | — | 2 | 48 | no | — |
| 98 | C22×D4 | — | 2 | 6 | no | — |
| 99 | C22×D4 | — | 2 | 12 | no | — |
| 100 | C22×D4 | — | 2 | 12 | no | — |
| 101 | C22×D4 | — | 2 | 24 | no | — |
| 102 | C22×D4 | — | 2 | 6 | no | — |
| 103 | C22×D4 | — | 2 | 24 | no | — |
| 104 | C22×D4 | — | 4 | 24 | no | — |
| 105 | C22×D4 | — | 4 | 48 | no | — |
| 106 | C22×D4 | — | 8 | 12 | no | — |
| 107 | C22×D4 | — | 8 | 12 | no | — |
| 108 | C22×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 109 | C22×D4 | C22×C4 | 1 | 6 | yes | — |
| 110 | C22×D4 | C22×C4 | 1 | 24 | yes | — |
| 111 | C23×C4 | — | 2 | 6 | no | — |
| 112 | C23×C4 | C22×C4 | 1 | 3 | yes | — |
| 113 | C23×C4 | C2×D4 | 1 | 6 | yes | — |
| 114 | C2×M16 | — | 2 | 24 | no | — |
| 115 | C2×M16 | — | 8 | 12 | no | — |
| 116 | C22×C8 | — | 2 | 6 | no | — |
| 117 | C22×C8 | — | 2 | 2 | no | — |
| 118 | C4⋊Q8 | — | 2 | 2 | no | — |
| 119 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 120 | C4⋊Q8 | C24 | 1 | 2 | yes | — |
| 121 | #127 | — | 2 | 12 | no | — |
| 122 | #127 | — | 2 | 24 | no | — |
| 123 | #127 | — | 2 | 6 | no | — |
| 124 | #127 | — | 2 | 24 | no | — |
| 125 | #127 | C22×C4 | 1 | 6 | yes | — |
| 126 | #127 | C22×C4 | 1 | 24 | yes | — |
| 127 | #127 | C24 | 1 | 6 | yes | — |
| 128 | #124 | — | 2 | 12 | no | — |
| 129 | #124 | C22×C4 | 1 | 24 | yes | — |
| 130 | #121 | — | 2 | 24 | no | — |
| 131 | #121 | — | 2 | 24 | no | — |
| 132 | #121 | — | 2 | 96 | no | — |
| 133 | #120 | — | 2 | 96 | no | — |
| 134 | C4×Q8 | C22⋊C4 | 1 | 8 | yes | — |
| 135 | C4×Q8 | C2×D4 | 1 | 8 | yes | — |
| 136 | C4×D4 | — | 2 | 24 | no | — |
| 137 | C4×D4 | — | 2 | 24 | no | — |
| 138 | C4×D4 | C22⋊C4 | 1 | 24 | yes | — |
| 139 | C4×D4 | C2×D4 | 1 | 24 | yes | — |
| 140 | #117 | C22⋊C4 | 1 | 12 | yes | — |
| 141 | #117 | C22×C4 | 1 | 12 | yes | — |
| 142 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 143 | C2×C4⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 144 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 145 | C2×C22⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 146 | C2×C42 | C22×C4 | 1 | 3 | yes | — |
| 147 | C2×C42 | C22×C4 | 1 | 6 | yes | — |
| 148 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 149 | C2×C42 | C2×D4 | 1 | 12 | yes | — |
| 150 | C2×C42 | C24 | 1 | 6 | yes | — |
| 151 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 152 | C4⋊C8 | — | 2 | 24 | no | — |
| 153 | C4⋊C8 | — | 2 | 6 | no | — |
| 154 | C4⋊C8 | — | 2 | 2 | no | — |
| 155 | C4⋊C8 | C24 | 1 | 12 | yes | — |
| 156 | C4⋊C8 | C24 | 1 | 4 | yes | — |
| 157 | C22⋊C8 | — | 2 | 24 | no | — |
| 158 | C4×C8 | — | 2 | 24 | no | — |
| 159 | C24 | — | 2 | 6 | no | — |
| 160 | C24 | — | 2 | 24 | no | — |
| 161 | C24 | — | 4 | 24 | no | — |
| 162 | C24 | — | 8 | 24 | no | — |
| 163 | C24 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 164 | #41 | — | 2 | 16 | no | — |
| 165 | #41 | — | 2 | 16 | no | — |
| 166 | #41 | — | 4 | 48 | no | — |
| 167 | #41 | — | 4 | 48 | no | — |
| 168 | #41 | — | 4 | 64 | no | — |
| 169 | C2×Q8 | — | 2 | 2 | no | — |
| 170 | C2×Q8 | — | 4 | 4 | no | — |
| 171 | C2×Q8 | #100 | 1 | 8 | yes | — |
| 172 | C2×Q8 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 173 | C2×Q8 | C23×C4 | 1 | 4 | yes | — |
| 174 | C2×D4 | — | 2 | 24 | no | — |
| 175 | C2×D4 | — | 2 | 12 | no | — |
| 176 | C2×D4 | — | 2 | 24 | no | — |
| 177 | C2×D4 | — | 2 | 6 | no | — |
| 178 | C2×D4 | — | 2 | 48 | no | — |
| 179 | C2×D4 | — | 2 | 12 | no | — |
| 180 | C2×D4 | — | 2 | 12 | no | — |
| 181 | C2×D4 | — | 4 | 12 | no | — |
| 182 | C2×D4 | — | 4 | 48 | no | — |
| 183 | C2×D4 | — | 4 | 48 | no | — |
| 184 | C2×D4 | — | 4 | 192 | no | — |
| 185 | C2×D4 | — | 4 | 24 | no | — |
| 186 | C2×D4 | — | 4 | 12 | no | — |
| 187 | C2×D4 | — | 4 | 12 | no | — |
| 188 | C2×D4 | — | 4 | 24 | no | — |
| 189 | C2×D4 | — | 8 | 48 | no | — |
| 190 | C2×D4 | — | 8 | 24 | no | — |
| 191 | C2×D4 | — | 8 | 24 | no | — |
| 192 | C2×D4 | — | 8 | 24 | no | — |
| 193 | C2×D4 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 194 | C2×D4 | C23×C4 | 1 | 12 | yes | — |
| 195 | C22×C4 | — | 2 | 12 | no | — |
| 196 | C22×C4 | — | 2 | 12 | no | — |
| 197 | C22×C4 | — | 2 | 12 | no | — |
| 198 | C22×C4 | — | 2 | 2 | no | — |
| 199 | C22×C4 | — | 2 | 24 | no | — |
| 200 | C22×C4 | — | 4 | 24 | no | — |
| 201 | C22×C4 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 202 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 203 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 204 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 205 | C22×C4 | C23×C4 | 1 | 6 | yes | — |
| 206 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 207 | C22×C4 | C22×D4 | 1 | 12 | yes | — |
| 208 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 209 | M16 | — | 8 | 24 | no | — |
| 210 | C2×C8 | — | 2 | 4 | no | — |
| 211 | C2×C8 | — | 2 | 12 | no | — |
| 212 | C2×C8 | — | 4 | 12 | no | — |
| 213 | C2×C8 | — | 4 | 4 | no | — |
| 214 | C4⋊C4 | — | 2 | 4 | no | — |
| 215 | C4⋊C4 | — | 2 | 4 | no | — |
| 216 | C4⋊C4 | C2×C22⋊C4 | 1 | 24 | yes | — |
| 217 | C22⋊C4 | — | 2 | 48 | no | — |
| 218 | C22⋊C4 | C2×C22⋊C4 | 1 | 24 | yes | — |
| 219 | C42 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 220 | C42 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 221 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 222 | C42 | C23×C4 | 1 | 6 | yes | — |
| 223 | C42 | C22×D4 | 1 | 12 | yes | — |
| 224 | C42 | C22×D4 | 1 | 2 | yes | — |
| 225 | C42 | C25 | 1 | 1 | yes | characteristic |
| 226 | C23 | — | 2 | 6 | no | — |
| 227 | C23 | — | 2 | 12 | no | — |
| 228 | C23 | — | 2 | 12 | no | — |
| 229 | C23 | — | 4 | 48 | no | — |
| 230 | C23 | — | 4 | 12 | no | — |
| 231 | C23 | — | 4 | 24 | no | — |
| 232 | C23 | — | 8 | 48 | no | — |
| 233 | C23 | — | 16 | 48 | no | — |
| 234 | C23 | C22×C22⋊C4 | 1 | 6 | yes | — |
| 235 | C23 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 236 | Q8 | — | 4 | 4 | no | — |
| 237 | Q8 | — | 4 | 4 | no | — |
| 238 | Q8 | C2×#100 | 1 | 16 | yes | — |
| 239 | D4 | — | 2 | 24 | no | — |
| 240 | D4 | — | 4 | 12 | no | — |
| 241 | D4 | — | 4 | 12 | no | — |
| 242 | D4 | — | 4 | 48 | no | — |
| 243 | D4 | — | 4 | 48 | no | — |
| 244 | D4 | — | 4 | 24 | no | — |
| 245 | D4 | — | 4 | 12 | no | — |
| 246 | D4 | — | 4 | 12 | no | — |
| 247 | D4 | — | 8 | 48 | no | — |
| 248 | C2×C4 | — | 2 | 2 | no | — |
| 249 | C2×C4 | — | 2 | 12 | no | — |
| 250 | C2×C4 | — | 2 | 12 | no | — |
| 251 | C2×C4 | — | 2 | 12 | no | — |
| 252 | C2×C4 | — | 2 | 24 | no | — |
| 253 | C2×C4 | — | 2 | 12 | no | — |
| 254 | C2×C4 | — | 2 | 12 | no | — |
| 255 | C2×C4 | — | 2 | 4 | no | — |
| 256 | C2×C4 | — | 4 | 4 | no | — |
| 257 | C2×C4 | — | 4 | 48 | no | — |
| 258 | C2×C4 | C2×#100 | 1 | 12 | yes | — |
| 259 | C2×C4 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 260 | C2×C4 | C22×C22⋊C4 | 1 | 12 | yes | — |
| 261 | C2×C4 | C22×C22⋊C4 | 1 | 6 | yes | — |
| 262 | C2×C4 | C22×C22⋊C4 | 1 | 2 | yes | — |
| 263 | C2×C4 | #576 | 1 | 4 | yes | — |
| 264 | C2×C4 | #577 | 1 | 4 | yes | — |
| 265 | C2×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | C23×D4 | 1 | 2 | yes | — |
| 267 | C8 | — | 4 | 12 | no | — |
| 268 | C8 | — | 4 | 4 | no | — |
| 269 | C22 | — | 2 | 6 | no | — |
| 270 | C22 | — | 2 | 12 | no | — |
| 271 | C22 | — | 4 | 12 | no | — |
| 272 | C22 | — | 4 | 12 | no | — |
| 273 | C22 | — | 4 | 24 | no | — |
| 274 | C22 | — | 8 | 24 | no | — |
| 275 | C22 | — | 16 | 48 | no | — |
| 276 | C22 | C22×#100 | 1 | 2 | yes | — |
| 277 | C22 | #2897 | 1 | 4 | yes | — |
| 278 | C22 | C23×C22⋊C4 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 2 | 1 | no | — |
| 280 | C4 | — | 2 | 1 | no | — |
| 281 | C4 | — | 2 | 12 | no | — |
| 282 | C4 | — | 4 | 4 | no | — |
| 283 | C4 | C22×#100 | 1 | 12 | yes | — |
| 284 | C4 | #2897 | 1 | 4 | yes | — |
| 285 | C4 | C2×#576 | 1 | 2 | yes | — |
| 286 | C4 | C2×#577 | 1 | 2 | yes | — |
| 287 | C2 | — | 4 | 6 | no | — |
| 288 | C2 | — | 8 | 12 | no | — |
| 289 | C2 | #17782 | 1 | 4 | yes | direct product factor |
| 290 | C2 | C23×#100 | 1 | 1 | yes | characteristic |
| 291 | C2 | C2×#2897 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2991 | C2 | semidirect |
| 3. #17782 | C2 | direct (N × H) |
| 4. C2×#1560 | C2 | semidirect |
| 5. C2×#1545 | C2 | semidirect |
| 6. C2×#1543 | C2 | semidirect |
| 18. #2991 | C22 | semidirect |
| 21. C2×#431 | C22 | semidirect |
| 22. C2×#423 | C22 | semidirect |
| 23. #1560 | C22 | semidirect |
| 24. #1545 | C22 | semidirect |
| 25. #1543 | C22 | semidirect |
| 26. C2×#333 | C22 | semidirect |
| 27. C2×#333 | C22 | semidirect |
| 28. C2×#331 | C22 | semidirect |
| 29. C2×#331 | C22 | semidirect |
| 71. #431 | C23 | semidirect |
| 73. #423 | C23 | semidirect |
| 79. #333 | C23 | semidirect |
| 80. #333 | C23 | semidirect |
| 81. #331 | C23 | semidirect |
| 82. #331 | C23 | semidirect |
| 289. C2 | #17782 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3498 | 2 |
| C2×#2991 | 2 |
| #17782 | 2 |
| C2×#1560 | 2 |
| C2×#1545 | 2 |
| C2×#1543 | 2 |