Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
22658 conjugacy classes of subgroups; only the 4002 normal subgroups are shown.
This group has 22658 conjugacy classes of subgroups — too many to draw. Only its 4002 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 | C23×2−1+4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C23×AGL(1,ℤ/8) | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C23×#124 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×#454 | C2 | 1 | 56 | yes | maximal, direct product factor |
| 5 | C23×#104 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C23×#101 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C22×2−1+4 | — | 2 | 28 | no | — |
| 8 | C22×2−1+4 | — | 2 | 14 | no | — |
| 9 | C22×2−1+4 | C22 | 1 | 28 | yes | complemented |
| 10 | C23×#41 | — | 2 | 2 | no | — |
| 11 | C23×#41 | — | 2 | 2 | no | — |
| 12 | C23×#41 | C22 | 1 | 2 | yes | — |
| 13 | C24×Q8 | — | 2 | 2 | no | — |
| 14 | C24×Q8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C24×D4 | C22 | 1 | 1 | yes | characteristic |
| 16 | C22×AGL(1,ℤ/8) | — | 2 | 28 | no | — |
| 17 | C22×AGL(1,ℤ/8) | C22 | 1 | 56 | yes | complemented |
| 18 | C23×SD16 | — | 2 | 2 | no | — |
| 19 | C23×D8 | — | 2 | 2 | no | — |
| 20 | C23×M16 | C22 | 1 | 2 | yes | — |
| 21 | C22×#124 | — | 2 | 14 | no | — |
| 22 | C22×#124 | C22 | 1 | 28 | yes | complemented |
| 23 | C23×C22⋊C4 | — | 2 | 2 | no | — |
| 24 | C23×C42 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#454 | C22 | 1 | 448 | yes | direct product factor |
| 26 | C22×#104 | C22 | 1 | 56 | yes | complemented |
| 27 | C22×#101 | C22 | 1 | 28 | yes | complemented |
| 28 | C26 | — | 2 | 1 | no | — |
| 29 | C2×2−1+4 | — | 2 | 336 | no | — |
| 30 | C2×2−1+4 | — | 2 | 336 | no | — |
| 31 | C2×2−1+4 | — | 2 | 168 | no | — |
| 32 | C2×2−1+4 | C23 | 1 | 112 | yes | complemented |
| 33 | C22×#41 | — | 2 | 56 | no | — |
| 34 | C22×#41 | — | 2 | 56 | no | — |
| 35 | C22×#41 | — | 2 | 56 | no | — |
| 36 | C22×#41 | — | 2 | 56 | no | — |
| 37 | C22×#41 | — | 2 | 14 | no | — |
| 38 | C22×#41 | — | 2 | 14 | no | — |
| 39 | C22×#41 | D4 | 1 | 28 | yes | — |
| 40 | C22×#41 | C23 | 1 | 28 | yes | — |
| 41 | C23×Q8 | — | 2 | 2 | no | — |
| 42 | C23×Q8 | — | 2 | 56 | no | — |
| 43 | C23×Q8 | — | 2 | 56 | no | — |
| 44 | C23×Q8 | — | 2 | 2 | no | — |
| 45 | C23×Q8 | — | 2 | 14 | no | — |
| 46 | C23×Q8 | D4 | 1 | 2 | yes | — |
| 47 | C23×Q8 | D4 | 1 | 14 | yes | — |
| 48 | C23×Q8 | C23 | 1 | 14 | yes | — |
| 49 | C23×D4 | — | 2 | 2 | no | — |
| 50 | C23×D4 | — | 2 | 2 | no | — |
| 51 | C23×D4 | — | 2 | 2 | no | — |
| 52 | C23×D4 | — | 2 | 14 | no | — |
| 53 | C23×D4 | — | 2 | 7 | no | — |
| 54 | C23×D4 | D4 | 1 | 2 | yes | — |
| 55 | C23×D4 | C23 | 1 | 14 | yes | — |
| 56 | C24×C4 | — | 2 | 4 | no | — |
| 57 | C24×C4 | — | 2 | 2 | no | — |
| 58 | C24×C4 | — | 2 | 1 | no | — |
| 59 | C24×C4 | D4 | 1 | 2 | yes | — |
| 60 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×AGL(1,ℤ/8) | — | 2 | 336 | no | — |
| 62 | C2×AGL(1,ℤ/8) | C23 | 1 | 224 | yes | complemented |
| 63 | C22×SD16 | — | 2 | 56 | no | — |
| 64 | C22×D8 | — | 2 | 56 | no | — |
| 65 | C22×M16 | — | 2 | 14 | no | — |
| 66 | C22×M16 | C23 | 1 | 28 | yes | — |
| 67 | C23×C8 | — | 2 | 2 | no | — |
| 68 | C2×#124 | — | 2 | 168 | no | — |
| 69 | C2×#124 | C23 | 1 | 112 | yes | complemented |
| 70 | C22×C22⋊C4 | — | 2 | 56 | no | — |
| 71 | C22×C42 | — | 2 | 7 | no | — |
| 72 | C22×C42 | C23 | 1 | 14 | yes | — |
| 73 | #454 | C23 | 1 | 512 | yes | direct product factor |
| 74 | C2×#104 | C23 | 1 | 224 | yes | complemented |
| 75 | C2×#101 | C23 | 1 | 112 | yes | complemented |
| 76 | C25 | — | 2 | 2 | no | — |
| 77 | C25 | — | 2 | 14 | no | — |
| 78 | C25 | — | 4 | 16 | no | — |
| 79 | C25 | — | 4 | 7 | no | — |
| 80 | C25 | — | 4 | 1 | no | — |
| 81 | C25 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | 2−1+4 | — | 2 | 1344 | no | — |
| 83 | 2−1+4 | — | 2 | 448 | no | — |
| 84 | 2−1+4 | — | 2 | 224 | no | — |
| 85 | 2−1+4 | C24 | 1 | 64 | yes | complemented |
| 86 | C2×#41 | — | 2 | 672 | no | — |
| 87 | C2×#41 | — | 2 | 672 | no | — |
| 88 | C2×#41 | — | 2 | 224 | no | — |
| 89 | C2×#41 | — | 2 | 168 | no | — |
| 90 | C2×#41 | — | 2 | 224 | no | — |
| 91 | C2×#41 | — | 2 | 84 | no | — |
| 92 | C2×#41 | — | 2 | 84 | no | — |
| 93 | C2×#41 | C2×D4 | 1 | 168 | yes | — |
| 94 | C2×#41 | C24 | 1 | 56 | yes | — |
| 95 | C22×Q8 | — | 2 | 56 | no | — |
| 96 | C22×Q8 | — | 2 | 672 | no | — |
| 97 | C22×Q8 | — | 2 | 224 | no | — |
| 98 | C22×Q8 | — | 2 | 14 | no | — |
| 99 | C22×Q8 | — | 2 | 28 | no | — |
| 100 | C22×Q8 | — | 2 | 84 | no | — |
| 101 | C22×Q8 | — | 2 | 28 | no | — |
| 102 | C22×Q8 | — | 2 | 84 | no | — |
| 103 | C22×Q8 | C2×D4 | 1 | 28 | yes | — |
| 104 | C22×Q8 | C2×D4 | 1 | 84 | yes | — |
| 105 | C22×Q8 | C24 | 1 | 28 | yes | — |
| 106 | C22×D4 | — | 2 | 56 | no | — |
| 107 | C22×D4 | — | 2 | 28 | no | — |
| 108 | C22×D4 | — | 2 | 28 | no | — |
| 109 | C22×D4 | — | 2 | 14 | no | — |
| 110 | C22×D4 | — | 2 | 28 | no | — |
| 111 | C22×D4 | — | 2 | 84 | no | — |
| 112 | C22×D4 | — | 2 | 42 | no | — |
| 113 | C22×D4 | — | 4 | 14 | no | — |
| 114 | C22×D4 | — | 4 | 42 | no | — |
| 115 | C22×D4 | C2×D4 | 1 | 28 | yes | — |
| 116 | C22×D4 | C24 | 1 | 28 | yes | — |
| 117 | C23×C4 | — | 2 | 112 | no | — |
| 118 | C23×C4 | — | 2 | 14 | no | — |
| 119 | C23×C4 | — | 2 | 2 | no | — |
| 120 | C23×C4 | — | 2 | 2 | no | — |
| 121 | C23×C4 | — | 2 | 28 | no | — |
| 122 | C23×C4 | — | 2 | 28 | no | — |
| 123 | C23×C4 | — | 2 | 14 | no | — |
| 124 | C23×C4 | — | 4 | 8 | no | — |
| 125 | C23×C4 | C2×D4 | 1 | 28 | yes | — |
| 126 | C23×C4 | C2×D4 | 1 | 7 | yes | — |
| 127 | C23×C4 | C2×D4 | 1 | 14 | yes | — |
| 128 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 129 | C23×C4 | C24 | 1 | 7 | yes | — |
| 130 | AGL(1,ℤ/8) | — | 2 | 448 | no | — |
| 131 | AGL(1,ℤ/8) | C24 | 1 | 128 | yes | complemented |
| 132 | C2×SD16 | — | 2 | 224 | no | — |
| 133 | C2×D8 | — | 2 | 224 | no | — |
| 134 | C2×M16 | — | 2 | 84 | no | — |
| 135 | C2×M16 | C24 | 1 | 56 | yes | — |
| 136 | C22×C8 | — | 2 | 28 | no | — |
| 137 | #124 | — | 2 | 224 | no | — |
| 138 | #124 | C24 | 1 | 64 | yes | complemented |
| 139 | C2×C22⋊C4 | — | 2 | 224 | no | — |
| 140 | C2×C42 | — | 2 | 42 | no | — |
| 141 | C2×C42 | C24 | 1 | 28 | yes | — |
| 142 | #104 | C24 | 1 | 128 | yes | complemented |
| 143 | #101 | C24 | 1 | 64 | yes | complemented |
| 144 | C24 | — | 2 | 8 | no | — |
| 145 | C24 | — | 2 | 28 | no | — |
| 146 | C24 | — | 2 | 28 | no | — |
| 147 | C24 | — | 4 | 112 | no | — |
| 148 | C24 | — | 4 | 16 | no | — |
| 149 | C24 | — | 4 | 42 | no | — |
| 150 | C24 | — | 4 | 14 | no | — |
| 151 | C24 | — | 8 | 56 | no | — |
| 152 | C24 | — | 8 | 8 | no | — |
| 153 | C24 | #120 | 1 | 7 | yes | — |
| 154 | C24 | #120 | 1 | 1 | yes | characteristic |
| 155 | C24 | C22×D4 | 1 | 7 | yes | — |
| 156 | #41 | — | 2 | 896 | no | — |
| 157 | #41 | — | 2 | 896 | no | — |
| 158 | #41 | — | 2 | 128 | no | — |
| 159 | #41 | — | 2 | 336 | no | — |
| 160 | #41 | — | 2 | 128 | no | — |
| 161 | #41 | — | 2 | 56 | no | — |
| 162 | #41 | — | 2 | 56 | no | — |
| 163 | #41 | C22×D4 | 1 | 112 | yes | — |
| 164 | #41 | C25 | 1 | 16 | yes | — |
| 165 | C2×Q8 | — | 2 | 224 | no | — |
| 166 | C2×Q8 | — | 2 | 896 | no | — |
| 167 | C2×Q8 | — | 2 | 128 | no | — |
| 168 | C2×Q8 | — | 2 | 84 | no | — |
| 169 | C2×Q8 | — | 2 | 168 | no | — |
| 170 | C2×Q8 | — | 2 | 168 | no | — |
| 171 | C2×Q8 | — | 2 | 56 | no | — |
| 172 | C2×Q8 | — | 2 | 56 | no | — |
| 173 | C2×Q8 | C22×D4 | 1 | 56 | yes | — |
| 174 | C2×Q8 | C22×D4 | 1 | 56 | yes | — |
| 175 | C2×Q8 | C25 | 1 | 8 | yes | — |
| 176 | C2×D4 | — | 2 | 224 | no | — |
| 177 | C2×D4 | — | 2 | 168 | no | — |
| 178 | C2×D4 | — | 2 | 56 | no | — |
| 179 | C2×D4 | — | 2 | 84 | no | — |
| 180 | C2×D4 | — | 2 | 56 | no | — |
| 181 | C2×D4 | — | 2 | 56 | no | — |
| 182 | C2×D4 | — | 2 | 28 | no | — |
| 183 | C2×D4 | — | 4 | 84 | no | — |
| 184 | C2×D4 | — | 4 | 84 | no | — |
| 185 | C2×D4 | C22×D4 | 1 | 56 | yes | — |
| 186 | C2×D4 | C25 | 1 | 8 | yes | — |
| 187 | C22×C4 | — | 2 | 448 | no | — |
| 188 | C22×C4 | — | 2 | 28 | no | — |
| 189 | C22×C4 | — | 2 | 28 | no | — |
| 190 | C22×C4 | — | 2 | 84 | no | — |
| 191 | C22×C4 | — | 2 | 168 | no | — |
| 192 | C22×C4 | — | 2 | 56 | no | — |
| 193 | C22×C4 | — | 2 | 28 | no | — |
| 194 | C22×C4 | — | 4 | 56 | no | — |
| 195 | C22×C4 | #120 | 1 | 14 | yes | — |
| 196 | C22×C4 | #120 | 1 | 42 | yes | — |
| 197 | C22×C4 | C22×D4 | 1 | 56 | yes | — |
| 198 | C22×C4 | C22×D4 | 1 | 21 | yes | — |
| 199 | C22×C4 | C22×D4 | 1 | 14 | yes | — |
| 200 | C22×C4 | C22×D4 | 1 | 42 | yes | — |
| 201 | C22×C4 | C25 | 1 | 7 | yes | — |
| 202 | SD16 | — | 2 | 128 | no | — |
| 203 | D8 | — | 2 | 128 | no | — |
| 204 | M16 | — | 2 | 56 | no | — |
| 205 | M16 | C25 | 1 | 16 | yes | — |
| 206 | C2×C8 | — | 2 | 56 | no | — |
| 207 | C22⋊C4 | — | 2 | 128 | no | — |
| 208 | C42 | — | 2 | 28 | no | — |
| 209 | C42 | C25 | 1 | 8 | yes | — |
| 210 | C23 | — | 2 | 28 | no | — |
| 211 | C23 | — | 2 | 28 | no | — |
| 212 | C23 | — | 2 | 56 | no | — |
| 213 | C23 | — | 2 | 8 | no | — |
| 214 | C23 | — | 4 | 112 | no | — |
| 215 | C23 | — | 4 | 112 | no | — |
| 216 | C23 | — | 4 | 28 | no | — |
| 217 | C23 | — | 4 | 28 | no | — |
| 218 | C23 | — | 8 | 168 | no | — |
| 219 | C23 | — | 8 | 56 | no | — |
| 220 | C23 | #454 | 1 | 8 | yes | direct product factor |
| 221 | C23 | C2×#120 | 1 | 21 | yes | — |
| 222 | C23 | C2×#120 | 1 | 7 | yes | — |
| 223 | C23 | C23×D4 | 1 | 7 | yes | — |
| 224 | Q8 | — | 2 | 128 | no | — |
| 225 | Q8 | — | 2 | 16 | no | — |
| 226 | Q8 | — | 2 | 56 | no | — |
| 227 | Q8 | — | 2 | 112 | no | — |
| 228 | Q8 | C23×D4 | 1 | 16 | yes | — |
| 229 | D4 | — | 2 | 128 | no | — |
| 230 | D4 | — | 2 | 16 | no | — |
| 231 | D4 | — | 2 | 112 | no | — |
| 232 | D4 | — | 2 | 16 | no | — |
| 233 | D4 | — | 2 | 56 | no | — |
| 234 | D4 | — | 4 | 56 | no | — |
| 235 | D4 | C23×D4 | 1 | 16 | yes | — |
| 236 | C2×C4 | — | 2 | 256 | no | — |
| 237 | C2×C4 | — | 2 | 56 | no | — |
| 238 | C2×C4 | — | 2 | 56 | no | — |
| 239 | C2×C4 | — | 2 | 56 | no | — |
| 240 | C2×C4 | — | 2 | 16 | no | — |
| 241 | C2×C4 | — | 2 | 112 | no | — |
| 242 | C2×C4 | — | 2 | 8 | no | — |
| 243 | C2×C4 | — | 4 | 56 | no | — |
| 244 | C2×C4 | C2×#120 | 1 | 42 | yes | — |
| 245 | C2×C4 | C2×#120 | 1 | 42 | yes | — |
| 246 | C2×C4 | C23×D4 | 1 | 16 | yes | — |
| 247 | C2×C4 | C23×D4 | 1 | 14 | yes | — |
| 248 | C2×C4 | C23×D4 | 1 | 14 | yes | — |
| 249 | C2×C4 | C23×D4 | 1 | 7 | yes | — |
| 250 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 251 | C8 | — | 2 | 16 | no | — |
| 252 | C22 | — | 2 | 14 | no | — |
| 253 | C22 | — | 2 | 42 | no | — |
| 254 | C22 | — | 2 | 16 | no | — |
| 255 | C22 | — | 4 | 16 | no | — |
| 256 | C22 | — | 4 | 112 | no | — |
| 257 | C22 | — | 4 | 8 | no | — |
| 258 | C22 | — | 8 | 56 | no | — |
| 259 | C22 | — | 8 | 56 | no | — |
| 260 | C22 | C2×#454 | 1 | 28 | yes | direct product factor |
| 261 | C22 | C22×#120 | 1 | 7 | yes | — |
| 262 | C22 | C22×#120 | 1 | 7 | yes | — |
| 263 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 264 | C4 | — | 2 | 16 | no | — |
| 265 | C4 | — | 2 | 16 | no | — |
| 266 | C4 | — | 4 | 8 | no | — |
| 267 | C4 | C22×#120 | 1 | 14 | yes | — |
| 268 | C4 | C24×D4 | 1 | 2 | yes | — |
| 269 | C2 | — | 2 | 1 | no | — |
| 270 | C2 | — | 2 | 7 | no | — |
| 271 | C2 | — | 4 | 16 | no | — |
| 272 | C2 | — | 8 | 8 | no | — |
| 273 | C2 | C22×#454 | 1 | 14 | yes | direct product factor |
| 274 | C2 | C23×#120 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×2−1+4 | C2 | semidirect |
| 2. C23×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C23×#124 | C2 | semidirect |
| 4. C22×#454 | C2 | direct (N × H) |
| 5. C23×#104 | C2 | semidirect |
| 6. C23×#101 | C2 | semidirect |
| 9. C22×2−1+4 | C22 | semidirect |
| 17. C22×AGL(1,ℤ/8) | C22 | semidirect |
| 22. C22×#124 | C22 | semidirect |
| 25. C2×#454 | C22 | direct (N × H) |
| 26. C22×#104 | C22 | semidirect |
| 27. C22×#101 | C22 | semidirect |
| 32. C2×2−1+4 | C23 | semidirect |
| 62. C2×AGL(1,ℤ/8) | C23 | semidirect |
| 69. C2×#124 | C23 | semidirect |
| 73. #454 | C23 | direct (N × H) |
| 74. C2×#104 | C23 | semidirect |
| 75. C2×#101 | C23 | semidirect |
| 85. 2−1+4 | C24 | semidirect |
| 131. AGL(1,ℤ/8) | C24 | semidirect |
| 138. #124 | C24 | semidirect |
| 142. #104 | C24 | semidirect |
| 143. #101 | C24 | semidirect |
| 220. C23 | #454 | direct (N × H) |
| 260. C22 | C2×#454 | direct (N × H) |
| 273. C2 | C22×#454 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×2−1+4 | 2 |
| C23×AGL(1,ℤ/8) | 2 |
| C23×#124 | 2 |
| C22×#454 | 2 |
| C23×#104 | 2 |
| C23×#101 | 2 |