Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2050 conjugacy classes of subgroups; only the 566 normal subgroups are shown.
This group has 2050 conjugacy classes of subgroups — too many to draw. Only its 566 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 | #60968 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #60801 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #19729 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #19727 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #19724 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #19721 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #17338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3444 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#532 | C22 | 1 | 3 | yes | — |
| 10 | #3253 | — | 2 | 6 | no | — |
| 11 | #3227 | C22 | 1 | 12 | yes | complemented |
| 12 | #3219 | — | 2 | 12 | no | — |
| 13 | #3210 | C22 | 1 | 12 | yes | complemented |
| 14 | #3207 | C22 | 1 | 12 | yes | complemented |
| 15 | #3205 | C22 | 1 | 12 | yes | complemented |
| 16 | #3192 | — | 2 | 6 | no | — |
| 17 | #3184 | C22 | 1 | 6 | yes | — |
| 18 | #3184 | C22 | 1 | 6 | yes | — |
| 19 | #3152 | C22 | 1 | 3 | yes | complemented |
| 20 | #3151 | C22 | 1 | 3 | yes | complemented |
| 21 | #3139 | C22 | 1 | 3 | yes | complemented |
| 22 | #3138 | C22 | 1 | 3 | yes | complemented |
| 23 | #3109 | — | 2 | 12 | no | — |
| 24 | #3040 | — | 2 | 6 | no | — |
| 25 | #3005 | — | 2 | 6 | no | — |
| 26 | #2920 | C22 | 1 | 3 | yes | — |
| 27 | #1721 | C22 | 1 | 6 | yes | complemented |
| 28 | #1721 | C22 | 1 | 6 | yes | complemented |
| 29 | #1720 | C22 | 1 | 12 | yes | complemented |
| 30 | #1717 | C22 | 1 | 12 | yes | complemented |
| 31 | #1716 | C22 | 1 | 6 | yes | complemented |
| 32 | #1715 | C22 | 1 | 6 | yes | complemented |
| 33 | #1712 | C22 | 1 | 2 | yes | complemented |
| 34 | #1709 | C22 | 1 | 6 | yes | complemented |
| 35 | #1708 | C22 | 1 | 6 | yes | complemented |
| 36 | #1705 | C22 | 1 | 2 | yes | complemented |
| 37 | #1453 | C22 | 1 | 6 | yes | complemented |
| 38 | #1453 | C22 | 1 | 6 | yes | complemented |
| 39 | C22×#41 | — | 2 | 3 | no | — |
| 40 | C2×#137 | — | 2 | 6 | no | — |
| 41 | C2×AGL(1,ℤ/8) | — | 2 | 6 | no | — |
| 42 | #533 | C23 | 1 | 3 | yes | — |
| 43 | #532 | — | 2 | 6 | no | — |
| 44 | #532 | — | 2 | 6 | no | — |
| 45 | #532 | — | 2 | 6 | no | — |
| 46 | #532 | C23 | 1 | 3 | yes | — |
| 47 | #532 | C23 | 1 | 12 | yes | — |
| 48 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 49 | C2×#127 | C23 | 1 | 3 | yes | — |
| 50 | #529 | — | 2 | 6 | no | — |
| 51 | C2×#124 | — | 2 | 3 | no | — |
| 52 | #525 | — | 2 | 3 | no | — |
| 53 | C2×#121 | — | 2 | 6 | no | — |
| 54 | C4×#41 | — | 2 | 3 | no | — |
| 55 | C2×C4×D4 | — | 2 | 6 | no | — |
| 56 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 57 | #498 | C23 | 1 | 6 | yes | — |
| 58 | #498 | C23 | 1 | 6 | yes | — |
| 59 | #497 | — | 2 | 6 | no | — |
| 60 | #496 | — | 2 | 6 | no | — |
| 61 | #494 | C23 | 1 | 6 | yes | — |
| 62 | #493 | C23 | 1 | 6 | yes | — |
| 63 | #493 | C23 | 1 | 6 | yes | — |
| 64 | #493 | C23 | 1 | 6 | yes | — |
| 65 | #492 | C23 | 1 | 6 | yes | — |
| 66 | #492 | C23 | 1 | 6 | yes | — |
| 67 | #491 | — | 2 | 6 | no | — |
| 68 | #490 | — | 2 | 6 | no | — |
| 69 | #487 | C23 | 1 | 12 | yes | — |
| 70 | #486 | C23 | 1 | 12 | yes | — |
| 71 | #486 | C23 | 1 | 12 | yes | — |
| 72 | #486 | C23 | 1 | 12 | yes | — |
| 73 | #481 | — | 2 | 24 | no | — |
| 74 | #476 | — | 2 | 24 | no | — |
| 75 | #469 | — | 2 | 12 | no | — |
| 76 | #469 | — | 2 | 12 | no | — |
| 77 | #464 | — | 2 | 24 | no | — |
| 78 | #463 | — | 2 | 24 | no | — |
| 79 | #459 | — | 2 | 24 | no | — |
| 80 | #452 | — | 2 | 24 | no | — |
| 81 | #449 | — | 2 | 24 | no | — |
| 82 | #447 | — | 2 | 24 | no | — |
| 83 | #442 | — | 2 | 12 | no | — |
| 84 | #442 | — | 2 | 12 | no | — |
| 85 | #432 | D4 | 1 | 12 | yes | — |
| 86 | #431 | C23 | 1 | 12 | yes | — |
| 87 | #423 | D4 | 1 | 6 | yes | — |
| 88 | #419 | — | 2 | 6 | no | — |
| 89 | #418 | — | 2 | 6 | no | — |
| 90 | #407 | D4 | 1 | 6 | yes | — |
| 91 | C4×M16 | C23 | 1 | 3 | yes | — |
| 92 | C4×M16 | C23 | 1 | 3 | yes | — |
| 93 | #322 | C23 | 1 | 6 | yes | — |
| 94 | #322 | C23 | 1 | 6 | yes | — |
| 95 | C82 | C23 | 1 | 2 | yes | — |
| 96 | C82 | C23 | 1 | 2 | yes | — |
| 97 | C2×#41 | — | 2 | 3 | no | — |
| 98 | C2×#41 | — | 2 | 12 | no | — |
| 99 | C2×#41 | — | 4 | 6 | no | — |
| 100 | C2×#41 | — | 8 | 12 | no | — |
| 101 | C22×Q8 | — | 2 | 3 | no | — |
| 102 | C22×D4 | — | 2 | 3 | no | — |
| 103 | C22×D4 | — | 2 | 6 | no | — |
| 104 | C22×D4 | — | 4 | 3 | no | — |
| 105 | C23×C4 | — | 4 | 3 | no | — |
| 106 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 107 | #137 | — | 8 | 12 | no | — |
| 108 | AGL(1,ℤ/8) | — | 8 | 12 | no | — |
| 109 | C2×Q16 | — | 2 | 12 | no | — |
| 110 | C2×SD16 | — | 2 | 12 | no | — |
| 111 | C2×SD16 | — | 2 | 12 | no | — |
| 112 | C2×D8 | — | 2 | 12 | no | — |
| 113 | C2×D8 | — | 2 | 12 | no | — |
| 114 | C2×D8 | — | 2 | 12 | no | — |
| 115 | C2×M16 | C2×D4 | 1 | 6 | yes | — |
| 116 | C2×M16 | C2×D4 | 1 | 6 | yes | — |
| 117 | C4⋊Q8 | — | 2 | 6 | no | — |
| 118 | C4⋊Q8 | — | 2 | 3 | no | — |
| 119 | C4⋊Q8 | C24 | 1 | 6 | yes | — |
| 120 | C4⋊Q8 | C24 | 1 | 3 | yes | — |
| 121 | #127 | — | 2 | 6 | no | — |
| 122 | #127 | — | 2 | 3 | no | — |
| 123 | #127 | C24 | 1 | 6 | yes | — |
| 124 | #127 | C24 | 1 | 1 | yes | characteristic |
| 125 | #126 | — | 4 | 2 | no | — |
| 126 | #125 | — | 2 | 3 | no | — |
| 127 | #124 | — | 2 | 12 | no | — |
| 128 | #124 | — | 2 | 3 | no | — |
| 129 | #124 | — | 4 | 3 | no | — |
| 130 | #124 | — | 4 | 3 | no | — |
| 131 | #123 | — | 4 | 3 | no | — |
| 132 | C22⋊Q8 | — | 2 | 6 | no | — |
| 133 | #121 | — | 2 | 12 | no | — |
| 134 | #121 | — | 2 | 12 | no | — |
| 135 | #121 | — | 2 | 6 | no | — |
| 136 | #121 | — | 4 | 6 | no | — |
| 137 | #121 | — | 4 | 6 | no | — |
| 138 | #120 | — | 4 | 1 | no | — |
| 139 | C4×Q8 | — | 2 | 6 | no | — |
| 140 | C4×D4 | — | 2 | 12 | no | — |
| 141 | C4×D4 | — | 2 | 12 | no | — |
| 142 | C4×D4 | — | 2 | 6 | no | — |
| 143 | C4×D4 | — | 4 | 6 | no | — |
| 144 | C4×D4 | — | 4 | 6 | no | — |
| 145 | C4×D4 | — | 4 | 6 | no | — |
| 146 | #117 | — | 2 | 3 | no | — |
| 147 | #117 | — | 4 | 3 | no | — |
| 148 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 149 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 150 | C2×C42 | — | 4 | 1 | no | — |
| 151 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 152 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 153 | C2×C42 | C24 | 1 | 3 | yes | — |
| 154 | #106 | — | 2 | 12 | no | — |
| 155 | #106 | — | 2 | 12 | no | — |
| 156 | C4⋊C8 | C2×D4 | 1 | 24 | yes | — |
| 157 | #103 | — | 2 | 24 | no | — |
| 158 | #102 | — | 2 | 24 | no | — |
| 159 | #102 | — | 2 | 24 | no | — |
| 160 | #102 | — | 2 | 24 | no | — |
| 161 | C22⋊C8 | C2×D4 | 1 | 24 | yes | — |
| 162 | #97 | C2×D4 | 1 | 6 | yes | — |
| 163 | #97 | C2×D4 | 1 | 6 | yes | — |
| 164 | C4×C8 | C24 | 1 | 6 | yes | — |
| 165 | C4×C8 | C24 | 1 | 6 | yes | — |
| 166 | C24 | — | 4 | 3 | no | — |
| 167 | C24 | #127 | 1 | 1 | yes | characteristic |
| 168 | #41 | — | 4 | 12 | no | — |
| 169 | #41 | — | 8 | 24 | no | — |
| 170 | #41 | — | 8 | 24 | no | — |
| 171 | C2×Q8 | — | 2 | 6 | no | — |
| 172 | C2×Q8 | — | 2 | 3 | no | — |
| 173 | C2×Q8 | — | 4 | 12 | no | — |
| 174 | C2×Q8 | — | 4 | 3 | no | — |
| 175 | C2×D4 | — | 2 | 12 | no | — |
| 176 | C2×D4 | — | 2 | 6 | no | — |
| 177 | C2×D4 | — | 2 | 3 | no | — |
| 178 | C2×D4 | — | 4 | 12 | no | — |
| 179 | C2×D4 | — | 4 | 3 | no | — |
| 180 | C2×D4 | — | 4 | 12 | no | — |
| 181 | C2×D4 | — | 4 | 6 | no | — |
| 182 | C2×D4 | — | 4 | 3 | no | — |
| 183 | C2×D4 | — | 8 | 12 | no | — |
| 184 | C2×D4 | — | 8 | 12 | no | — |
| 185 | C22×C4 | — | 4 | 6 | no | — |
| 186 | C22×C4 | — | 4 | 3 | no | — |
| 187 | C22×C4 | — | 4 | 3 | no | — |
| 188 | C22×C4 | — | 4 | 12 | no | — |
| 189 | C22×C4 | — | 8 | 12 | no | — |
| 190 | C22×C4 | #127 | 1 | 6 | yes | — |
| 191 | C22×C4 | #127 | 1 | 3 | yes | — |
| 192 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 193 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 194 | Q16 | — | 8 | 12 | no | — |
| 195 | SD16 | — | 8 | 12 | no | — |
| 196 | SD16 | — | 8 | 12 | no | — |
| 197 | D8 | — | 8 | 12 | no | — |
| 198 | D8 | — | 8 | 12 | no | — |
| 199 | D8 | — | 8 | 12 | no | — |
| 200 | M16 | — | 4 | 6 | no | — |
| 201 | M16 | — | 4 | 6 | no | — |
| 202 | C2×C8 | C22×D4 | 1 | 12 | yes | — |
| 203 | C2×C8 | C22×D4 | 1 | 12 | yes | — |
| 204 | C4⋊C4 | — | 2 | 6 | no | — |
| 205 | C4⋊C4 | — | 2 | 12 | no | — |
| 206 | C4⋊C4 | — | 4 | 6 | no | — |
| 207 | C4⋊C4 | — | 4 | 6 | no | — |
| 208 | C22⋊C4 | — | 4 | 3 | no | — |
| 209 | C22⋊C4 | — | 4 | 12 | no | — |
| 210 | C22⋊C4 | — | 4 | 12 | no | — |
| 211 | C22⋊C4 | — | 4 | 6 | no | — |
| 212 | C42 | — | 4 | 3 | no | — |
| 213 | C42 | #127 | 1 | 1 | yes | characteristic |
| 214 | C42 | #127 | 1 | 3 | yes | — |
| 215 | C42 | #127 | 1 | 2 | yes | — |
| 216 | C42 | C22×D4 | 1 | 3 | yes | — |
| 217 | C42 | C22×D4 | 1 | 6 | yes | — |
| 218 | C42 | C25 | 1 | 1 | yes | characteristic |
| 219 | C23 | — | 4 | 3 | no | — |
| 220 | C23 | — | 4 | 1 | no | — |
| 221 | C23 | — | 4 | 6 | no | — |
| 222 | C23 | — | 8 | 6 | no | — |
| 223 | C23 | — | 8 | 12 | no | — |
| 224 | C23 | C2×#127 | 1 | 3 | yes | — |
| 225 | Q8 | — | 4 | 6 | no | — |
| 226 | Q8 | — | 4 | 12 | no | — |
| 227 | Q8 | — | 8 | 6 | no | — |
| 228 | D4 | — | 4 | 6 | no | — |
| 229 | D4 | — | 4 | 24 | no | — |
| 230 | D4 | — | 4 | 12 | no | — |
| 231 | D4 | — | 8 | 6 | no | — |
| 232 | D4 | — | 8 | 12 | no | — |
| 233 | D4 | — | 8 | 24 | no | — |
| 234 | D4 | — | 8 | 6 | no | — |
| 235 | C2×C4 | — | 2 | 6 | no | — |
| 236 | C2×C4 | — | 2 | 12 | no | — |
| 237 | C2×C4 | — | 4 | 3 | no | — |
| 238 | C2×C4 | — | 4 | 6 | no | — |
| 239 | C2×C4 | — | 4 | 3 | no | — |
| 240 | C2×C4 | — | 4 | 6 | no | — |
| 241 | C2×C4 | — | 8 | 24 | no | — |
| 242 | C2×C4 | — | 8 | 6 | no | — |
| 243 | C2×C4 | — | 8 | 6 | no | — |
| 244 | C2×C4 | C2×#127 | 1 | 3 | yes | — |
| 245 | C2×C4 | C2×#127 | 1 | 6 | yes | — |
| 246 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 247 | C8 | — | 2 | 12 | no | — |
| 248 | C8 | — | 2 | 12 | no | — |
| 249 | C22 | — | 2 | 3 | no | — |
| 250 | C22 | — | 2 | 6 | no | — |
| 251 | C22 | — | 8 | 2 | no | — |
| 252 | C22 | — | 8 | 6 | no | — |
| 253 | C22 | — | 8 | 3 | no | — |
| 254 | C22 | — | 8 | 12 | no | — |
| 255 | C22 | — | 16 | 12 | no | — |
| 256 | C22 | C22×#127 | 1 | 1 | yes | characteristic |
| 257 | C4 | — | 2 | 6 | no | — |
| 258 | C4 | — | 2 | 3 | no | — |
| 259 | C4 | — | 8 | 6 | no | — |
| 260 | C4 | — | 8 | 3 | no | — |
| 261 | C4 | #3585 | 1 | 6 | yes | — |
| 262 | C2 | — | 4 | 3 | no | — |
| 263 | C2 | — | 16 | 1 | no | — |
| 264 | C2 | — | 16 | 6 | no | — |
| 265 | C2 | #60644 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #60968 | C2 | semidirect |
| 2. #60801 | C2 | semidirect |
| 3. #19729 | C2 | semidirect |
| 4. #19727 | C2 | semidirect |
| 5. #19724 | C2 | semidirect |
| 6. #19721 | C2 | semidirect |
| 7. #17338 | C2 | semidirect |
| 11. #3227 | C22 | semidirect |
| 13. #3210 | C22 | semidirect |
| 14. #3207 | C22 | semidirect |
| 15. #3205 | C22 | semidirect |
| 19. #3152 | C22 | semidirect |
| 20. #3151 | C22 | semidirect |
| 21. #3139 | C22 | semidirect |
| 22. #3138 | C22 | semidirect |
| 27. #1721 | C22 | semidirect |
| 28. #1721 | C22 | semidirect |
| 29. #1720 | C22 | semidirect |
| 30. #1717 | C22 | semidirect |
| 31. #1716 | C22 | semidirect |
| 32. #1715 | C22 | semidirect |
| 33. #1712 | C22 | semidirect |
| 34. #1709 | C22 | semidirect |
| 35. #1708 | C22 | semidirect |
| 36. #1705 | C22 | semidirect |
| 37. #1453 | C22 | semidirect |
| 38. #1453 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60968 | 2 |
| #60801 | 2 |
| #19729 | 2 |
| #19727 | 2 |
| #19724 | 2 |
| #19721 | 2 |
| #17338 | 2 |