Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
700 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 170 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 | #33717 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33707 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #19603 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#1706 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #12527 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #12526 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12525 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11402 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#509 | — | 2 | 4 | no | — |
| 10 | #3169 | — | 2 | 1 | no | — |
| 11 | #3169 | — | 2 | 1 | no | — |
| 12 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 13 | #3147 | C22 | 1 | 2 | yes | — |
| 14 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#492 | — | 2 | 1 | no | — |
| 16 | C2×#486 | — | 2 | 1 | no | — |
| 17 | #2250 | — | 2 | 4 | no | — |
| 18 | #2248 | C22 | 1 | 4 | yes | — |
| 19 | #2246 | C22 | 1 | 4 | yes | — |
| 20 | #2243 | — | 2 | 4 | no | — |
| 21 | #2241 | C22 | 1 | 4 | yes | complemented |
| 22 | #2240 | C22 | 1 | 4 | yes | complemented |
| 23 | #2139 | — | 2 | 4 | no | — |
| 24 | C4×M32 | C22 | 1 | 2 | yes | — |
| 25 | #1713 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #1712 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | #1709 | C22 | 1 | 2 | yes | complemented |
| 28 | #1706 | — | 2 | 1 | no | — |
| 29 | #1706 | — | 2 | 1 | no | — |
| 30 | #1706 | — | 2 | 1 | no | — |
| 31 | #1706 | — | 2 | 1 | no | — |
| 32 | #1706 | C22 | 1 | 2 | yes | — |
| 33 | #1706 | C22 | 1 | 2 | yes | — |
| 34 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1312 | C22 | 1 | 2 | yes | complemented |
| 36 | #1312 | C22 | 1 | 2 | yes | complemented |
| 37 | C2×#135 | — | 2 | 2 | no | — |
| 38 | C2×#135 | — | 2 | 2 | no | — |
| 39 | C22×SD16 | — | 4 | 2 | no | — |
| 40 | C22×D8 | — | 2 | 2 | no | — |
| 41 | #532 | — | 2 | 2 | no | — |
| 42 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 43 | C2×#127 | — | 2 | 1 | no | — |
| 44 | #509 | — | 4 | 16 | no | — |
| 45 | C2×SD32 | — | 2 | 8 | no | — |
| 46 | C2×D16 | — | 2 | 8 | no | — |
| 47 | C2×M32 | D4 | 1 | 4 | yes | — |
| 48 | #500 | — | 2 | 2 | no | — |
| 49 | #500 | — | 2 | 1 | no | — |
| 50 | #500 | — | 2 | 1 | no | — |
| 51 | #500 | — | 2 | 1 | no | — |
| 52 | #500 | C23 | 1 | 2 | yes | — |
| 53 | #495 | — | 2 | 2 | no | — |
| 54 | #495 | — | 2 | 2 | no | — |
| 55 | #494 | C23 | 1 | 2 | yes | — |
| 56 | #493 | — | 2 | 1 | no | — |
| 57 | #493 | — | 2 | 1 | no | — |
| 58 | #493 | — | 2 | 1 | no | — |
| 59 | #493 | — | 2 | 2 | no | — |
| 60 | #493 | C23 | 1 | 2 | yes | — |
| 61 | #493 | C23 | 1 | 2 | yes | — |
| 62 | #492 | — | 2 | 2 | no | — |
| 63 | #492 | — | 2 | 2 | no | — |
| 64 | #492 | — | 2 | 2 | no | — |
| 65 | #492 | — | 4 | 2 | no | — |
| 66 | #486 | — | 2 | 2 | no | — |
| 67 | #486 | — | 2 | 2 | no | — |
| 68 | #486 | — | 2 | 2 | no | — |
| 69 | #486 | — | 2 | 2 | no | — |
| 70 | #486 | — | 4 | 2 | no | — |
| 71 | #466 | — | 4 | 2 | no | — |
| 72 | #466 | — | 4 | 2 | no | — |
| 73 | C4×D8 | — | 4 | 2 | no | — |
| 74 | C4×D8 | — | 4 | 2 | no | — |
| 75 | C2×#107 | — | 2 | 2 | no | — |
| 76 | C2×#102 | — | 4 | 2 | no | — |
| 77 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 78 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 80 | #358 | — | 2 | 8 | no | — |
| 81 | #357 | — | 2 | 8 | no | — |
| 82 | #346 | D4 | 1 | 4 | yes | — |
| 83 | C4×C16 | C23 | 1 | 4 | yes | — |
| 84 | C82 | D4 | 1 | 1 | yes | characteristic |
| 85 | C82 | D4 | 1 | 1 | yes | characteristic |
| 86 | C82 | C23 | 1 | 1 | yes | characteristic |
| 87 | C82 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×#41 | — | 4 | 2 | no | — |
| 89 | C22×Q8 | — | 4 | 1 | no | — |
| 90 | C22×D4 | — | 4 | 1 | no | — |
| 91 | C22×D4 | — | 4 | 1 | no | — |
| 92 | C22×D4 | — | 4 | 1 | no | — |
| 93 | #135 | — | 4 | 8 | no | — |
| 94 | #135 | — | 4 | 8 | no | — |
| 95 | C2×Q16 | — | 2 | 2 | no | — |
| 96 | C2×Q16 | — | 2 | 2 | no | — |
| 97 | C2×SD16 | — | 4 | 2 | no | — |
| 98 | C2×SD16 | — | 4 | 2 | no | — |
| 99 | C2×SD16 | — | 4 | 4 | no | — |
| 100 | C2×SD16 | — | 4 | 4 | no | — |
| 101 | C2×SD16 | — | 8 | 8 | no | — |
| 102 | C2×D8 | — | 2 | 4 | no | — |
| 103 | C2×D8 | — | 2 | 2 | no | — |
| 104 | C2×D8 | — | 2 | 2 | no | — |
| 105 | C2×D8 | — | 4 | 2 | no | — |
| 106 | C2×D8 | — | 4 | 2 | no | — |
| 107 | C2×D8 | — | 4 | 2 | no | — |
| 108 | C2×D8 | — | 4 | 8 | no | — |
| 109 | C22×C8 | D8 | 1 | 2 | yes | — |
| 110 | C22×C8 | D8 | 1 | 2 | yes | — |
| 111 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 112 | C4⋊Q8 | — | 2 | 2 | no | — |
| 113 | C4⋊Q8 | — | 2 | 2 | no | — |
| 114 | C4⋊Q8 | — | 4 | 1 | no | — |
| 115 | C4⋊Q8 | — | 4 | 1 | no | — |
| 116 | C4⋊Q8 | — | 4 | 1 | no | — |
| 117 | #127 | — | 2 | 2 | no | — |
| 118 | #127 | — | 2 | 2 | no | — |
| 119 | #127 | — | 4 | 1 | no | — |
| 120 | #127 | — | 4 | 1 | no | — |
| 121 | #127 | — | 4 | 1 | no | — |
| 122 | #124 | — | 4 | 2 | no | — |
| 123 | #121 | — | 4 | 2 | no | — |
| 124 | #121 | — | 4 | 2 | no | — |
| 125 | C4×D4 | — | 4 | 2 | no | — |
| 126 | C4×D4 | — | 4 | 2 | no | — |
| 127 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 128 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 129 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | SD32 | — | 4 | 16 | no | — |
| 131 | D16 | — | 4 | 16 | no | — |
| 132 | M32 | — | 2 | 8 | no | — |
| 133 | C2×C16 | C2×D4 | 1 | 8 | yes | — |
| 134 | #107 | — | 2 | 4 | no | — |
| 135 | #107 | — | 4 | 2 | no | — |
| 136 | #107 | — | 4 | 2 | no | — |
| 137 | #107 | — | 4 | 2 | no | — |
| 138 | #102 | — | 4 | 4 | no | — |
| 139 | #102 | — | 4 | 4 | no | — |
| 140 | #102 | — | 4 | 4 | no | — |
| 141 | #102 | — | 4 | 4 | no | — |
| 142 | C4×C8 | D8 | 1 | 2 | yes | — |
| 143 | C4×C8 | D8 | 1 | 2 | yes | — |
| 144 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 145 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 146 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 147 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 148 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 149 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 150 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 151 | C24 | — | 8 | 1 | no | — |
| 152 | #41 | — | 8 | 4 | no | — |
| 153 | #41 | — | 8 | 4 | no | — |
| 154 | C2×Q8 | — | 4 | 2 | no | — |
| 155 | C2×Q8 | — | 4 | 2 | no | — |
| 156 | C2×Q8 | — | 8 | 4 | no | — |
| 157 | C2×Q8 | — | 8 | 1 | no | — |
| 158 | C2×D4 | — | 4 | 2 | no | — |
| 159 | C2×D4 | — | 4 | 2 | no | — |
| 160 | C2×D4 | — | 4 | 2 | no | — |
| 161 | C2×D4 | — | 4 | 2 | no | — |
| 162 | C2×D4 | — | 4 | 2 | no | — |
| 163 | C2×D4 | — | 4 | 2 | no | — |
| 164 | C2×D4 | — | 8 | 4 | no | — |
| 165 | C2×D4 | — | 8 | 4 | no | — |
| 166 | C2×D4 | — | 8 | 2 | no | — |
| 167 | C2×D4 | — | 8 | 1 | no | — |
| 168 | C2×D4 | — | 8 | 1 | no | — |
| 169 | C2×D4 | — | 8 | 1 | no | — |
| 170 | C2×D4 | — | 8 | 4 | no | — |
| 171 | C22×C4 | — | 8 | 1 | no | — |
| 172 | C22×C4 | — | 8 | 2 | no | — |
| 173 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 174 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 175 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 176 | Q16 | — | 4 | 4 | no | — |
| 177 | Q16 | — | 4 | 4 | no | — |
| 178 | SD16 | — | 8 | 8 | no | — |
| 179 | SD16 | — | 8 | 4 | no | — |
| 180 | SD16 | — | 8 | 4 | no | — |
| 181 | SD16 | — | 8 | 8 | no | — |
| 182 | D8 | — | 4 | 4 | no | — |
| 183 | D8 | — | 4 | 8 | no | — |
| 184 | D8 | — | 4 | 4 | no | — |
| 185 | D8 | — | 8 | 4 | no | — |
| 186 | D8 | — | 8 | 4 | no | — |
| 187 | D8 | — | 8 | 4 | no | — |
| 188 | C2×C8 | — | 2 | 4 | no | — |
| 189 | C2×C8 | — | 4 | 2 | no | — |
| 190 | C2×C8 | #127 | 1 | 2 | yes | — |
| 191 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 192 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 193 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 194 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 195 | C2×C8 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 196 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 197 | C4⋊C4 | — | 4 | 2 | no | — |
| 198 | C4⋊C4 | — | 4 | 2 | no | — |
| 199 | C4⋊C4 | — | 4 | 2 | no | — |
| 200 | C4⋊C4 | — | 4 | 2 | no | — |
| 201 | C4⋊C4 | — | 8 | 1 | no | — |
| 202 | C4⋊C4 | — | 8 | 1 | no | — |
| 203 | C22⋊C4 | — | 8 | 2 | no | — |
| 204 | C22⋊C4 | — | 8 | 2 | no | — |
| 205 | C42 | #127 | 1 | 1 | yes | characteristic |
| 206 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 207 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 208 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 209 | C16 | — | 2 | 8 | no | — |
| 210 | C23 | — | 8 | 2 | no | — |
| 211 | C23 | — | 8 | 2 | no | — |
| 212 | C23 | — | 16 | 2 | no | — |
| 213 | C23 | — | 16 | 2 | no | — |
| 214 | C23 | — | 16 | 2 | no | — |
| 215 | C23 | #493 | 1 | 1 | yes | characteristic |
| 216 | Q8 | — | 8 | 2 | no | — |
| 217 | Q8 | — | 8 | 2 | no | — |
| 218 | Q8 | — | 8 | 4 | no | — |
| 219 | Q8 | — | 16 | 2 | no | — |
| 220 | D4 | — | 8 | 4 | no | — |
| 221 | D4 | — | 8 | 2 | no | — |
| 222 | D4 | — | 8 | 2 | no | — |
| 223 | D4 | — | 8 | 4 | no | — |
| 224 | D4 | — | 8 | 4 | no | — |
| 225 | D4 | — | 8 | 4 | no | — |
| 226 | D4 | — | 8 | 4 | no | — |
| 227 | D4 | — | 16 | 2 | no | — |
| 228 | D4 | — | 16 | 2 | no | — |
| 229 | D4 | — | 16 | 4 | no | — |
| 230 | D4 | — | 16 | 2 | no | — |
| 231 | C2×C4 | — | 2 | 2 | no | — |
| 232 | C2×C4 | — | 2 | 2 | no | — |
| 233 | C2×C4 | — | 8 | 2 | no | — |
| 234 | C2×C4 | — | 8 | 2 | no | — |
| 235 | C2×C4 | — | 16 | 2 | no | — |
| 236 | C2×C4 | — | 16 | 4 | no | — |
| 237 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 240 | C2×C4 | #496 | 1 | 4 | yes | — |
| 241 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 242 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 243 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 244 | C8 | — | 2 | 2 | no | — |
| 245 | C8 | — | 2 | 2 | no | — |
| 246 | C8 | — | 2 | 2 | no | — |
| 247 | C8 | #496 | 1 | 4 | yes | — |
| 248 | C8 | #509 | 1 | 4 | yes | complemented |
| 249 | C8 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 250 | C22 | — | 2 | 1 | no | — |
| 251 | C22 | — | 2 | 1 | no | — |
| 252 | C22 | — | 16 | 2 | no | — |
| 253 | C22 | — | 16 | 2 | no | — |
| 254 | C22 | — | 16 | 2 | no | — |
| 255 | C22 | — | 16 | 2 | no | — |
| 256 | C22 | — | 16 | 2 | no | — |
| 257 | C22 | — | 16 | 2 | no | — |
| 258 | C22 | — | 32 | 4 | no | — |
| 259 | C22 | #1721 | 1 | 2 | yes | — |
| 260 | C22 | C2×#493 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 2 | 1 | no | — |
| 262 | C4 | — | 2 | 1 | no | — |
| 263 | C4 | — | 16 | 2 | no | — |
| 264 | C4 | — | 16 | 2 | no | — |
| 265 | C4 | #1721 | 1 | 2 | yes | — |
| 266 | C4 | C2×#496 | 1 | 2 | yes | — |
| 267 | C4 | C2×#509 | 1 | 2 | yes | — |
| 268 | C4 | #3414 | 1 | 2 | yes | — |
| 269 | C2 | — | 2 | 2 | no | — |
| 270 | C2 | — | 32 | 2 | no | — |
| 271 | C2 | — | 32 | 2 | no | — |
| 272 | C2 | C2×#1721 | 1 | 1 | yes | characteristic |
| 273 | C2 | #33717 | 1 | 1 | yes | characteristic |
| 274 | C2 | #33719 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33717 | C2 | semidirect |
| 2. #33707 | C2 | semidirect |
| 3. #19603 | C2 | semidirect |
| 4. C2×#1706 | C2 | semidirect |
| 5. #12527 | C2 | semidirect |
| 6. #12526 | C2 | semidirect |
| 7. #12525 | C2 | semidirect |
| 8. #11402 | C2 | semidirect |
| 21. #2241 | C22 | semidirect |
| 22. #2240 | C22 | semidirect |
| 25. #1713 | C22 | semidirect |
| 26. #1712 | C22 | semidirect |
| 27. #1709 | C22 | semidirect |
| 35. #1312 | C22 | semidirect |
| 36. #1312 | C22 | semidirect |
| 248. C8 | #509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33717 | 2 |
| #33707 | 2 |
| #19603 | 2 |
| C2×#1706 | 2 |
| #12527 | 2 |
| #12526 | 2 |
| #12525 | 2 |
| #11402 | 2 |