Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1011 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 150 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 | #60307 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33564 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #24287 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #13052 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #12809 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #3584 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | #3529 | C22 | 1 | 1 | yes | characteristic |
| 8 | #3067 | — | 2 | 3 | no | — |
| 9 | #3018 | C22 | 1 | 6 | yes | — |
| 10 | #2995 | — | 2 | 3 | no | — |
| 11 | #2980 | C22 | 1 | 3 | yes | — |
| 12 | #2978 | — | 2 | 3 | no | — |
| 13 | #2966 | — | 2 | 3 | no | — |
| 14 | #2931 | C22 | 1 | 3 | yes | — |
| 15 | #2929 | — | 2 | 3 | no | — |
| 16 | #2203 | — | 2 | 6 | no | — |
| 17 | #2202 | — | 2 | 2 | no | — |
| 18 | #2132 | C22 | 1 | 6 | yes | — |
| 19 | #2050 | — | 2 | 3 | no | — |
| 20 | #2017 | — | 2 | 3 | no | — |
| 21 | #1974 | — | 2 | 3 | no | — |
| 22 | #1813 | C22 | 1 | 3 | yes | complemented |
| 23 | #1791 | C22 | 1 | 6 | yes | complemented |
| 24 | #1392 | C22 | 1 | 6 | yes | complemented |
| 25 | #585 | C23 | 1 | 2 | yes | — |
| 26 | #578 | — | 4 | 1 | no | — |
| 27 | #577 | — | 4 | 2 | no | — |
| 28 | #576 | — | 4 | 1 | no | — |
| 29 | #575 | — | 2 | 6 | no | — |
| 30 | #575 | — | 2 | 3 | no | — |
| 31 | C2×#135 | — | 2 | 3 | no | — |
| 32 | C2×#131 | D4 | 1 | 3 | yes | — |
| 33 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 34 | C23×C8 | — | 2 | 1 | no | — |
| 35 | #563 | D4 | 1 | 3 | yes | — |
| 36 | #562 | — | 2 | 3 | no | — |
| 37 | H(3,𝔽4) | D4 | 1 | 1 | yes | characteristic |
| 38 | #560 | — | 2 | 3 | no | — |
| 39 | #532 | C23 | 1 | 3 | yes | — |
| 40 | #529 | — | 2 | 3 | no | — |
| 41 | #525 | — | 2 | 3 | no | — |
| 42 | #473 | — | 4 | 3 | no | — |
| 43 | #472 | — | 4 | 3 | no | — |
| 44 | #471 | — | 4 | 6 | no | — |
| 45 | #456 | — | 2 | 6 | no | — |
| 46 | #455 | — | 2 | 6 | no | — |
| 47 | #454 | — | 2 | 6 | no | — |
| 48 | #453 | — | 2 | 6 | no | — |
| 49 | #446 | — | 2 | 3 | no | — |
| 50 | C8×Q8 | — | 2 | 1 | no | — |
| 51 | #436 | — | 2 | 3 | no | — |
| 52 | #436 | — | 2 | 6 | no | — |
| 53 | #435 | — | 2 | 6 | no | — |
| 54 | #435 | — | 2 | 6 | no | — |
| 55 | #435 | — | 2 | 6 | no | — |
| 56 | C8×D4 | — | 2 | 3 | no | — |
| 57 | C8×D4 | — | 2 | 6 | no | — |
| 58 | #433 | — | 2 | 6 | no | — |
| 59 | #432 | — | 2 | 3 | no | — |
| 60 | C2×#108 | — | 2 | 3 | no | — |
| 61 | #424 | — | 2 | 3 | no | — |
| 62 | #423 | D4 | 1 | 3 | yes | — |
| 63 | C2×#104 | — | 2 | 6 | no | — |
| 64 | #413 | — | 2 | 6 | no | — |
| 65 | #413 | C23 | 1 | 6 | yes | — |
| 66 | #408 | — | 2 | 3 | no | — |
| 67 | #408 | — | 2 | 6 | no | — |
| 68 | #408 | D4 | 1 | 3 | yes | — |
| 69 | #408 | C23 | 1 | 3 | yes | — |
| 70 | #407 | — | 2 | 3 | no | — |
| 71 | #405 | — | 2 | 3 | no | — |
| 72 | C2×C4×C8 | D4 | 1 | 3 | yes | — |
| 73 | #356 | — | 2 | 6 | no | — |
| 74 | #355 | — | 2 | 6 | no | — |
| 75 | #338 | — | 2 | 3 | no | — |
| 76 | 2−1+4 | — | 2 | 2 | no | — |
| 77 | 2−1+4 | — | 4 | 2 | no | — |
| 78 | H(5,𝔽2) | — | 2 | 6 | no | — |
| 79 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 80 | C2×#41 | — | 2 | 6 | no | — |
| 81 | C2×#41 | — | 8 | 2 | no | — |
| 82 | C2×#41 | C2×D4 | 1 | 3 | yes | — |
| 83 | C2×#41 | C24 | 1 | 1 | yes | characteristic |
| 84 | C23×C4 | — | 2 | 1 | no | — |
| 85 | #137 | — | 4 | 6 | no | — |
| 86 | #137 | — | 4 | 6 | no | — |
| 87 | #137 | — | 4 | 6 | no | — |
| 88 | AGL(1,ℤ/8) | — | 4 | 6 | no | — |
| 89 | AGL(1,ℤ/8) | — | 4 | 6 | no | — |
| 90 | AGL(1,ℤ/8) | — | 4 | 6 | no | — |
| 91 | #135 | — | 4 | 3 | no | — |
| 92 | #135 | — | 4 | 3 | no | — |
| 93 | #135 | — | 4 | 1 | no | — |
| 94 | #135 | — | 4 | 3 | no | — |
| 95 | #135 | — | 8 | 6 | no | — |
| 96 | C2×Q16 | — | 2 | 3 | no | — |
| 97 | C2×SD16 | — | 2 | 6 | no | — |
| 98 | C2×D8 | — | 2 | 3 | no | — |
| 99 | #131 | — | 4 | 1 | no | — |
| 100 | #131 | — | 4 | 1 | no | — |
| 101 | #131 | — | 4 | 3 | no | — |
| 102 | #131 | — | 4 | 3 | no | — |
| 103 | C2×M16 | — | 2 | 3 | no | — |
| 104 | C2×M16 | — | 2 | 3 | no | — |
| 105 | C2×M16 | — | 2 | 6 | no | — |
| 106 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 107 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 108 | C22×C8 | — | 2 | 3 | no | — |
| 109 | C22×C8 | — | 2 | 3 | no | — |
| 110 | C22×C8 | — | 8 | 1 | no | — |
| 111 | C22×C8 | — | 8 | 1 | no | — |
| 112 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 113 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 114 | C4⋊Q8 | C2×D4 | 1 | 3 | yes | — |
| 115 | #127 | #41 | 1 | 3 | yes | — |
| 116 | #126 | — | 2 | 6 | no | — |
| 117 | #125 | — | 2 | 3 | no | — |
| 118 | #124 | — | 2 | 3 | no | — |
| 119 | #124 | C2×D4 | 1 | 3 | yes | — |
| 120 | #124 | #41 | 1 | 3 | yes | — |
| 121 | #123 | — | 2 | 3 | no | — |
| 122 | #123 | — | 2 | 6 | no | — |
| 123 | #123 | — | 2 | 3 | no | — |
| 124 | C22⋊Q8 | — | 2 | 3 | no | — |
| 125 | C22⋊Q8 | — | 2 | 6 | no | — |
| 126 | #121 | — | 2 | 6 | no | — |
| 127 | #121 | — | 2 | 3 | no | — |
| 128 | #121 | — | 2 | 6 | no | — |
| 129 | #120 | — | 2 | 3 | no | — |
| 130 | #120 | — | 2 | 3 | no | — |
| 131 | C4×Q8 | — | 2 | 1 | no | — |
| 132 | C4×D4 | — | 2 | 3 | no | — |
| 133 | C4×D4 | — | 2 | 6 | no | — |
| 134 | #117 | — | 2 | 3 | no | — |
| 135 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 136 | #108 | — | 4 | 3 | no | — |
| 137 | #108 | — | 4 | 3 | no | — |
| 138 | C4⋊C8 | — | 2 | 3 | no | — |
| 139 | C4⋊C8 | — | 2 | 3 | no | — |
| 140 | C4⋊C8 | — | 2 | 6 | no | — |
| 141 | #104 | — | 4 | 6 | no | — |
| 142 | #104 | — | 4 | 6 | no | — |
| 143 | #101 | — | 2 | 6 | no | — |
| 144 | #101 | — | 4 | 6 | no | — |
| 145 | #100 | — | 2 | 6 | no | — |
| 146 | #100 | — | 4 | 6 | no | — |
| 147 | C22⋊C8 | — | 2 | 6 | no | — |
| 148 | C22⋊C8 | — | 2 | 6 | no | — |
| 149 | C22⋊C8 | — | 2 | 6 | no | — |
| 150 | C22⋊C8 | — | 8 | 6 | no | — |
| 151 | #97 | — | 2 | 6 | no | — |
| 152 | C4×C8 | — | 2 | 3 | no | — |
| 153 | C4×C8 | — | 2 | 3 | no | — |
| 154 | C24 | — | 2 | 1 | no | — |
| 155 | #41 | — | 4 | 1 | no | — |
| 156 | #41 | — | 4 | 1 | no | — |
| 157 | #41 | — | 4 | 3 | no | — |
| 158 | #41 | — | 4 | 3 | no | — |
| 159 | #41 | — | 4 | 6 | no | — |
| 160 | #41 | — | 4 | 6 | no | — |
| 161 | #41 | — | 4 | 2 | no | — |
| 162 | #41 | — | 4 | 6 | no | — |
| 163 | #41 | — | 8 | 6 | no | — |
| 164 | #41 | — | 8 | 2 | no | — |
| 165 | C2×Q8 | — | 2 | 1 | no | — |
| 166 | C2×Q8 | — | 2 | 6 | no | — |
| 167 | C2×Q8 | — | 8 | 2 | no | — |
| 168 | C2×Q8 | #121 | 1 | 3 | yes | — |
| 169 | C2×Q8 | C22×D4 | 1 | 1 | yes | characteristic |
| 170 | C2×D4 | — | 2 | 6 | no | — |
| 171 | C2×D4 | — | 2 | 3 | no | — |
| 172 | C2×D4 | — | 2 | 6 | no | — |
| 173 | C2×D4 | — | 2 | 3 | no | — |
| 174 | C2×D4 | — | 4 | 6 | no | — |
| 175 | C2×D4 | — | 8 | 6 | no | — |
| 176 | C2×D4 | #121 | 1 | 3 | yes | — |
| 177 | C2×D4 | C2×#41 | 1 | 3 | yes | — |
| 178 | C22×C4 | — | 2 | 3 | no | — |
| 179 | C22×C4 | — | 2 | 3 | no | — |
| 180 | C22×C4 | — | 2 | 3 | no | — |
| 181 | C22×C4 | — | 4 | 6 | no | — |
| 182 | C22×C4 | — | 8 | 1 | no | — |
| 183 | C22×C4 | — | 8 | 1 | no | — |
| 184 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 185 | Q16 | — | 4 | 3 | no | — |
| 186 | Q16 | — | 4 | 1 | no | — |
| 187 | Q16 | — | 8 | 6 | no | — |
| 188 | SD16 | — | 4 | 6 | no | — |
| 189 | SD16 | — | 4 | 2 | no | — |
| 190 | SD16 | — | 8 | 6 | no | — |
| 191 | SD16 | — | 8 | 6 | no | — |
| 192 | D8 | — | 4 | 3 | no | — |
| 193 | D8 | — | 4 | 1 | no | — |
| 194 | D8 | — | 8 | 6 | no | — |
| 195 | M16 | — | 4 | 3 | no | — |
| 196 | M16 | — | 4 | 3 | no | — |
| 197 | M16 | — | 4 | 6 | no | — |
| 198 | M16 | — | 4 | 6 | no | — |
| 199 | M16 | — | 8 | 3 | no | — |
| 200 | M16 | — | 8 | 3 | no | — |
| 201 | C2×C8 | — | 2 | 3 | no | — |
| 202 | C2×C8 | — | 2 | 3 | no | — |
| 203 | C2×C8 | — | 2 | 3 | no | — |
| 204 | C2×C8 | — | 2 | 3 | no | — |
| 205 | C2×C8 | — | 4 | 3 | no | — |
| 206 | C2×C8 | — | 4 | 3 | no | — |
| 207 | C2×C8 | — | 4 | 3 | no | — |
| 208 | C2×C8 | — | 4 | 6 | no | — |
| 209 | C2×C8 | — | 8 | 3 | no | — |
| 210 | C2×C8 | #120 | 1 | 3 | yes | — |
| 211 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 212 | C4⋊C4 | — | 2 | 6 | no | — |
| 213 | C4⋊C4 | — | 2 | 3 | no | — |
| 214 | C4⋊C4 | — | 2 | 3 | no | — |
| 215 | C4⋊C4 | — | 2 | 6 | no | — |
| 216 | C22⋊C4 | — | 2 | 3 | no | — |
| 217 | C22⋊C4 | — | 2 | 3 | no | — |
| 218 | C22⋊C4 | — | 2 | 6 | no | — |
| 219 | C22⋊C4 | — | 2 | 6 | no | — |
| 220 | C22⋊C4 | — | 2 | 6 | no | — |
| 221 | C42 | — | 2 | 3 | no | — |
| 222 | C42 | #121 | 1 | 3 | yes | — |
| 223 | C42 | #121 | 1 | 3 | yes | — |
| 224 | C23 | — | 2 | 3 | no | — |
| 225 | C23 | — | 4 | 6 | no | — |
| 226 | C23 | — | 8 | 1 | no | — |
| 227 | C23 | — | 8 | 2 | no | — |
| 228 | C23 | #525 | 1 | 3 | yes | — |
| 229 | Q8 | — | 4 | 1 | no | — |
| 230 | Q8 | — | 4 | 3 | no | — |
| 231 | Q8 | — | 4 | 6 | no | — |
| 232 | Q8 | — | 4 | 2 | no | — |
| 233 | Q8 | — | 8 | 1 | no | — |
| 234 | D4 | — | 4 | 3 | no | — |
| 235 | D4 | — | 4 | 3 | no | — |
| 236 | D4 | — | 4 | 6 | no | — |
| 237 | D4 | — | 4 | 2 | no | — |
| 238 | D4 | — | 8 | 3 | no | — |
| 239 | D4 | — | 8 | 6 | no | — |
| 240 | D4 | — | 8 | 3 | no | — |
| 241 | D4 | — | 8 | 6 | no | — |
| 242 | D4 | — | 8 | 2 | no | — |
| 243 | C2×C4 | — | 2 | 3 | no | — |
| 244 | C2×C4 | — | 2 | 3 | no | — |
| 245 | C2×C4 | — | 2 | 3 | no | — |
| 246 | C2×C4 | — | 2 | 3 | no | — |
| 247 | C2×C4 | — | 2 | 3 | no | — |
| 248 | C2×C4 | — | 4 | 3 | no | — |
| 249 | C2×C4 | — | 4 | 3 | no | — |
| 250 | C2×C4 | — | 4 | 6 | no | — |
| 251 | C2×C4 | — | 8 | 6 | no | — |
| 252 | C2×C4 | — | 8 | 3 | no | — |
| 253 | C2×C4 | — | 8 | 3 | no | — |
| 254 | C2×C4 | — | 8 | 2 | no | — |
| 255 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 257 | C8 | — | 2 | 3 | no | — |
| 258 | C8 | — | 4 | 3 | no | — |
| 259 | C8 | — | 4 | 3 | no | — |
| 260 | C8 | — | 8 | 6 | no | — |
| 261 | C8 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 262 | C8 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 263 | C22 | — | 2 | 3 | no | — |
| 264 | C22 | — | 4 | 3 | no | — |
| 265 | C22 | — | 4 | 6 | no | — |
| 266 | C22 | — | 4 | 3 | no | — |
| 267 | C22 | — | 8 | 4 | no | — |
| 268 | C22 | — | 8 | 2 | no | — |
| 269 | C22 | — | 16 | 6 | no | — |
| 270 | C22 | #2847 | 1 | 1 | yes | characteristic |
| 271 | C4 | — | 2 | 3 | no | — |
| 272 | C4 | — | 4 | 3 | no | — |
| 273 | C4 | — | 4 | 3 | no | — |
| 274 | C4 | — | 4 | 3 | no | — |
| 275 | C4 | — | 8 | 2 | no | — |
| 276 | C4 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 277 | C4 | #3027 | 1 | 1 | yes | characteristic |
| 278 | C2 | — | 2 | 1 | no | — |
| 279 | C2 | — | 4 | 3 | no | — |
| 280 | C2 | — | 8 | 3 | no | — |
| 281 | C2 | — | 16 | 2 | no | — |
| 282 | C2 | #32926 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32926 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #60307 | C2 | semidirect |
| 2. #33564 | C2 | semidirect |
| 3. #24287 | C2 | semidirect |
| 4. #13052 | C2 | semidirect |
| 5. #12809 | C2 | semidirect |
| 6. #3584 | C22 | semidirect |
| 22. #1813 | C22 | semidirect |
| 23. #1791 | C22 | semidirect |
| 24. #1392 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60307 | 2 |
| #33564 | 2 |
| #24287 | 2 |
| #13052 | 2 |
| #12809 | 2 |