Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
350 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #33800 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2255 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19638 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11974 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11678 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #9788 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7165 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3206 | — | 2 | 1 | no | — |
| 9 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#499 | C22 | 1 | 1 | yes | characteristic |
| 12 | #3141 | — | 2 | 1 | no | — |
| 13 | #2255 | — | 2 | 1 | no | — |
| 14 | #2255 | — | 2 | 2 | no | — |
| 15 | #2255 | — | 2 | 2 | no | — |
| 16 | #2244 | — | 2 | 1 | no | — |
| 17 | #2233 | — | 2 | 1 | no | — |
| 18 | #2232 | — | 2 | 1 | no | — |
| 19 | #2228 | — | 2 | 2 | no | — |
| 20 | #2163 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#367 | — | 2 | 1 | no | — |
| 22 | C2×#366 | — | 2 | 1 | no | — |
| 23 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1719 | — | 2 | 2 | no | — |
| 25 | #1710 | — | 2 | 1 | no | — |
| 26 | #1706 | — | 2 | 1 | no | — |
| 27 | #1568 | C22 | 1 | 1 | yes | characteristic |
| 28 | #1528 | — | 2 | 1 | no | — |
| 29 | #1470 | — | 2 | 1 | no | — |
| 30 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1364 | — | 2 | 2 | no | — |
| 32 | #1333 | — | 2 | 2 | no | — |
| 33 | #532 | — | 2 | 1 | no | — |
| 34 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 35 | C2×#125 | — | 2 | 1 | no | — |
| 36 | C22×C16 | — | 2 | 1 | no | — |
| 37 | #500 | — | 2 | 1 | no | — |
| 38 | #500 | — | 2 | 1 | no | — |
| 39 | #500 | — | 2 | 1 | no | — |
| 40 | #500 | D4 | 1 | 2 | yes | — |
| 41 | #500 | D4 | 1 | 1 | yes | characteristic |
| 42 | #499 | — | 2 | 1 | no | — |
| 43 | #499 | — | 2 | 1 | no | — |
| 44 | #499 | — | 2 | 1 | no | — |
| 45 | #499 | — | 2 | 1 | no | — |
| 46 | #493 | D4 | 1 | 1 | yes | characteristic |
| 47 | #492 | — | 2 | 2 | no | — |
| 48 | #489 | — | 4 | 2 | no | — |
| 49 | #487 | — | 2 | 2 | no | — |
| 50 | #487 | — | 2 | 2 | no | — |
| 51 | #486 | — | 2 | 2 | no | — |
| 52 | #486 | — | 2 | 1 | no | — |
| 53 | #466 | — | 2 | 1 | no | — |
| 54 | #466 | — | 2 | 1 | no | — |
| 55 | #461 | — | 4 | 2 | no | — |
| 56 | #448 | — | 4 | 2 | no | — |
| 57 | C4×D8 | — | 2 | 1 | no | — |
| 58 | C4×D8 | — | 2 | 1 | no | — |
| 59 | #432 | — | 2 | 1 | no | — |
| 60 | #431 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×#107 | — | 2 | 1 | no | — |
| 62 | C2×#107 | — | 2 | 1 | no | — |
| 63 | C2×#107 | D4 | 1 | 1 | yes | characteristic |
| 64 | C2×#107 | D4 | 1 | 1 | yes | characteristic |
| 65 | C2×#106 | — | 2 | 1 | no | — |
| 66 | #417 | — | 4 | 2 | no | — |
| 67 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 68 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 69 | #367 | — | 2 | 4 | no | — |
| 70 | #367 | — | 2 | 2 | no | — |
| 71 | #366 | — | 2 | 4 | no | — |
| 72 | #366 | — | 2 | 2 | no | — |
| 73 | C4⋊C16 | D4 | 1 | 2 | yes | — |
| 74 | #357 | — | 2 | 2 | no | — |
| 75 | #357 | — | 2 | 2 | no | — |
| 76 | C22⋊C16 | — | 2 | 1 | no | — |
| 77 | #346 | — | 2 | 1 | no | — |
| 78 | #346 | — | 2 | 1 | no | — |
| 79 | #346 | D4 | 1 | 2 | yes | — |
| 80 | #335 | — | 2 | 1 | no | — |
| 81 | #334 | — | 2 | 1 | no | — |
| 82 | #330 | — | 4 | 2 | no | — |
| 83 | #324 | — | 4 | 2 | no | — |
| 84 | #322 | — | 2 | 1 | no | — |
| 85 | C82 | — | 2 | 1 | no | — |
| 86 | C2×#41 | — | 4 | 1 | no | — |
| 87 | C22×Q8 | — | 4 | 1 | no | — |
| 88 | C2×SD16 | — | 4 | 4 | no | — |
| 89 | C2×D8 | — | 2 | 1 | no | — |
| 90 | C2×D8 | — | 2 | 1 | no | — |
| 91 | C2×M16 | — | 2 | 1 | no | — |
| 92 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 94 | C4⋊Q8 | — | 2 | 2 | no | — |
| 95 | C4⋊Q8 | — | 2 | 1 | no | — |
| 96 | C4⋊Q8 | — | 4 | 1 | no | — |
| 97 | C4⋊Q8 | — | 4 | 1 | no | — |
| 98 | C4⋊Q8 | — | 4 | 1 | no | — |
| 99 | #127 | — | 2 | 1 | no | — |
| 100 | #125 | — | 4 | 1 | no | — |
| 101 | #125 | — | 4 | 1 | no | — |
| 102 | #125 | — | 4 | 1 | no | — |
| 103 | #125 | — | 4 | 1 | no | — |
| 104 | #124 | — | 4 | 1 | no | — |
| 105 | #121 | — | 4 | 1 | no | — |
| 106 | #121 | — | 4 | 1 | no | — |
| 107 | C4×D4 | — | 4 | 1 | no | — |
| 108 | C4×D4 | — | 4 | 1 | no | — |
| 109 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 110 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 111 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 112 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 113 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 114 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 115 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×C16 | — | 2 | 4 | no | — |
| 117 | C2×C16 | — | 2 | 2 | no | — |
| 118 | C2×C16 | — | 2 | 2 | no | — |
| 119 | #107 | — | 2 | 1 | no | — |
| 120 | #107 | — | 2 | 1 | no | — |
| 121 | #107 | — | 2 | 1 | no | — |
| 122 | #107 | — | 2 | 2 | no | — |
| 123 | #107 | — | 2 | 1 | no | — |
| 124 | #107 | — | 2 | 1 | no | — |
| 125 | #107 | — | 2 | 2 | no | — |
| 126 | #107 | — | 2 | 1 | no | — |
| 127 | #107 | — | 4 | 1 | no | — |
| 128 | #107 | — | 4 | 1 | no | — |
| 129 | #106 | — | 4 | 1 | no | — |
| 130 | #106 | — | 4 | 1 | no | — |
| 131 | C4⋊C8 | — | 2 | 1 | no | — |
| 132 | C4⋊C8 | — | 4 | 1 | no | — |
| 133 | #103 | — | 4 | 4 | no | — |
| 134 | #103 | — | 4 | 4 | no | — |
| 135 | #102 | — | 4 | 4 | no | — |
| 136 | #102 | — | 4 | 1 | no | — |
| 137 | #102 | — | 4 | 1 | no | — |
| 138 | C22⋊C8 | — | 4 | 1 | no | — |
| 139 | C22⋊C8 | D8 | 1 | 2 | yes | — |
| 140 | #97 | — | 4 | 1 | no | — |
| 141 | #97 | D8 | 1 | 2 | yes | — |
| 142 | C4×C8 | — | 2 | 1 | no | — |
| 143 | C4×C8 | — | 2 | 1 | no | — |
| 144 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 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 | #41 | — | 8 | 4 | no | — |
| 149 | C2×Q8 | — | 4 | 1 | no | — |
| 150 | C2×Q8 | — | 4 | 2 | no | — |
| 151 | C2×Q8 | — | 8 | 1 | no | — |
| 152 | C2×Q8 | — | 16 | 2 | no | — |
| 153 | C2×D4 | — | 4 | 1 | no | — |
| 154 | C2×D4 | — | 4 | 1 | no | — |
| 155 | C2×D4 | — | 4 | 1 | no | — |
| 156 | C2×D4 | — | 8 | 1 | no | — |
| 157 | C22×C4 | — | 4 | 1 | no | — |
| 158 | C22×C4 | — | 4 | 1 | no | — |
| 159 | C22×C4 | — | 8 | 1 | no | — |
| 160 | C22×C4 | — | 8 | 1 | no | — |
| 161 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 162 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 163 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 164 | SD16 | — | 16 | 4 | no | — |
| 165 | D8 | — | 4 | 2 | no | — |
| 166 | D8 | — | 4 | 2 | no | — |
| 167 | M16 | — | 4 | 2 | no | — |
| 168 | C2×C8 | — | 2 | 2 | no | — |
| 169 | C2×C8 | — | 2 | 2 | no | — |
| 170 | C2×C8 | — | 2 | 2 | no | — |
| 171 | C2×C8 | — | 2 | 2 | no | — |
| 172 | C2×C8 | — | 8 | 1 | no | — |
| 173 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 175 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 176 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 177 | C4⋊C4 | — | 4 | 1 | no | — |
| 178 | C4⋊C4 | — | 4 | 1 | no | — |
| 179 | C4⋊C4 | — | 4 | 1 | no | — |
| 180 | C4⋊C4 | — | 4 | 2 | no | — |
| 181 | C4⋊C4 | — | 4 | 2 | no | — |
| 182 | C4⋊C4 | — | 4 | 1 | no | — |
| 183 | C4⋊C4 | — | 4 | 1 | no | — |
| 184 | C4⋊C4 | — | 4 | 1 | no | — |
| 185 | C4⋊C4 | — | 8 | 1 | no | — |
| 186 | C4⋊C4 | — | 8 | 1 | no | — |
| 187 | C4⋊C4 | — | 8 | 1 | no | — |
| 188 | C4⋊C4 | — | 8 | 1 | no | — |
| 189 | C4⋊C4 | — | 8 | 1 | no | — |
| 190 | C4⋊C4 | — | 8 | 1 | no | — |
| 191 | C22⋊C4 | — | 8 | 1 | no | — |
| 192 | C22⋊C4 | — | 8 | 1 | no | — |
| 193 | C42 | #120 | 1 | 1 | yes | characteristic |
| 194 | C42 | #127 | 1 | 1 | yes | characteristic |
| 195 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 196 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 197 | C16 | — | 2 | 4 | no | — |
| 198 | C16 | — | 4 | 2 | no | — |
| 199 | C23 | — | 8 | 1 | no | — |
| 200 | C23 | #459 | 1 | 1 | yes | characteristic |
| 201 | Q8 | — | 8 | 4 | no | — |
| 202 | Q8 | — | 8 | 2 | no | — |
| 203 | Q8 | — | 16 | 2 | no | — |
| 204 | D4 | — | 8 | 2 | no | — |
| 205 | D4 | — | 8 | 2 | no | — |
| 206 | D4 | — | 8 | 1 | no | — |
| 207 | D4 | — | 8 | 1 | no | — |
| 208 | D4 | — | 16 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 2 | no | — |
| 210 | C2×C4 | — | 2 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 8 | 2 | no | — |
| 214 | C2×C4 | — | 8 | 1 | no | — |
| 215 | C2×C4 | — | 8 | 1 | no | — |
| 216 | C2×C4 | — | 8 | 1 | no | — |
| 217 | C2×C4 | — | 8 | 2 | no | — |
| 218 | C2×C4 | — | 16 | 2 | no | — |
| 219 | C2×C4 | — | 16 | 2 | no | — |
| 220 | C2×C4 | — | 16 | 1 | no | — |
| 221 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | #496 | 1 | 1 | yes | characteristic |
| 227 | C8 | — | 2 | 1 | no | — |
| 228 | C8 | — | 2 | 1 | no | — |
| 229 | C8 | — | 4 | 2 | no | — |
| 230 | C8 | — | 4 | 2 | no | — |
| 231 | C8 | — | 16 | 1 | no | — |
| 232 | C8 | #456 | 1 | 2 | yes | — |
| 233 | C8 | #473 | 1 | 2 | yes | — |
| 234 | C22 | — | 2 | 1 | no | — |
| 235 | C22 | — | 2 | 1 | no | — |
| 236 | C22 | — | 16 | 1 | no | — |
| 237 | C22 | — | 16 | 1 | no | — |
| 238 | C22 | — | 16 | 1 | no | — |
| 239 | C22 | #2011 | 1 | 1 | yes | characteristic |
| 240 | C22 | #2212 | 1 | 2 | yes | — |
| 241 | C4 | — | 2 | 1 | no | — |
| 242 | C4 | — | 2 | 1 | no | — |
| 243 | C4 | — | 2 | 1 | no | — |
| 244 | C4 | — | 2 | 1 | no | — |
| 245 | C4 | — | 2 | 1 | no | — |
| 246 | C4 | — | 16 | 1 | no | — |
| 247 | C4 | — | 16 | 1 | no | — |
| 248 | C4 | — | 16 | 1 | no | — |
| 249 | C4 | — | 16 | 2 | no | — |
| 250 | C4 | #2005 | 1 | 1 | yes | characteristic |
| 251 | C4 | #2020 | 1 | 1 | yes | characteristic |
| 252 | C2 | — | 2 | 2 | no | — |
| 253 | C2 | — | 32 | 1 | no | — |
| 254 | C2 | #10174 | 1 | 1 | yes | characteristic |
| 255 | C2 | #13332 | 1 | 1 | yes | characteristic |
| 256 | C2 | #13334 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2255 | C2 | semidirect |
| 4. #11974 | C2 | semidirect |
| 6. #9788 | C2 | semidirect |
| 7. #7165 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33800 | 2 |
| C2×#2255 | 2 |
| #19638 | 2 |
| #11974 | 2 |
| #11678 | 2 |
| #9788 | 2 |
| #7165 | 2 |