Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
328 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 64 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 | #33771 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13257 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #12018 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #9802 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #9724 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7298 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 10 | #2246 | — | 2 | 1 | no | — |
| 11 | #2240 | — | 2 | 1 | no | — |
| 12 | #2206 | — | 2 | 1 | no | — |
| 13 | #2206 | — | 2 | 1 | no | — |
| 14 | #2184 | — | 2 | 2 | no | — |
| 15 | #2162 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#361 | — | 2 | 2 | no | — |
| 18 | #1965 | — | 2 | 1 | no | — |
| 19 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1567 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1545 | — | 2 | 1 | no | — |
| 22 | #1525 | — | 2 | 1 | no | — |
| 23 | #1484 | — | 2 | 1 | no | — |
| 24 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1370 | — | 2 | 1 | no | — |
| 26 | #1367 | — | 2 | 1 | no | — |
| 27 | #1365 | — | 2 | 1 | no | — |
| 28 | #1361 | — | 2 | 1 | no | — |
| 29 | C22×D8 | — | 2 | 1 | no | — |
| 30 | C22×D8 | D4 | 1 | 2 | yes | — |
| 31 | #532 | — | 2 | 1 | no | — |
| 32 | C2×#127 | — | 2 | 1 | no | — |
| 33 | C2×D16 | — | 2 | 2 | no | — |
| 34 | C2×D16 | — | 2 | 2 | no | — |
| 35 | C2×M32 | — | 2 | 1 | no | — |
| 36 | #500 | D4 | 1 | 1 | yes | characteristic |
| 37 | #493 | — | 2 | 1 | no | — |
| 38 | #493 | — | 2 | 1 | no | — |
| 39 | #493 | — | 2 | 1 | no | — |
| 40 | #493 | — | 2 | 1 | no | — |
| 41 | #493 | D4 | 1 | 1 | yes | characteristic |
| 42 | #486 | — | 2 | 1 | no | — |
| 43 | #466 | — | 2 | 1 | no | — |
| 44 | #466 | — | 2 | 1 | no | — |
| 45 | C4×D8 | — | 2 | 1 | no | — |
| 46 | C4×D8 | — | 2 | 1 | no | — |
| 47 | #431 | D4 | 1 | 1 | yes | characteristic |
| 48 | C2×#108 | — | 2 | 1 | no | — |
| 49 | #423 | — | 2 | 1 | no | — |
| 50 | C2×#100 | — | 4 | 2 | no | — |
| 51 | C4×M16 | — | 2 | 1 | no | — |
| 52 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 53 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 54 | C4⋊C16 | — | 2 | 1 | no | — |
| 55 | C4⋊C16 | — | 2 | 1 | no | — |
| 56 | C4⋊C16 | D4 | 1 | 1 | yes | characteristic |
| 57 | C4⋊C16 | D4 | 1 | 1 | yes | characteristic |
| 58 | #361 | — | 4 | 4 | no | — |
| 59 | #357 | — | 2 | 2 | no | — |
| 60 | #357 | — | 2 | 2 | no | — |
| 61 | C22⋊C16 | — | 2 | 1 | no | — |
| 62 | C4×C16 | — | 2 | 1 | no | — |
| 63 | #341 | — | 2 | 1 | no | — |
| 64 | #335 | — | 2 | 1 | no | — |
| 65 | #335 | — | 2 | 1 | no | — |
| 66 | #331 | — | 4 | 2 | no | — |
| 67 | #329 | — | 4 | 2 | no | — |
| 68 | D4⋊C8 | — | 4 | 2 | no | — |
| 69 | #322 | — | 2 | 1 | no | — |
| 70 | C82 | — | 2 | 1 | no | — |
| 71 | C2×#41 | — | 4 | 1 | no | — |
| 72 | C22×D4 | — | 2 | 2 | no | — |
| 73 | C22×D4 | — | 4 | 1 | no | — |
| 74 | C22×D4 | — | 4 | 1 | no | — |
| 75 | C2×D8 | — | 2 | 2 | no | — |
| 76 | C2×D8 | — | 2 | 1 | no | — |
| 77 | C2×D8 | — | 2 | 2 | no | — |
| 78 | C2×D8 | — | 2 | 1 | no | — |
| 79 | C2×D8 | — | 4 | 2 | no | — |
| 80 | C2×D8 | — | 4 | 2 | no | — |
| 81 | C2×D8 | — | 4 | 1 | no | — |
| 82 | C2×D8 | — | 4 | 1 | no | — |
| 83 | C2×D8 | — | 8 | 2 | no | — |
| 84 | C2×M16 | — | 2 | 1 | no | — |
| 85 | C2×M16 | — | 4 | 1 | no | — |
| 86 | C2×M16 | — | 4 | 1 | no | — |
| 87 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 89 | C4⋊Q8 | — | 2 | 1 | no | — |
| 90 | #127 | — | 2 | 1 | no | — |
| 91 | #127 | — | 4 | 1 | no | — |
| 92 | #127 | — | 4 | 1 | no | — |
| 93 | #127 | — | 4 | 1 | no | — |
| 94 | #127 | — | 4 | 1 | no | — |
| 95 | #124 | — | 4 | 1 | no | — |
| 96 | #121 | — | 4 | 1 | no | — |
| 97 | #121 | — | 4 | 1 | no | — |
| 98 | C4×D4 | — | 4 | 1 | no | — |
| 99 | C4×D4 | — | 4 | 1 | no | — |
| 100 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | D16 | — | 8 | 2 | no | — |
| 102 | D16 | — | 8 | 2 | no | — |
| 103 | M32 | — | 4 | 2 | no | — |
| 104 | C2×C16 | — | 2 | 1 | no | — |
| 105 | C2×C16 | — | 2 | 1 | no | — |
| 106 | C2×C16 | — | 4 | 1 | no | — |
| 107 | #108 | — | 2 | 2 | no | — |
| 108 | #108 | — | 4 | 1 | no | — |
| 109 | #107 | — | 2 | 1 | no | — |
| 110 | #107 | — | 2 | 1 | no | — |
| 111 | C4⋊C8 | — | 4 | 1 | no | — |
| 112 | C4⋊C8 | — | 4 | 1 | no | — |
| 113 | C4⋊C8 | D8 | 1 | 2 | yes | — |
| 114 | #102 | — | 4 | 1 | no | — |
| 115 | #102 | — | 4 | 1 | no | — |
| 116 | #100 | — | 8 | 4 | no | — |
| 117 | C22⋊C8 | — | 2 | 1 | no | — |
| 118 | #97 | — | 2 | 1 | no | — |
| 119 | #97 | — | 4 | 1 | no | — |
| 120 | C4×C8 | — | 2 | 1 | no | — |
| 121 | C4×C8 | — | 2 | 1 | no | — |
| 122 | C4×C8 | — | 4 | 1 | no | — |
| 123 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 125 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 126 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 127 | C24 | — | 4 | 2 | no | — |
| 128 | #41 | — | 16 | 2 | no | — |
| 129 | C2×Q8 | — | 4 | 1 | no | — |
| 130 | C2×D4 | — | 4 | 1 | no | — |
| 131 | C2×D4 | — | 4 | 2 | no | — |
| 132 | C2×D4 | — | 4 | 1 | no | — |
| 133 | C2×D4 | — | 4 | 2 | no | — |
| 134 | C2×D4 | — | 4 | 1 | no | — |
| 135 | C2×D4 | — | 8 | 2 | no | — |
| 136 | C2×D4 | — | 8 | 2 | no | — |
| 137 | C2×D4 | — | 8 | 2 | no | — |
| 138 | C2×D4 | — | 8 | 2 | no | — |
| 139 | C2×D4 | — | 8 | 1 | no | — |
| 140 | C2×D4 | — | 8 | 1 | no | — |
| 141 | C2×D4 | — | 8 | 1 | no | — |
| 142 | C2×D4 | — | 8 | 1 | no | — |
| 143 | C2×D4 | — | 8 | 1 | no | — |
| 144 | C2×D4 | — | 16 | 2 | no | — |
| 145 | C22×C4 | — | 8 | 1 | no | — |
| 146 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 147 | C22×C4 | #135 | 1 | 1 | yes | characteristic |
| 148 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 149 | D8 | — | 4 | 4 | no | — |
| 150 | D8 | — | 4 | 1 | no | — |
| 151 | D8 | — | 4 | 2 | no | — |
| 152 | D8 | — | 4 | 1 | no | — |
| 153 | D8 | — | 4 | 2 | no | — |
| 154 | D8 | — | 8 | 1 | no | — |
| 155 | D8 | — | 8 | 2 | no | — |
| 156 | D8 | — | 8 | 2 | no | — |
| 157 | M16 | — | 2 | 2 | no | — |
| 158 | M16 | — | 4 | 1 | no | — |
| 159 | M16 | — | 4 | 2 | no | — |
| 160 | M16 | — | 8 | 1 | no | — |
| 161 | M16 | — | 8 | 2 | no | — |
| 162 | C2×C8 | — | 2 | 1 | no | — |
| 163 | C2×C8 | — | 2 | 1 | no | — |
| 164 | C2×C8 | — | 2 | 2 | no | — |
| 165 | C2×C8 | — | 4 | 1 | no | — |
| 166 | C2×C8 | — | 4 | 1 | no | — |
| 167 | C2×C8 | — | 8 | 1 | no | — |
| 168 | C2×C8 | — | 8 | 1 | no | — |
| 169 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | #123 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | #123 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | #124 | 1 | 1 | yes | characteristic |
| 173 | C4⋊C4 | — | 4 | 1 | no | — |
| 174 | C4⋊C4 | — | 4 | 1 | no | — |
| 175 | C22⋊C4 | — | 8 | 1 | no | — |
| 176 | C22⋊C4 | — | 8 | 1 | no | — |
| 177 | C42 | #121 | 1 | 1 | yes | characteristic |
| 178 | C42 | #121 | 1 | 1 | yes | characteristic |
| 179 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 180 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 181 | C16 | — | 4 | 1 | no | — |
| 182 | C16 | — | 4 | 1 | no | — |
| 183 | C16 | — | 8 | 1 | no | — |
| 184 | C23 | — | 8 | 4 | no | — |
| 185 | C23 | — | 8 | 1 | no | — |
| 186 | C23 | — | 8 | 2 | no | — |
| 187 | C23 | — | 16 | 2 | no | — |
| 188 | C23 | — | 16 | 2 | no | — |
| 189 | C23 | #463 | 1 | 1 | yes | characteristic |
| 190 | Q8 | — | 16 | 1 | no | — |
| 191 | D4 | — | 8 | 2 | no | — |
| 192 | D4 | — | 8 | 1 | no | — |
| 193 | D4 | — | 8 | 1 | no | — |
| 194 | D4 | — | 8 | 1 | no | — |
| 195 | D4 | — | 8 | 2 | no | — |
| 196 | D4 | — | 8 | 1 | no | — |
| 197 | D4 | — | 16 | 2 | no | — |
| 198 | D4 | — | 16 | 1 | no | — |
| 199 | D4 | — | 16 | 2 | no | — |
| 200 | D4 | — | 16 | 2 | no | — |
| 201 | D4 | — | 16 | 2 | no | — |
| 202 | D4 | — | 16 | 1 | no | — |
| 203 | D4 | — | 16 | 1 | no | — |
| 204 | D4 | — | 16 | 2 | no | — |
| 205 | C2×C4 | — | 2 | 1 | no | — |
| 206 | C2×C4 | — | 2 | 1 | no | — |
| 207 | C2×C4 | — | 2 | 1 | no | — |
| 208 | C2×C4 | — | 2 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 1 | no | — |
| 210 | C2×C4 | — | 8 | 1 | no | — |
| 211 | C2×C4 | — | 16 | 2 | no | — |
| 212 | C2×C4 | #394 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | #463 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | #466 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | #469 | 1 | 1 | yes | characteristic |
| 218 | C8 | — | 2 | 1 | no | — |
| 219 | C8 | — | 2 | 1 | no | — |
| 220 | C8 | — | 2 | 1 | no | — |
| 221 | C8 | — | 4 | 2 | no | — |
| 222 | C8 | — | 4 | 2 | no | — |
| 223 | C8 | — | 8 | 2 | no | — |
| 224 | C8 | — | 16 | 1 | no | — |
| 225 | C8 | #453 | 1 | 1 | yes | characteristic |
| 226 | C8 | #455 | 1 | 1 | yes | characteristic |
| 227 | C22 | — | 2 | 1 | no | — |
| 228 | C22 | — | 2 | 1 | no | — |
| 229 | C22 | — | 2 | 1 | no | — |
| 230 | C22 | — | 16 | 2 | no | — |
| 231 | C22 | — | 16 | 1 | no | — |
| 232 | C22 | — | 16 | 1 | no | — |
| 233 | C22 | — | 16 | 2 | no | — |
| 234 | C22 | — | 16 | 1 | no | — |
| 235 | C22 | — | 16 | 2 | no | — |
| 236 | C22 | — | 32 | 4 | no | — |
| 237 | C22 | #2054 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 2 | 1 | no | — |
| 239 | C4 | — | 2 | 1 | no | — |
| 240 | C4 | — | 2 | 1 | no | — |
| 241 | C4 | — | 16 | 1 | no | — |
| 242 | C4 | #2037 | 1 | 1 | yes | characteristic |
| 243 | C4 | #2048 | 1 | 1 | yes | characteristic |
| 244 | C4 | #2212 | 1 | 1 | yes | characteristic |
| 245 | C4 | #2213 | 1 | 1 | yes | characteristic |
| 246 | C4 | #2221 | 1 | 2 | yes | — |
| 247 | C2 | — | 4 | 1 | no | — |
| 248 | C2 | — | 32 | 1 | no | — |
| 249 | C2 | — | 32 | 2 | no | — |
| 250 | C2 | #10500 | 1 | 1 | yes | characteristic |
| 251 | C2 | #13439 | 1 | 1 | yes | characteristic |
| 252 | C2 | #13445 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2054 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33771 | C2 |
| 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. #13257 | C2 | semidirect |
| 3. #12018 | C2 | semidirect |
| 4. #11986 | C2 | semidirect |
| 5. #9802 | C2 | semidirect |
| 6. #9724 | C2 | semidirect |
| 7. #7298 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33771 | 2 |
| #13257 | 2 |
| #12018 | 2 |
| #11986 | 2 |
| #9802 | 2 |
| #9724 | 2 |
| #7298 | 2 |