Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
271 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 50 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 | #29821 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12864 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11524 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #7401 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7400 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #7399 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7398 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#495 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | #3040 | — | 2 | 1 | no | — |
| 10 | #2922 | C22 | 1 | 1 | yes | characteristic |
| 11 | #2165 | — | 2 | 1 | no | — |
| 12 | #2047 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | #2046 | — | 2 | 1 | no | — |
| 14 | #2045 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #2020 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2019 | — | 2 | 1 | no | — |
| 17 | #2018 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1967 | — | 2 | 1 | no | — |
| 19 | #1837 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #1827 | — | 2 | 1 | no | — |
| 21 | #1395 | — | 2 | 1 | no | — |
| 22 | #1392 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1391 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1328 | — | 2 | 1 | no | — |
| 25 | #1321 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1321 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#137 | — | 2 | 1 | no | — |
| 28 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 29 | C22×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C22×SD16 | — | 2 | 1 | no | — |
| 31 | C22×D8 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×#124 | — | 2 | 1 | no | — |
| 33 | C4×#41 | D4 | 1 | 1 | yes | characteristic |
| 34 | #495 | — | 4 | 1 | no | — |
| 35 | #495 | — | 4 | 1 | no | — |
| 36 | #464 | — | 4 | 2 | no | — |
| 37 | #463 | — | 4 | 2 | no | — |
| 38 | #433 | — | 4 | 1 | no | — |
| 39 | #431 | — | 4 | 1 | no | — |
| 40 | #424 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | #424 | D4 | 1 | 1 | yes | characteristic |
| 42 | #423 | — | 2 | 1 | no | — |
| 43 | #419 | — | 2 | 1 | no | — |
| 44 | #418 | — | 2 | 1 | no | — |
| 45 | #413 | — | 2 | 1 | no | — |
| 46 | C2×#101 | — | 2 | 1 | no | — |
| 47 | C2×#101 | — | 2 | 1 | no | — |
| 48 | C2×#100 | — | 2 | 1 | no | — |
| 49 | C2×#100 | — | 2 | 1 | no | — |
| 50 | #410 | — | 2 | 1 | no | — |
| 51 | #408 | D4 | 1 | 1 | yes | characteristic |
| 52 | #408 | C23 | 1 | 1 | yes | characteristic |
| 53 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 54 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #364 | — | 4 | 1 | no | — |
| 56 | #349 | — | 2 | 1 | no | — |
| 57 | #349 | — | 2 | 1 | no | — |
| 58 | #347 | — | 4 | 1 | no | — |
| 59 | #341 | — | 2 | 1 | no | — |
| 60 | #341 | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | #341 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | #339 | — | 4 | 1 | no | — |
| 63 | H(3,ℤ/4) | — | 4 | 1 | no | — |
| 64 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C22×Q8 | — | 2 | 1 | no | — |
| 66 | C22×D4 | — | 2 | 1 | no | — |
| 67 | #137 | — | 8 | 1 | no | — |
| 68 | #137 | — | 8 | 1 | no | — |
| 69 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 70 | AGL(1,ℤ/8) | — | 8 | 1 | no | — |
| 71 | C2×Q16 | — | 4 | 1 | no | — |
| 72 | C2×Q16 | — | 4 | 1 | no | — |
| 73 | C2×Q16 | — | 8 | 1 | no | — |
| 74 | C2×SD16 | — | 4 | 2 | no | — |
| 75 | C2×SD16 | — | 4 | 1 | no | — |
| 76 | C2×SD16 | — | 4 | 1 | no | — |
| 77 | C2×SD16 | — | 8 | 1 | no | — |
| 78 | C2×SD16 | — | 8 | 1 | no | — |
| 79 | C2×D8 | — | 4 | 1 | no | — |
| 80 | C2×D8 | — | 4 | 1 | no | — |
| 81 | C2×D8 | — | 8 | 1 | no | — |
| 82 | C2×M16 | — | 2 | 1 | no | — |
| 83 | C2×M16 | — | 2 | 1 | no | — |
| 84 | C2×M16 | — | 4 | 1 | no | — |
| 85 | C2×M16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 86 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | #124 | — | 4 | 2 | no | — |
| 90 | #124 | — | 4 | 1 | no | — |
| 91 | #124 | — | 4 | 1 | no | — |
| 92 | C4×Q8 | — | 2 | 1 | no | — |
| 93 | C4×D4 | — | 2 | 1 | no | — |
| 94 | C4×D4 | — | 4 | 1 | no | — |
| 95 | #117 | — | 2 | 1 | no | — |
| 96 | #117 | — | 4 | 1 | no | — |
| 97 | #117 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 99 | C2×C42 | — | 2 | 1 | no | — |
| 100 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 101 | M32 | — | 4 | 1 | no | — |
| 102 | M32 | — | 4 | 1 | no | — |
| 103 | C4⋊C8 | — | 4 | 1 | no | — |
| 104 | C4⋊C8 | — | 4 | 1 | no | — |
| 105 | C4⋊C8 | — | 4 | 2 | no | — |
| 106 | #103 | — | 4 | 2 | no | — |
| 107 | #102 | — | 4 | 2 | no | — |
| 108 | #101 | — | 4 | 1 | no | — |
| 109 | #101 | — | 8 | 1 | no | — |
| 110 | #101 | — | 8 | 1 | no | — |
| 111 | #100 | — | 4 | 1 | no | — |
| 112 | #100 | — | 8 | 1 | no | — |
| 113 | #100 | — | 8 | 1 | no | — |
| 114 | #99 | — | 4 | 1 | no | — |
| 115 | #99 | — | 4 | 1 | no | — |
| 116 | C22⋊C8 | — | 4 | 1 | no | — |
| 117 | C22⋊C8 | — | 4 | 1 | no | — |
| 118 | #97 | — | 2 | 1 | no | — |
| 119 | C4×C8 | — | 2 | 1 | no | — |
| 120 | C4×C8 | — | 2 | 1 | no | — |
| 121 | C4×C8 | — | 2 | 1 | no | — |
| 122 | C24 | — | 4 | 1 | no | — |
| 123 | #41 | — | 4 | 1 | no | — |
| 124 | #41 | — | 4 | 1 | no | — |
| 125 | C2×Q8 | — | 4 | 1 | no | — |
| 126 | C2×Q8 | — | 4 | 1 | no | — |
| 127 | C2×Q8 | — | 4 | 1 | no | — |
| 128 | C2×Q8 | — | 4 | 1 | no | — |
| 129 | C2×Q8 | — | 4 | 1 | no | — |
| 130 | C2×Q8 | — | 4 | 1 | no | — |
| 131 | C2×Q8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 132 | C2×D4 | — | 2 | 1 | no | — |
| 133 | C2×D4 | — | 4 | 1 | no | — |
| 134 | C2×D4 | — | 4 | 1 | no | — |
| 135 | C2×D4 | — | 4 | 1 | no | — |
| 136 | C2×D4 | — | 4 | 1 | no | — |
| 137 | C2×D4 | — | 4 | 1 | no | — |
| 138 | C2×D4 | — | 4 | 1 | no | — |
| 139 | C2×D4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 140 | C22×C4 | — | 2 | 1 | no | — |
| 141 | C22×C4 | — | 2 | 1 | no | — |
| 142 | C22×C4 | — | 4 | 1 | no | — |
| 143 | C22×C4 | — | 4 | 1 | no | — |
| 144 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 145 | Q16 | — | 8 | 2 | no | — |
| 146 | Q16 | — | 8 | 2 | no | — |
| 147 | SD16 | — | 8 | 4 | no | — |
| 148 | SD16 | — | 8 | 2 | no | — |
| 149 | SD16 | — | 8 | 2 | no | — |
| 150 | D8 | — | 8 | 2 | no | — |
| 151 | D8 | — | 8 | 2 | no | — |
| 152 | M16 | — | 4 | 1 | no | — |
| 153 | M16 | — | 4 | 1 | no | — |
| 154 | M16 | — | 4 | 1 | no | — |
| 155 | M16 | — | 4 | 1 | no | — |
| 156 | M16 | — | 4 | 1 | no | — |
| 157 | M16 | — | 4 | 1 | no | — |
| 158 | M16 | — | 8 | 1 | no | — |
| 159 | M16 | — | 8 | 1 | no | — |
| 160 | C2×C8 | — | 2 | 1 | no | — |
| 161 | C2×C8 | — | 2 | 1 | no | — |
| 162 | C2×C8 | — | 2 | 1 | no | — |
| 163 | C2×C8 | — | 2 | 1 | no | — |
| 164 | C2×C8 | — | 4 | 1 | no | — |
| 165 | C2×C8 | — | 4 | 1 | no | — |
| 166 | C2×C8 | — | 4 | 1 | no | — |
| 167 | C2×C8 | — | 4 | 1 | no | — |
| 168 | C2×C8 | — | 8 | 1 | no | — |
| 169 | C4⋊C4 | — | 2 | 1 | no | — |
| 170 | C4⋊C4 | — | 4 | 1 | no | — |
| 171 | C4⋊C4 | — | 8 | 1 | no | — |
| 172 | C22⋊C4 | — | 2 | 1 | no | — |
| 173 | C22⋊C4 | — | 4 | 1 | no | — |
| 174 | C22⋊C4 | — | 8 | 2 | no | — |
| 175 | C22⋊C4 | — | 8 | 2 | no | — |
| 176 | C22⋊C4 | — | 8 | 1 | no | — |
| 177 | C42 | — | 2 | 1 | no | — |
| 178 | C42 | — | 2 | 1 | no | — |
| 179 | C42 | — | 4 | 1 | no | — |
| 180 | C42 | — | 8 | 1 | no | — |
| 181 | C42 | #99 | 1 | 1 | yes | characteristic |
| 182 | C42 | #99 | 1 | 1 | yes | characteristic |
| 183 | C16 | — | 8 | 1 | no | — |
| 184 | C16 | — | 8 | 1 | no | — |
| 185 | C23 | — | 2 | 1 | no | — |
| 186 | C23 | — | 8 | 1 | no | — |
| 187 | C23 | — | 8 | 1 | no | — |
| 188 | C23 | — | 8 | 1 | no | — |
| 189 | C23 | — | 16 | 2 | no | — |
| 190 | C23 | #421 | 1 | 1 | yes | characteristic |
| 191 | Q8 | — | 4 | 1 | no | — |
| 192 | Q8 | — | 8 | 1 | no | — |
| 193 | Q8 | — | 8 | 1 | no | — |
| 194 | Q8 | — | 8 | 1 | no | — |
| 195 | Q8 | — | 8 | 1 | no | — |
| 196 | D4 | — | 4 | 1 | no | — |
| 197 | D4 | — | 8 | 1 | no | — |
| 198 | D4 | — | 8 | 1 | no | — |
| 199 | D4 | — | 8 | 1 | no | — |
| 200 | D4 | — | 8 | 1 | no | — |
| 201 | D4 | — | 8 | 1 | no | — |
| 202 | C2×C4 | — | 2 | 1 | no | — |
| 203 | C2×C4 | — | 2 | 1 | no | — |
| 204 | C2×C4 | — | 2 | 1 | no | — |
| 205 | C2×C4 | — | 2 | 1 | no | — |
| 206 | C2×C4 | — | 2 | 1 | no | — |
| 207 | C2×C4 | — | 4 | 1 | no | — |
| 208 | C2×C4 | — | 4 | 1 | no | — |
| 209 | C2×C4 | — | 4 | 1 | no | — |
| 210 | C2×C4 | — | 8 | 1 | no | — |
| 211 | C2×C4 | — | 8 | 2 | no | — |
| 212 | C2×C4 | — | 8 | 1 | no | — |
| 213 | C2×C4 | — | 8 | 1 | no | — |
| 214 | C2×C4 | — | 8 | 1 | no | — |
| 215 | C2×C4 | — | 16 | 1 | no | — |
| 216 | C2×C4 | C2×#99 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | #418 | 1 | 1 | yes | characteristic |
| 218 | C8 | — | 4 | 1 | no | — |
| 219 | C8 | — | 4 | 1 | no | — |
| 220 | C8 | — | 4 | 1 | no | — |
| 221 | C8 | — | 4 | 1 | no | — |
| 222 | C8 | — | 8 | 1 | no | — |
| 223 | C8 | — | 8 | 1 | no | — |
| 224 | C8 | — | 16 | 1 | no | — |
| 225 | C22 | — | 2 | 1 | no | — |
| 226 | C22 | — | 4 | 1 | no | — |
| 227 | C22 | — | 4 | 1 | no | — |
| 228 | C22 | — | 8 | 1 | no | — |
| 229 | C22 | — | 16 | 2 | no | — |
| 230 | C22 | — | 16 | 1 | no | — |
| 231 | C22 | — | 32 | 1 | no | — |
| 232 | C22 | — | 32 | 1 | no | — |
| 233 | C22 | #1510 | 1 | 1 | yes | characteristic |
| 234 | C4 | — | 2 | 1 | no | — |
| 235 | C4 | — | 4 | 1 | no | — |
| 236 | C4 | — | 4 | 1 | no | — |
| 237 | C4 | — | 4 | 1 | no | — |
| 238 | C4 | — | 8 | 1 | no | — |
| 239 | C4 | — | 16 | 1 | no | — |
| 240 | C4 | — | 32 | 1 | no | — |
| 241 | C4 | #2122 | 1 | 1 | yes | characteristic |
| 242 | C4 | #2128 | 1 | 1 | yes | characteristic |
| 243 | C2 | — | 2 | 1 | no | — |
| 244 | C2 | — | 4 | 1 | no | — |
| 245 | C2 | — | 8 | 1 | no | — |
| 246 | C2 | — | 32 | 1 | no | — |
| 247 | C2 | #11800 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11800 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| prime radical | #2018 | C22 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2922 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12864 | C2 | semidirect |
| 3. #11524 | C2 | semidirect |
| 4. #7401 | C2 | semidirect |
| 6. #7399 | C2 | semidirect |
| 8. C2×#495 | C4 | semidirect |
| 12. #2047 | C4 | semidirect |
| 14. #2045 | C4 | semidirect |
| 19. #1837 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #29821 | 2 |
| #12864 | 2 |
| #11524 | 2 |
| #7401 | 2 |
| #7400 | 2 |
| #7399 | 2 |
| #7398 | 2 |