Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
389 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 112 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 | #27691 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #27673 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13522 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13498 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13497 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13200 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #13175 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12933 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12930 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #3093 | — | 2 | 1 | no | — |
| 11 | #2929 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2916 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#427 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2161 | C22 | 1 | 2 | yes | — |
| 15 | C2×#365 | C22 | 1 | 2 | yes | — |
| 16 | C2×#365 | C22 | 1 | 2 | yes | — |
| 17 | #2153 | C22 | 1 | 1 | yes | characteristic |
| 18 | #2117 | C22 | 1 | 2 | yes | — |
| 19 | #2112 | C22 | 1 | 2 | yes | — |
| 20 | #2110 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2103 | — | 2 | 1 | no | — |
| 22 | #2091 | — | 2 | 1 | no | — |
| 23 | #2089 | — | 2 | 1 | no | — |
| 24 | #2078 | — | 2 | 1 | no | — |
| 25 | #1953 | C22 | 1 | 2 | yes | — |
| 26 | #1951 | C22 | 1 | 2 | yes | — |
| 27 | #1891 | — | 2 | 1 | no | — |
| 28 | #1858 | — | 2 | 1 | no | — |
| 29 | #1822 | — | 2 | 1 | no | — |
| 30 | #1815 | C22 | 1 | 2 | yes | — |
| 31 | #1813 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1810 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1810 | C22 | 1 | 2 | yes | — |
| 34 | #1388 | C22 | 1 | 2 | yes | — |
| 35 | #1388 | C22 | 1 | 2 | yes | — |
| 36 | #1385 | C22 | 1 | 2 | yes | — |
| 37 | #1383 | C22 | 1 | 2 | yes | — |
| 38 | #1383 | C22 | 1 | 2 | yes | — |
| 39 | #568 | — | 2 | 1 | no | — |
| 40 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 41 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 42 | #525 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×#117 | — | 2 | 1 | no | — |
| 44 | C2×M32 | C23 | 1 | 2 | yes | — |
| 45 | C2×M32 | C23 | 1 | 2 | yes | — |
| 46 | #485 | — | 4 | 2 | no | — |
| 47 | #482 | — | 4 | 2 | no | — |
| 48 | #435 | — | 4 | 1 | no | — |
| 49 | C8×D4 | — | 4 | 1 | no | — |
| 50 | #430 | — | 2 | 1 | no | — |
| 51 | #430 | — | 2 | 2 | no | — |
| 52 | #430 | — | 4 | 1 | no | — |
| 53 | C2×#108 | — | 4 | 1 | no | — |
| 54 | C2×#108 | C23 | 1 | 2 | yes | — |
| 55 | C2×#108 | C23 | 1 | 2 | yes | — |
| 56 | C2×#108 | C23 | 1 | 2 | yes | — |
| 57 | #428 | — | 2 | 2 | no | — |
| 58 | #427 | — | 4 | 1 | no | — |
| 59 | #427 | — | 4 | 1 | no | — |
| 60 | C2×#107 | — | 2 | 1 | no | — |
| 61 | C2×#106 | C23 | 1 | 2 | yes | — |
| 62 | #424 | D4 | 1 | 1 | yes | characteristic |
| 63 | #417 | — | 2 | 2 | no | — |
| 64 | #413 | — | 2 | 1 | no | — |
| 65 | #408 | D4 | 1 | 1 | yes | characteristic |
| 66 | #407 | — | 2 | 1 | no | — |
| 67 | #407 | — | 2 | 2 | no | — |
| 68 | #407 | D4 | 1 | 1 | yes | characteristic |
| 69 | #405 | C23 | 1 | 1 | yes | characteristic |
| 70 | #368 | — | 2 | 4 | no | — |
| 71 | #365 | — | 4 | 2 | no | — |
| 72 | #365 | — | 4 | 2 | no | — |
| 73 | #364 | — | 4 | 1 | no | — |
| 74 | #350 | — | 2 | 4 | no | — |
| 75 | C22⋊C16 | — | 2 | 4 | no | — |
| 76 | #347 | — | 4 | 1 | no | — |
| 77 | #344 | — | 2 | 2 | no | — |
| 78 | #343 | — | 2 | 2 | no | — |
| 79 | #343 | — | 2 | 2 | no | — |
| 80 | #341 | — | 2 | 1 | no | — |
| 81 | #341 | — | 2 | 2 | no | — |
| 82 | #338 | — | 2 | 1 | no | — |
| 83 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | #131 | — | 2 | 2 | no | — |
| 86 | #131 | — | 2 | 2 | no | — |
| 87 | #131 | — | 8 | 2 | no | — |
| 88 | C2×M16 | — | 2 | 1 | no | — |
| 89 | C2×M16 | — | 2 | 2 | no | — |
| 90 | C2×M16 | — | 2 | 2 | no | — |
| 91 | C2×M16 | — | 2 | 2 | no | — |
| 92 | C2×M16 | — | 4 | 2 | no | — |
| 93 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C22×C8 | — | 2 | 2 | no | — |
| 96 | C22×C8 | — | 2 | 1 | no | — |
| 97 | C22×C8 | — | 2 | 1 | no | — |
| 98 | C22×C8 | — | 2 | 1 | no | — |
| 99 | C22×C8 | — | 2 | 2 | no | — |
| 100 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 102 | #123 | — | 2 | 1 | no | — |
| 103 | C22⋊Q8 | — | 2 | 1 | no | — |
| 104 | #121 | — | 2 | 1 | no | — |
| 105 | #120 | — | 2 | 1 | no | — |
| 106 | C4×D4 | — | 4 | 1 | no | — |
| 107 | #117 | — | 4 | 2 | no | — |
| 108 | #117 | — | 4 | 2 | no | — |
| 109 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 111 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 112 | C2×C42 | — | 4 | 1 | no | — |
| 113 | M32 | — | 2 | 2 | no | — |
| 114 | M32 | — | 2 | 2 | no | — |
| 115 | M32 | — | 2 | 2 | no | — |
| 116 | M32 | — | 2 | 2 | no | — |
| 117 | C2×C16 | — | 2 | 2 | no | — |
| 118 | C2×C16 | #41 | 1 | 4 | yes | — |
| 119 | #108 | — | 2 | 2 | no | — |
| 120 | #108 | — | 2 | 2 | no | — |
| 121 | #108 | — | 2 | 2 | no | — |
| 122 | #108 | — | 2 | 2 | no | — |
| 123 | #108 | — | 2 | 2 | no | — |
| 124 | #108 | — | 2 | 2 | no | — |
| 125 | #108 | — | 4 | 2 | no | — |
| 126 | #108 | — | 8 | 2 | no | — |
| 127 | #107 | — | 4 | 2 | no | — |
| 128 | #107 | — | 4 | 2 | no | — |
| 129 | #106 | — | 2 | 2 | no | — |
| 130 | #106 | — | 2 | 2 | no | — |
| 131 | #106 | — | 4 | 2 | no | — |
| 132 | C4⋊C8 | — | 4 | 1 | no | — |
| 133 | #103 | — | 4 | 2 | no | — |
| 134 | #102 | — | 4 | 2 | no | — |
| 135 | #101 | — | 4 | 1 | no | — |
| 136 | #100 | — | 4 | 1 | no | — |
| 137 | C22⋊C8 | — | 4 | 1 | no | — |
| 138 | C22⋊C8 | — | 4 | 1 | no | — |
| 139 | C22⋊C8 | — | 4 | 2 | no | — |
| 140 | C22⋊C8 | — | 8 | 2 | no | — |
| 141 | #97 | — | 2 | 1 | no | — |
| 142 | C4×C8 | — | 2 | 1 | no | — |
| 143 | C24 | #137 | 1 | 1 | yes | characteristic |
| 144 | #41 | — | 2 | 2 | no | — |
| 145 | #41 | — | 2 | 2 | no | — |
| 146 | C2×Q8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 147 | C2×D4 | — | 2 | 1 | no | — |
| 148 | C2×D4 | — | 4 | 1 | no | — |
| 149 | C2×D4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 150 | C22×C4 | — | 2 | 1 | no | — |
| 151 | C22×C4 | — | 2 | 1 | no | — |
| 152 | C22×C4 | — | 2 | 2 | no | — |
| 153 | C22×C4 | — | 2 | 1 | no | — |
| 154 | C22×C4 | — | 2 | 2 | no | — |
| 155 | C22×C4 | — | 4 | 2 | no | — |
| 156 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 157 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 158 | M16 | — | 2 | 2 | no | — |
| 159 | M16 | — | 2 | 2 | no | — |
| 160 | M16 | — | 4 | 2 | no | — |
| 161 | M16 | — | 4 | 2 | no | — |
| 162 | M16 | — | 4 | 2 | no | — |
| 163 | M16 | — | 4 | 2 | no | — |
| 164 | M16 | — | 4 | 2 | no | — |
| 165 | M16 | — | 4 | 2 | no | — |
| 166 | M16 | — | 4 | 2 | no | — |
| 167 | M16 | — | 8 | 2 | no | — |
| 168 | C2×C8 | — | 2 | 1 | no | — |
| 169 | C2×C8 | — | 2 | 1 | no | — |
| 170 | C2×C8 | — | 2 | 1 | no | — |
| 171 | C2×C8 | — | 2 | 2 | no | — |
| 172 | C2×C8 | — | 4 | 2 | no | — |
| 173 | C2×C8 | — | 4 | 1 | no | — |
| 174 | C2×C8 | — | 4 | 1 | no | — |
| 175 | C2×C8 | — | 4 | 1 | no | — |
| 176 | C2×C8 | — | 4 | 1 | no | — |
| 177 | C2×C8 | — | 4 | 2 | no | — |
| 178 | C2×C8 | — | 4 | 2 | no | — |
| 179 | C2×C8 | — | 4 | 2 | no | — |
| 180 | C2×C8 | — | 8 | 2 | no | — |
| 181 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 182 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 185 | C4⋊C4 | — | 2 | 1 | no | — |
| 186 | C4⋊C4 | — | 4 | 1 | no | — |
| 187 | C4⋊C4 | — | 4 | 2 | no | — |
| 188 | C4⋊C4 | — | 4 | 2 | no | — |
| 189 | C22⋊C4 | — | 2 | 1 | no | — |
| 190 | C22⋊C4 | — | 4 | 1 | no | — |
| 191 | C22⋊C4 | — | 4 | 1 | no | — |
| 192 | C22⋊C4 | — | 8 | 2 | no | — |
| 193 | C42 | — | 2 | 1 | no | — |
| 194 | C42 | — | 8 | 2 | no | — |
| 195 | C16 | — | 2 | 4 | no | — |
| 196 | C16 | — | 4 | 2 | no | — |
| 197 | C23 | — | 2 | 2 | no | — |
| 198 | C23 | — | 2 | 1 | no | — |
| 199 | C23 | — | 2 | 1 | no | — |
| 200 | C23 | — | 8 | 1 | no | — |
| 201 | C23 | C2×#137 | 1 | 1 | yes | characteristic |
| 202 | Q8 | — | 2 | 2 | no | — |
| 203 | D4 | — | 2 | 2 | no | — |
| 204 | D4 | — | 4 | 2 | no | — |
| 205 | D4 | — | 8 | 2 | no | — |
| 206 | C2×C4 | — | 2 | 2 | no | — |
| 207 | C2×C4 | — | 2 | 1 | no | — |
| 208 | C2×C4 | — | 2 | 1 | no | — |
| 209 | C2×C4 | — | 4 | 2 | no | — |
| 210 | C2×C4 | — | 4 | 1 | no | — |
| 211 | C2×C4 | — | 4 | 1 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 4 | 1 | no | — |
| 214 | C2×C4 | — | 4 | 1 | no | — |
| 215 | C2×C4 | — | 8 | 2 | no | — |
| 216 | C2×C4 | — | 16 | 2 | no | — |
| 217 | C2×C4 | #534 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | #547 | 1 | 2 | yes | — |
| 219 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | #577 | 1 | 2 | yes | — |
| 221 | C8 | — | 4 | 1 | no | — |
| 222 | C8 | — | 4 | 1 | no | — |
| 223 | C8 | — | 4 | 1 | no | — |
| 224 | C8 | — | 8 | 2 | no | — |
| 225 | C8 | — | 8 | 2 | no | — |
| 226 | C8 | — | 8 | 2 | no | — |
| 227 | C8 | #539 | 1 | 2 | yes | — |
| 228 | C8 | #559 | 1 | 1 | yes | characteristic |
| 229 | C8 | #563 | 1 | 1 | yes | characteristic |
| 230 | C22 | — | 4 | 1 | no | — |
| 231 | C22 | — | 4 | 1 | no | — |
| 232 | C22 | — | 4 | 1 | no | — |
| 233 | C22 | — | 4 | 4 | no | — |
| 234 | C22 | — | 4 | 1 | no | — |
| 235 | C22 | — | 4 | 1 | no | — |
| 236 | C22 | — | 8 | 1 | no | — |
| 237 | C22 | #3193 | 1 | 1 | yes | characteristic |
| 238 | C22 | #3344 | 1 | 2 | yes | — |
| 239 | C4 | — | 4 | 1 | no | — |
| 240 | C4 | — | 4 | 1 | no | — |
| 241 | C4 | — | 8 | 1 | no | — |
| 242 | C4 | — | 16 | 2 | no | — |
| 243 | C4 | #2724 | 1 | 1 | yes | characteristic |
| 244 | C4 | #3200 | 1 | 1 | yes | characteristic |
| 245 | C4 | #3345 | 1 | 2 | yes | — |
| 246 | C2 | — | 2 | 1 | no | — |
| 247 | C2 | — | 2 | 2 | no | — |
| 248 | C2 | — | 8 | 1 | no | — |
| 249 | C2 | — | 8 | 1 | no | — |
| 250 | C2 | #32468 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32468 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13522 | C2 | semidirect |
| 4. #13498 | C2 | semidirect |
| 5. #13497 | C2 | semidirect |
| 7. #13175 | C2 | semidirect |
| 8. #12933 | C2 | semidirect |
| 9. #12930 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27691 | 2 |
| #27673 | 2 |
| #13522 | 2 |
| #13498 | 2 |
| #13497 | 2 |
| #13200 | 2 |
| #13175 | 2 |
| #12933 | 2 |
| #12930 | 2 |