Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
336 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #30300 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11724 | C2 | 1 | 2 | yes | maximal |
| 3 | #11297 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #9193 | C2 | 1 | 2 | yes | maximal |
| 5 | #7188 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #2905 | C22 | 1 | 1 | yes | characteristic |
| 7 | #2628 | C22 | 1 | 2 | yes | — |
| 8 | #2085 | — | 2 | 2 | no | — |
| 9 | #2057 | — | 2 | 2 | no | — |
| 10 | #1809 | — | 2 | 1 | no | — |
| 11 | #1808 | — | 2 | 1 | no | — |
| 12 | #1742 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1738 | — | 2 | 1 | no | — |
| 14 | C2×#341 | — | 2 | 1 | no | — |
| 15 | #1729 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1549 | — | 2 | 2 | no | — |
| 17 | #1458 | C22 | 1 | 2 | yes | — |
| 18 | #1354 | — | 2 | 4 | no | — |
| 19 | #1286 | — | 2 | 2 | no | — |
| 20 | C22×M16 | — | 2 | 1 | no | — |
| 21 | C22×M16 | — | 2 | 1 | no | — |
| 22 | C22×M16 | D4 | 1 | 1 | yes | characteristic |
| 23 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 25 | C2×#117 | C23 | 1 | 1 | yes | characteristic |
| 26 | #431 | — | 2 | 2 | no | — |
| 27 | C2×#107 | — | 2 | 1 | no | — |
| 28 | C2×#107 | — | 2 | 1 | no | — |
| 29 | C2×#107 | — | 2 | 1 | no | — |
| 30 | C2×#106 | — | 2 | 1 | no | — |
| 31 | #424 | — | 4 | 2 | no | — |
| 32 | #423 | — | 2 | 2 | no | — |
| 33 | #400 | — | 2 | 2 | no | — |
| 34 | #399 | — | 2 | 2 | no | — |
| 35 | #395 | — | 2 | 2 | no | — |
| 36 | #393 | — | 2 | 2 | no | — |
| 37 | #389 | — | 2 | 2 | no | — |
| 38 | #382 | D4 | 1 | 2 | yes | — |
| 39 | #380 | — | 2 | 2 | no | — |
| 40 | #343 | — | 2 | 1 | no | — |
| 41 | #343 | D4 | 1 | 2 | yes | — |
| 42 | #341 | — | 4 | 2 | no | — |
| 43 | #341 | — | 4 | 2 | no | — |
| 44 | #340 | — | 2 | 2 | no | — |
| 45 | #340 | — | 2 | 2 | no | — |
| 46 | #336 | — | 2 | 2 | no | — |
| 47 | #332 | — | 4 | 4 | no | — |
| 48 | #324 | — | 2 | 2 | no | — |
| 49 | #323 | — | 2 | 2 | no | — |
| 50 | C22×Q8 | — | 2 | 2 | no | — |
| 51 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C2×M16 | — | 2 | 1 | no | — |
| 53 | C2×M16 | — | 2 | 2 | no | — |
| 54 | C2×M16 | — | 2 | 2 | no | — |
| 55 | C2×M16 | — | 4 | 1 | no | — |
| 56 | C2×M16 | — | 4 | 1 | no | — |
| 57 | C2×M16 | — | 4 | 1 | no | — |
| 58 | C2×M16 | — | 4 | 1 | no | — |
| 59 | C2×M16 | — | 4 | 1 | no | — |
| 60 | C2×M16 | — | 4 | 1 | no | — |
| 61 | C2×M16 | — | 4 | 1 | no | — |
| 62 | C2×M16 | — | 4 | 1 | no | — |
| 63 | C2×M16 | — | 4 | 1 | no | — |
| 64 | C2×M16 | — | 4 | 1 | no | — |
| 65 | C2×M16 | — | 4 | 1 | no | — |
| 66 | C2×M16 | — | 4 | 1 | no | — |
| 67 | C2×M16 | Q16 | 1 | 2 | yes | — |
| 68 | C22×C8 | — | 2 | 1 | no | — |
| 69 | C22×C8 | — | 2 | 1 | no | — |
| 70 | C22×C8 | — | 2 | 1 | no | — |
| 71 | C22×C8 | — | 4 | 1 | no | — |
| 72 | C22×C8 | — | 4 | 1 | no | — |
| 73 | C22×C8 | — | 4 | 1 | no | — |
| 74 | C22×C8 | — | 4 | 1 | no | — |
| 75 | C22×C8 | SD16 | 1 | 2 | yes | — |
| 76 | C4⋊Q8 | — | 4 | 2 | no | — |
| 77 | C4⋊Q8 | — | 4 | 2 | no | — |
| 78 | #117 | — | 4 | 1 | no | — |
| 79 | #117 | — | 4 | 1 | no | — |
| 80 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 81 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 82 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 83 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 84 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C4⋊C4 | #41 | 1 | 1 | yes | characteristic |
| 86 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 87 | C2×C22⋊C4 | #41 | 1 | 2 | yes | — |
| 88 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 89 | #107 | — | 8 | 1 | no | — |
| 90 | #107 | — | 8 | 1 | no | — |
| 91 | #107 | — | 8 | 1 | no | — |
| 92 | #106 | — | 8 | 1 | no | — |
| 93 | C4⋊C8 | — | 2 | 4 | no | — |
| 94 | C4⋊C8 | — | 4 | 2 | no | — |
| 95 | C4⋊C8 | — | 4 | 2 | no | — |
| 96 | C22⋊C8 | — | 4 | 2 | no | — |
| 97 | C4×C8 | — | 4 | 2 | no | — |
| 98 | #95 | — | 2 | 2 | no | — |
| 99 | #95 | — | 2 | 2 | no | — |
| 100 | #95 | — | 4 | 2 | no | — |
| 101 | #95 | — | 4 | 2 | no | — |
| 102 | C24 | #124 | 1 | 1 | yes | characteristic |
| 103 | C2×Q8 | — | 8 | 2 | no | — |
| 104 | C2×Q8 | — | 16 | 2 | no | — |
| 105 | C22×C4 | — | 2 | 1 | no | — |
| 106 | C22×C4 | — | 2 | 1 | no | — |
| 107 | C22×C4 | — | 2 | 1 | no | — |
| 108 | C22×C4 | — | 2 | 1 | no | — |
| 109 | C22×C4 | — | 4 | 2 | no | — |
| 110 | C22×C4 | — | 4 | 2 | no | — |
| 111 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 112 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 113 | C22×C4 | #121 | 1 | 2 | yes | — |
| 114 | C22×C4 | #123 | 1 | 2 | yes | — |
| 115 | C22×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 116 | C22×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 117 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 118 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 119 | M16 | — | 4 | 2 | no | — |
| 120 | M16 | — | 4 | 2 | no | — |
| 121 | M16 | — | 4 | 2 | no | — |
| 122 | M16 | — | 4 | 2 | no | — |
| 123 | M16 | — | 4 | 2 | no | — |
| 124 | M16 | — | 8 | 1 | no | — |
| 125 | M16 | — | 8 | 1 | no | — |
| 126 | M16 | — | 8 | 1 | no | — |
| 127 | M16 | — | 8 | 1 | no | — |
| 128 | M16 | — | 8 | 1 | no | — |
| 129 | C2×C8 | — | 2 | 2 | no | — |
| 130 | C2×C8 | — | 2 | 2 | no | — |
| 131 | C2×C8 | — | 4 | 1 | no | — |
| 132 | C2×C8 | — | 4 | 1 | no | — |
| 133 | C2×C8 | — | 4 | 1 | no | — |
| 134 | C2×C8 | — | 4 | 1 | no | — |
| 135 | C2×C8 | — | 4 | 2 | no | — |
| 136 | C2×C8 | — | 4 | 2 | no | — |
| 137 | C2×C8 | — | 4 | 2 | no | — |
| 138 | C2×C8 | — | 8 | 1 | no | — |
| 139 | C2×C8 | — | 8 | 1 | no | — |
| 140 | C2×C8 | — | 8 | 1 | no | — |
| 141 | C2×C8 | — | 8 | 1 | no | — |
| 142 | C2×C8 | — | 8 | 1 | no | — |
| 143 | C2×C8 | — | 8 | 2 | no | — |
| 144 | C4⋊C4 | — | 4 | 1 | no | — |
| 145 | C4⋊C4 | — | 4 | 1 | no | — |
| 146 | C4⋊C4 | — | 4 | 4 | no | — |
| 147 | C4⋊C4 | — | 4 | 2 | no | — |
| 148 | C4⋊C4 | — | 4 | 2 | no | — |
| 149 | C4⋊C4 | — | 4 | 2 | no | — |
| 150 | C4⋊C4 | — | 4 | 2 | no | — |
| 151 | C4⋊C4 | — | 16 | 2 | no | — |
| 152 | C22⋊C4 | — | 4 | 2 | no | — |
| 153 | C22⋊C4 | — | 16 | 2 | no | — |
| 154 | C42 | — | 2 | 2 | no | — |
| 155 | C42 | — | 2 | 2 | no | — |
| 156 | C23 | — | 2 | 1 | no | — |
| 157 | C23 | — | 2 | 1 | no | — |
| 158 | C23 | — | 4 | 1 | no | — |
| 159 | C23 | — | 4 | 1 | no | — |
| 160 | C23 | #394 | 1 | 1 | yes | characteristic |
| 161 | C23 | #487 | 1 | 1 | yes | characteristic |
| 162 | C23 | #491 | 1 | 1 | yes | characteristic |
| 163 | Q8 | — | 16 | 4 | no | — |
| 164 | C2×C4 | — | 2 | 1 | no | — |
| 165 | C2×C4 | — | 2 | 1 | no | — |
| 166 | C2×C4 | — | 2 | 1 | no | — |
| 167 | C2×C4 | — | 2 | 1 | no | — |
| 168 | C2×C4 | — | 2 | 1 | no | — |
| 169 | C2×C4 | — | 2 | 1 | no | — |
| 170 | C2×C4 | — | 2 | 1 | no | — |
| 171 | C2×C4 | — | 2 | 1 | no | — |
| 172 | C2×C4 | — | 2 | 1 | no | — |
| 173 | C2×C4 | — | 2 | 1 | no | — |
| 174 | C2×C4 | — | 2 | 2 | no | — |
| 175 | C2×C4 | — | 2 | 2 | no | — |
| 176 | C2×C4 | — | 4 | 2 | no | — |
| 177 | C2×C4 | — | 4 | 2 | no | — |
| 178 | C2×C4 | — | 8 | 2 | no | — |
| 179 | C2×C4 | — | 8 | 2 | no | — |
| 180 | C2×C4 | — | 8 | 2 | no | — |
| 181 | C2×C4 | — | 8 | 2 | no | — |
| 182 | C2×C4 | — | 8 | 2 | no | — |
| 183 | C2×C4 | — | 8 | 2 | no | — |
| 184 | C2×C4 | #448 | 1 | 1 | yes | characteristic |
| 185 | C2×C4 | #451 | 1 | 1 | yes | characteristic |
| 186 | C2×C4 | #454 | 1 | 1 | yes | characteristic |
| 187 | C2×C4 | #455 | 1 | 1 | yes | characteristic |
| 188 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 189 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 190 | C2×C4 | #461 | 1 | 1 | yes | characteristic |
| 191 | C2×C4 | #462 | 1 | 1 | yes | characteristic |
| 192 | C8 | — | 4 | 2 | no | — |
| 193 | C8 | — | 4 | 2 | no | — |
| 194 | C8 | — | 8 | 2 | no | — |
| 195 | C8 | — | 8 | 2 | no | — |
| 196 | C8 | — | 16 | 2 | no | — |
| 197 | C22 | — | 2 | 1 | no | — |
| 198 | C22 | — | 2 | 1 | no | — |
| 199 | C22 | — | 2 | 1 | no | — |
| 200 | C22 | — | 2 | 1 | no | — |
| 201 | C22 | — | 4 | 1 | no | — |
| 202 | C22 | — | 4 | 1 | no | — |
| 203 | C22 | — | 4 | 1 | no | — |
| 204 | C22 | — | 4 | 1 | no | — |
| 205 | C22 | — | 8 | 1 | no | — |
| 206 | C22 | #2040 | 1 | 1 | yes | characteristic |
| 207 | C22 | #2043 | 1 | 1 | yes | characteristic |
| 208 | C22 | #2044 | 1 | 1 | yes | characteristic |
| 209 | C4 | — | 2 | 1 | no | — |
| 210 | C4 | — | 2 | 1 | no | — |
| 211 | C4 | — | 4 | 2 | no | — |
| 212 | C4 | — | 4 | 2 | no | — |
| 213 | C4 | — | 16 | 2 | no | — |
| 214 | C4 | — | 16 | 2 | no | — |
| 215 | C4 | #1599 | 1 | 1 | yes | characteristic |
| 216 | C4 | #1602 | 1 | 1 | yes | characteristic |
| 217 | C4 | #1608 | 1 | 1 | yes | characteristic |
| 218 | C4 | #1617 | 1 | 1 | yes | characteristic |
| 219 | C2 | — | 2 | 1 | no | — |
| 220 | C2 | — | 2 | 1 | no | — |
| 221 | C2 | — | 4 | 1 | no | — |
| 222 | C2 | — | 4 | 1 | no | — |
| 223 | C2 | #10407 | 1 | 1 | yes | characteristic |
| 224 | C2 | #13431 | 1 | 1 | yes | characteristic |
| 225 | C2 | #13436 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator, Frattini, cosocle | C2×#117 | C23 |
| prime radical | C24 | #124 |
| LCS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30300 | 2 |
| #11724 | 2 |
| #11297 | 2 |
| #9193 | 2 |
| #7188 | 2 |