Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
676 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 162 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 | #21715 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #21708 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #17901 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #10666 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10663 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10034 | C2 | 1 | 2 | yes | maximal |
| 7 | #10014 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #8137 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #3150 | C22 | 1 | 1 | yes | characteristic |
| 10 | #3103 | C22 | 1 | 4 | yes | — |
| 11 | C2×C8×Q8 | C22 | 1 | 2 | yes | — |
| 12 | C2×#443 | — | 2 | 2 | no | — |
| 13 | #2931 | C22 | 1 | 2 | yes | — |
| 14 | #2399 | C22 | 1 | 1 | yes | characteristic |
| 15 | #1985 | C22 | 1 | 2 | yes | — |
| 16 | #1975 | — | 2 | 2 | no | — |
| 17 | #1963 | C22 | 1 | 2 | yes | — |
| 18 | #1940 | — | 2 | 2 | no | — |
| 19 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1843 | C22 | 1 | 1 | yes | characteristic |
| 21 | C8×C4⋊C4 | — | 2 | 2 | no | — |
| 22 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1583 | — | 2 | 8 | no | — |
| 24 | C8×SD16 | — | 2 | 8 | no | — |
| 25 | #1570 | C22 | 1 | 4 | yes | — |
| 26 | #1487 | C22 | 1 | 4 | yes | — |
| 27 | #1480 | C22 | 1 | 4 | yes | — |
| 28 | #1450 | C22 | 1 | 2 | yes | — |
| 29 | #1275 | C22 | 1 | 2 | yes | — |
| 30 | #1274 | C22 | 1 | 2 | yes | — |
| 31 | C2×#135 | — | 2 | 2 | no | — |
| 32 | C2×#131 | — | 2 | 2 | no | — |
| 33 | #532 | C23 | 1 | 2 | yes | — |
| 34 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 35 | #497 | — | 2 | 2 | no | — |
| 36 | #496 | — | 2 | 2 | no | — |
| 37 | #481 | — | 2 | 4 | no | — |
| 38 | #475 | — | 2 | 4 | no | — |
| 39 | #460 | — | 2 | 4 | no | — |
| 40 | #448 | — | 2 | 4 | no | — |
| 41 | C8×Q8 | — | 2 | 4 | no | — |
| 42 | C8×Q8 | — | 2 | 4 | no | — |
| 43 | #443 | — | 4 | 8 | no | — |
| 44 | C4×SD16 | — | 2 | 8 | no | — |
| 45 | #436 | — | 2 | 4 | no | — |
| 46 | C8×D4 | — | 2 | 4 | no | — |
| 47 | #431 | C23 | 1 | 4 | yes | — |
| 48 | C2×#108 | — | 2 | 2 | no | — |
| 49 | #423 | C23 | 1 | 2 | yes | — |
| 50 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 51 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 52 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 53 | C2×#104 | — | 2 | 4 | no | — |
| 54 | #408 | — | 2 | 2 | no | — |
| 55 | C2×#97 | D4 | 1 | 2 | yes | — |
| 56 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 57 | C2×C4×C8 | — | 2 | 2 | no | — |
| 58 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 59 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 60 | #398 | — | 2 | 2 | no | — |
| 61 | #395 | — | 2 | 2 | no | — |
| 62 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 63 | #376 | C23 | 1 | 1 | yes | characteristic |
| 64 | #336 | — | 2 | 4 | no | — |
| 65 | #334 | — | 2 | 8 | no | — |
| 66 | Q8⋊C8 | — | 2 | 8 | no | — |
| 67 | D4⋊C8 | — | 2 | 8 | no | — |
| 68 | #322 | D4 | 1 | 4 | yes | — |
| 69 | C82 | D4 | 1 | 4 | yes | — |
| 70 | C2×#41 | — | 2 | 2 | no | — |
| 71 | C22×Q8 | #41 | 1 | 2 | yes | — |
| 72 | #135 | — | 4 | 4 | no | — |
| 73 | #135 | — | 4 | 4 | no | — |
| 74 | C2×Q16 | — | 2 | 2 | no | — |
| 75 | C2×SD16 | — | 2 | 4 | no | — |
| 76 | C2×SD16 | — | 2 | 8 | no | — |
| 77 | C2×D8 | — | 2 | 2 | no | — |
| 78 | #131 | — | 4 | 8 | no | — |
| 79 | C2×M16 | — | 2 | 2 | no | — |
| 80 | C2×M16 | — | 4 | 2 | no | — |
| 81 | C22×C8 | — | 2 | 2 | no | — |
| 82 | C22×C8 | — | 4 | 2 | no | — |
| 83 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 85 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C4⋊Q8 | #41 | 1 | 2 | yes | — |
| 87 | #127 | #41 | 1 | 2 | yes | — |
| 88 | #124 | — | 2 | 2 | no | — |
| 89 | #121 | — | 2 | 4 | no | — |
| 90 | C4×Q8 | — | 2 | 4 | no | — |
| 91 | C4×Q8 | — | 2 | 4 | no | — |
| 92 | C4×D4 | — | 2 | 4 | no | — |
| 93 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 94 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 95 | C2×C4⋊C4 | #41 | 1 | 2 | yes | — |
| 96 | C2×C42 | — | 2 | 2 | no | — |
| 97 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 99 | #108 | — | 4 | 4 | no | — |
| 100 | #106 | — | 2 | 8 | no | — |
| 101 | C4⋊C8 | — | 2 | 4 | no | — |
| 102 | C4⋊C8 | — | 2 | 4 | no | — |
| 103 | C4⋊C8 | — | 2 | 4 | no | — |
| 104 | C4⋊C8 | — | 4 | 4 | no | — |
| 105 | C4⋊C8 | — | 4 | 4 | no | — |
| 106 | #104 | — | 4 | 8 | no | — |
| 107 | #103 | — | 2 | 8 | no | — |
| 108 | #102 | — | 2 | 8 | no | — |
| 109 | C22⋊C8 | — | 2 | 4 | no | — |
| 110 | C22⋊C8 | — | 4 | 4 | no | — |
| 111 | #97 | — | 2 | 2 | no | — |
| 112 | #97 | — | 4 | 2 | no | — |
| 113 | C4×C8 | — | 4 | 2 | no | — |
| 114 | C4×C8 | — | 4 | 2 | no | — |
| 115 | C4×C8 | SD16 | 1 | 8 | yes | complemented |
| 116 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 117 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 118 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 119 | #95 | — | 2 | 2 | no | — |
| 120 | #95 | — | 2 | 2 | no | — |
| 121 | #95 | — | 2 | 2 | no | — |
| 122 | #41 | — | 4 | 4 | no | — |
| 123 | #41 | — | 4 | 4 | no | — |
| 124 | C2×Q8 | — | 2 | 4 | no | — |
| 125 | C2×Q8 | — | 2 | 2 | no | — |
| 126 | C2×Q8 | — | 4 | 4 | no | — |
| 127 | C2×D4 | — | 2 | 4 | no | — |
| 128 | C2×D4 | — | 2 | 2 | no | — |
| 129 | C2×D4 | — | 4 | 2 | no | — |
| 130 | C22×C4 | — | 2 | 2 | no | — |
| 131 | C22×C4 | — | 2 | 2 | no | — |
| 132 | C22×C4 | — | 2 | 2 | no | — |
| 133 | C22×C4 | — | 4 | 2 | no | — |
| 134 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 135 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 136 | Q16 | — | 4 | 4 | no | — |
| 137 | SD16 | — | 4 | 8 | no | — |
| 138 | SD16 | — | 4 | 16 | no | — |
| 139 | D8 | — | 4 | 4 | no | — |
| 140 | M16 | — | 4 | 4 | no | — |
| 141 | M16 | — | 8 | 4 | no | — |
| 142 | C2×C8 | — | 2 | 2 | no | — |
| 143 | C2×C8 | — | 2 | 2 | no | — |
| 144 | C2×C8 | — | 2 | 2 | no | — |
| 145 | C2×C8 | — | 2 | 8 | no | — |
| 146 | C2×C8 | — | 4 | 2 | no | — |
| 147 | C2×C8 | — | 4 | 2 | no | — |
| 148 | C2×C8 | — | 4 | 4 | no | — |
| 149 | C2×C8 | — | 8 | 4 | no | — |
| 150 | C2×C8 | #127 | 1 | 2 | yes | — |
| 151 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 152 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 153 | C2×C8 | C2×SD16 | 1 | 4 | yes | — |
| 154 | C2×C8 | C2×SD16 | 1 | 4 | yes | — |
| 155 | C4⋊C4 | — | 2 | 4 | no | — |
| 156 | C4⋊C4 | — | 2 | 4 | no | — |
| 157 | C4⋊C4 | — | 4 | 8 | no | — |
| 158 | C4⋊C4 | — | 4 | 4 | no | — |
| 159 | C22⋊C4 | — | 4 | 4 | no | — |
| 160 | C42 | — | 2 | 2 | no | — |
| 161 | C42 | — | 4 | 2 | no | — |
| 162 | C42 | — | 4 | 2 | no | — |
| 163 | C42 | C2×SD16 | 1 | 4 | yes | — |
| 164 | C42 | C22×D4 | 1 | 2 | yes | — |
| 165 | C42 | C2×#41 | 1 | 2 | yes | — |
| 166 | C23 | — | 4 | 2 | no | — |
| 167 | C23 | #525 | 1 | 1 | yes | characteristic |
| 168 | Q8 | — | 2 | 8 | no | — |
| 169 | Q8 | — | 4 | 4 | no | — |
| 170 | Q8 | — | 4 | 4 | no | — |
| 171 | D4 | — | 4 | 8 | no | — |
| 172 | D4 | — | 4 | 4 | no | — |
| 173 | D4 | — | 8 | 4 | no | — |
| 174 | C2×C4 | — | 2 | 4 | no | — |
| 175 | C2×C4 | — | 2 | 4 | no | — |
| 176 | C2×C4 | — | 4 | 2 | no | — |
| 177 | C2×C4 | — | 4 | 4 | no | — |
| 178 | C2×C4 | — | 4 | 2 | no | — |
| 179 | C2×C4 | — | 4 | 2 | no | — |
| 180 | C2×C4 | — | 4 | 4 | no | — |
| 181 | C2×C4 | — | 4 | 2 | no | — |
| 182 | C2×C4 | — | 4 | 2 | no | — |
| 183 | C2×C4 | — | 8 | 2 | no | — |
| 184 | C2×C4 | — | 8 | 2 | no | — |
| 185 | C2×C4 | #525 | 1 | 4 | yes | — |
| 186 | C2×C4 | #525 | 1 | 1 | yes | characteristic |
| 187 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 188 | C2×C4 | C22×SD16 | 1 | 2 | yes | — |
| 189 | C2×C4 | #575 | 1 | 2 | yes | — |
| 190 | C8 | — | 2 | 4 | no | — |
| 191 | C8 | — | 2 | 2 | no | — |
| 192 | C8 | — | 4 | 8 | no | — |
| 193 | C8 | — | 8 | 2 | no | — |
| 194 | C8 | — | 8 | 2 | no | — |
| 195 | C8 | #492 | 1 | 2 | yes | — |
| 196 | C8 | #492 | 1 | 2 | yes | — |
| 197 | C22 | — | 2 | 2 | no | — |
| 198 | C22 | — | 8 | 4 | no | — |
| 199 | C22 | — | 8 | 2 | no | — |
| 200 | C22 | #2397 | 1 | 1 | yes | characteristic |
| 201 | C22 | #3103 | 1 | 2 | yes | — |
| 202 | C4 | — | 2 | 4 | no | — |
| 203 | C4 | — | 4 | 4 | no | — |
| 204 | C4 | — | 4 | 4 | no | — |
| 205 | C4 | — | 8 | 2 | no | — |
| 206 | C4 | — | 8 | 2 | no | — |
| 207 | C4 | #3101 | 1 | 2 | yes | — |
| 208 | C4 | C2×#492 | 1 | 1 | yes | characteristic |
| 209 | C4 | #3150 | 1 | 1 | yes | characteristic |
| 210 | C2 | — | 2 | 2 | no | — |
| 211 | C2 | — | 16 | 2 | no | — |
| 212 | C2 | #23794 | 1 | 2 | yes | — |
| 213 | C2 | #23799 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2397 |
| prime radical | #3150 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21715 | C2 | semidirect |
| 3. #17901 | C2 | semidirect |
| 4. #10666 | C2 | semidirect |
| 5. #10663 | C2 | semidirect |
| 7. #10014 | C2 | semidirect |
| 8. #8137 | C2 | semidirect |
| 115. C4×C8 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21715 | 2 |
| #21708 | 2 |
| #17901 | 2 |
| #10666 | 2 |
| #10663 | 2 |
| #10034 | 2 |
| #10014 | 2 |
| #8137 | 2 |