Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
734 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 194 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 | #23223 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | C2×#1709 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #9747 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #9066 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #9061 | C2 | 1 | 2 | yes | maximal |
| 6 | #8282 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#486 | C22 | 1 | 4 | yes | — |
| 10 | #2369 | C22 | 1 | 2 | yes | — |
| 11 | #2366 | C22 | 1 | 2 | yes | — |
| 12 | #1987 | C22 | 1 | 4 | yes | — |
| 13 | #1982 | C22 | 1 | 4 | yes | — |
| 14 | #1936 | — | 2 | 4 | no | — |
| 15 | #1925 | — | 2 | 4 | no | — |
| 16 | #1862 | — | 2 | 4 | no | — |
| 17 | C4×#107 | C22 | 1 | 2 | yes | — |
| 18 | #1770 | C22 | 1 | 4 | yes | — |
| 19 | #1762 | — | 2 | 4 | no | — |
| 20 | #1709 | C4 | 1 | 8 | yes | complemented |
| 21 | C2×#335 | C22 | 1 | 2 | yes | — |
| 22 | C2×#333 | C22 | 1 | 2 | yes | — |
| 23 | C2×#331 | C22 | 1 | 2 | yes | — |
| 24 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1279 | C22 | 1 | 4 | yes | — |
| 26 | C22×Q16 | — | 4 | 1 | no | — |
| 27 | C22×D8 | — | 4 | 1 | no | — |
| 28 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 29 | C2×#127 | C23 | 1 | 2 | yes | — |
| 30 | C2×C4×D4 | — | 2 | 4 | no | — |
| 31 | #494 | — | 4 | 1 | no | — |
| 32 | #494 | C2×C4 | 1 | 4 | yes | complemented |
| 33 | #493 | — | 4 | 1 | no | — |
| 34 | #493 | C2×C4 | 1 | 4 | yes | — |
| 35 | #486 | — | 2 | 8 | no | — |
| 36 | #486 | C2×C4 | 1 | 16 | yes | complemented |
| 37 | C2×#107 | D4 | 1 | 2 | yes | — |
| 38 | C2×#107 | D4 | 1 | 2 | yes | — |
| 39 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 40 | C2×#102 | — | 2 | 4 | no | — |
| 41 | C2×#102 | — | 2 | 4 | no | — |
| 42 | C2×#102 | — | 2 | 4 | no | — |
| 43 | C2×#102 | — | 2 | 4 | no | — |
| 44 | C2×#97 | — | 2 | 4 | no | — |
| 45 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 47 | #391 | — | 4 | 2 | no | — |
| 48 | #390 | — | 2 | 4 | no | — |
| 49 | #389 | — | 2 | 4 | no | — |
| 50 | #386 | — | 4 | 2 | no | — |
| 51 | C4×C4⋊C4 | — | 2 | 4 | no | — |
| 52 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 53 | #340 | — | 2 | 4 | no | — |
| 54 | #340 | — | 2 | 4 | no | — |
| 55 | #340 | D4 | 1 | 4 | yes | — |
| 56 | #336 | — | 2 | 4 | no | — |
| 57 | #335 | — | 2 | 4 | no | — |
| 58 | #333 | — | 4 | 2 | no | — |
| 59 | #331 | — | 4 | 2 | no | — |
| 60 | C82 | C2×C4 | 1 | 4 | yes | complemented |
| 61 | C22×Q8 | — | 2 | 2 | no | — |
| 62 | C22×D4 | — | 2 | 4 | no | — |
| 63 | C22×D4 | — | 2 | 2 | no | — |
| 64 | C23×C4 | — | 4 | 2 | no | — |
| 65 | C2×Q16 | — | 4 | 4 | no | — |
| 66 | C2×Q16 | — | 8 | 2 | no | — |
| 67 | C2×Q16 | — | 8 | 2 | no | — |
| 68 | C2×D8 | — | 4 | 4 | no | — |
| 69 | C2×D8 | — | 8 | 2 | no | — |
| 70 | C2×D8 | — | 8 | 2 | no | — |
| 71 | C22×C8 | — | 2 | 1 | no | — |
| 72 | C22×C8 | — | 2 | 4 | no | — |
| 73 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 74 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 75 | C4⋊Q8 | — | 2 | 4 | no | — |
| 76 | C4⋊Q8 | — | 2 | 2 | no | — |
| 77 | C4⋊Q8 | C22×C4 | 1 | 4 | yes | — |
| 78 | #127 | — | 2 | 4 | no | — |
| 79 | #127 | — | 2 | 2 | no | — |
| 80 | #127 | C22×C4 | 1 | 4 | yes | — |
| 81 | C4×D4 | — | 4 | 8 | no | — |
| 82 | C4×D4 | — | 8 | 4 | no | — |
| 83 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 84 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 85 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 86 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 87 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 88 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 89 | C2×C42 | — | 2 | 4 | no | — |
| 90 | C2×C42 | — | 4 | 2 | no | — |
| 91 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 92 | #107 | — | 4 | 2 | no | — |
| 93 | #107 | — | 4 | 2 | no | — |
| 94 | C4⋊C8 | — | 2 | 4 | no | — |
| 95 | C4⋊C8 | — | 2 | 4 | no | — |
| 96 | #102 | — | 2 | 16 | no | — |
| 97 | #102 | — | 2 | 16 | no | — |
| 98 | #102 | — | 4 | 8 | no | — |
| 99 | #102 | — | 8 | 4 | no | — |
| 100 | #97 | — | 8 | 4 | no | — |
| 101 | C4×C8 | — | 2 | 1 | no | — |
| 102 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 103 | C4×C8 | C22×C4 | 1 | 4 | yes | — |
| 104 | C4×C8 | #41 | 1 | 4 | yes | — |
| 105 | #95 | — | 2 | 4 | no | — |
| 106 | #95 | — | 4 | 4 | no | — |
| 107 | C24 | — | 4 | 2 | no | — |
| 108 | C2×Q8 | — | 2 | 4 | no | — |
| 109 | C2×Q8 | — | 4 | 2 | no | — |
| 110 | C2×Q8 | — | 8 | 2 | no | — |
| 111 | C2×Q8 | — | 8 | 2 | no | — |
| 112 | C2×D4 | — | 2 | 8 | no | — |
| 113 | C2×D4 | — | 2 | 4 | no | — |
| 114 | C2×D4 | — | 4 | 4 | no | — |
| 115 | C2×D4 | — | 4 | 2 | no | — |
| 116 | C2×D4 | — | 4 | 4 | no | — |
| 117 | C2×D4 | — | 4 | 4 | no | — |
| 118 | C2×D4 | — | 4 | 4 | no | — |
| 119 | C2×D4 | — | 4 | 4 | no | — |
| 120 | C2×D4 | — | 8 | 2 | no | — |
| 121 | C2×D4 | — | 8 | 2 | no | — |
| 122 | C22×C4 | — | 2 | 4 | no | — |
| 123 | C22×C4 | — | 2 | 4 | no | — |
| 124 | C22×C4 | — | 4 | 2 | no | — |
| 125 | C22×C4 | — | 8 | 8 | no | — |
| 126 | C22×C4 | — | 16 | 2 | no | — |
| 127 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 128 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 129 | Q16 | — | 8 | 8 | no | — |
| 130 | D8 | — | 8 | 8 | no | — |
| 131 | C2×C8 | — | 2 | 2 | no | — |
| 132 | C2×C8 | — | 2 | 2 | no | — |
| 133 | C2×C8 | — | 2 | 2 | no | — |
| 134 | C2×C8 | — | 2 | 2 | no | — |
| 135 | C2×C8 | — | 2 | 2 | no | — |
| 136 | C2×C8 | — | 2 | 2 | no | — |
| 137 | C2×C8 | — | 2 | 2 | no | — |
| 138 | C2×C8 | — | 4 | 4 | no | — |
| 139 | C2×C8 | — | 4 | 4 | no | — |
| 140 | C2×C8 | — | 4 | 4 | no | — |
| 141 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 142 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 143 | C4⋊C4 | — | 2 | 8 | no | — |
| 144 | C4⋊C4 | — | 2 | 4 | no | — |
| 145 | C4⋊C4 | — | 2 | 4 | no | — |
| 146 | C4⋊C4 | — | 2 | 4 | no | — |
| 147 | C4⋊C4 | — | 2 | 4 | no | — |
| 148 | C4⋊C4 | — | 4 | 4 | no | — |
| 149 | C4⋊C4 | — | 16 | 2 | no | — |
| 150 | C22⋊C4 | — | 8 | 8 | no | — |
| 151 | C22⋊C4 | — | 16 | 2 | no | — |
| 152 | C42 | — | 4 | 4 | no | — |
| 153 | C42 | — | 4 | 4 | no | — |
| 154 | C42 | — | 16 | 2 | no | — |
| 155 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 156 | C42 | C2×#41 | 1 | 2 | yes | — |
| 157 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 158 | C23 | — | 4 | 4 | no | — |
| 159 | C23 | — | 8 | 4 | no | — |
| 160 | C23 | — | 8 | 4 | no | — |
| 161 | C23 | — | 8 | 2 | no | — |
| 162 | C23 | — | 8 | 2 | no | — |
| 163 | C23 | D42 | 1 | 1 | yes | characteristic |
| 164 | Q8 | — | 8 | 4 | no | — |
| 165 | Q8 | — | 16 | 2 | no | — |
| 166 | D4 | — | 4 | 8 | no | — |
| 167 | D4 | — | 4 | 8 | no | — |
| 168 | D4 | — | 8 | 4 | no | — |
| 169 | D4 | — | 8 | 4 | no | — |
| 170 | D4 | — | 8 | 4 | no | — |
| 171 | D4 | — | 16 | 2 | no | — |
| 172 | C2×C4 | — | 2 | 1 | no | — |
| 173 | C2×C4 | — | 2 | 1 | no | — |
| 174 | C2×C4 | — | 4 | 4 | no | — |
| 175 | C2×C4 | — | 4 | 4 | no | — |
| 176 | C2×C4 | — | 4 | 4 | no | — |
| 177 | C2×C4 | — | 4 | 4 | no | — |
| 178 | C2×C4 | — | 4 | 4 | no | — |
| 179 | C2×C4 | — | 4 | 4 | no | — |
| 180 | C2×C4 | — | 4 | 4 | no | — |
| 181 | C2×C4 | — | 8 | 2 | no | — |
| 182 | C2×C4 | — | 8 | 2 | no | — |
| 183 | C2×C4 | — | 16 | 8 | no | — |
| 184 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 185 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 186 | C2×C4 | #550 | 1 | 2 | yes | — |
| 187 | C2×C4 | #559 | 1 | 1 | yes | characteristic |
| 188 | C2×C4 | #575 | 1 | 2 | yes | — |
| 189 | C2×C4 | #575 | 1 | 2 | yes | — |
| 190 | C2×C4 | #576 | 1 | 2 | yes | — |
| 191 | C2×C4 | #578 | 1 | 2 | yes | — |
| 192 | C8 | — | 2 | 4 | no | — |
| 193 | C8 | — | 2 | 4 | no | — |
| 194 | C8 | — | 2 | 4 | no | — |
| 195 | C8 | — | 8 | 4 | no | — |
| 196 | C22 | — | 8 | 8 | no | — |
| 197 | C22 | — | 8 | 4 | no | — |
| 198 | C22 | — | 16 | 2 | no | — |
| 199 | C22 | — | 16 | 4 | no | — |
| 200 | C22 | — | 16 | 2 | no | — |
| 201 | C22 | #2366 | 1 | 1 | yes | characteristic |
| 202 | C22 | #3273 | 1 | 2 | yes | — |
| 203 | C22 | #3277 | 1 | 2 | yes | — |
| 204 | C22 | #3292 | 1 | 1 | yes | characteristic |
| 205 | C22 | #3292 | 1 | 1 | yes | characteristic |
| 206 | C4 | — | 2 | 1 | no | — |
| 207 | C4 | — | 2 | 1 | no | — |
| 208 | C4 | — | 8 | 4 | no | — |
| 209 | C4 | — | 8 | 4 | no | — |
| 210 | C4 | — | 8 | 4 | no | — |
| 211 | C4 | #2943 | 1 | 2 | yes | — |
| 212 | C4 | #2951 | 1 | 2 | yes | — |
| 213 | C4 | #3402 | 1 | 2 | yes | — |
| 214 | C4 | #3403 | 1 | 2 | yes | — |
| 215 | C2 | — | 16 | 4 | no | — |
| 216 | C2 | #20302 | 1 | 1 | yes | characteristic |
| 217 | C2 | #20308 | 1 | 2 | yes | — |
| 218 | C2 | #20319 | 1 | 1 | yes | characteristic |
| 219 | C2 | #23191 | 1 | 2 | yes | — |
| 220 | C2 | #23201 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23223 | C2 | semidirect |
| 3. #9747 | C2 | semidirect |
| 4. #9066 | C2 | semidirect |
| 6. #8282 | C2 | semidirect |
| 20. #1709 | C4 | semidirect |
| 32. #494 | C2×C4 | semidirect |
| 36. #486 | C2×C4 | semidirect |
| 60. C82 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23223 | 2 |
| C2×#1709 | 2 |
| #9747 | 2 |
| #9066 | 2 |
| #9061 | 2 |
| #8282 | 2 |