Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
443 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 116 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 | #33964 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #19995 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #12518 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #12444 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #12428 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12335 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12328 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12054 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C8×D16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3393 | C22 | 1 | 2 | yes | — |
| 11 | #3391 | C22 | 1 | 2 | yes | — |
| 12 | #3304 | — | 2 | 2 | no | — |
| 13 | #2250 | — | 2 | 2 | no | — |
| 14 | #2248 | C22 | 1 | 2 | yes | — |
| 15 | #2247 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2246 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2210 | — | 2 | 2 | no | — |
| 18 | #2208 | C22 | 1 | 2 | yes | — |
| 19 | #2207 | — | 2 | 2 | no | — |
| 20 | #2206 | C22 | 1 | 2 | yes | — |
| 21 | #2183 | C22 | 1 | 2 | yes | — |
| 22 | #2175 | — | 2 | 2 | no | — |
| 23 | C4×D16 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1709 | C22 | 1 | 2 | yes | — |
| 25 | #1681 | — | 2 | 2 | no | — |
| 26 | #1670 | C22 | 1 | 2 | yes | — |
| 27 | #1668 | C22 | 1 | 2 | yes | — |
| 28 | #1642 | — | 2 | 2 | no | — |
| 29 | #1628 | — | 2 | 2 | no | — |
| 30 | C8×Q16 | C22 | 1 | 1 | yes | characteristic |
| 31 | C8×D8 | C22 | 1 | 2 | yes | — |
| 32 | #1371 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1370 | C22 | 1 | 1 | yes | characteristic |
| 34 | #1362 | C22 | 1 | 2 | yes | — |
| 35 | #1361 | C22 | 1 | 2 | yes | — |
| 36 | Q8⋊C16 | C22 | 1 | 2 | yes | — |
| 37 | #1331 | C22 | 1 | 2 | yes | — |
| 38 | C8×C16 | C22 | 1 | 1 | yes | characteristic |
| 39 | #556 | — | 2 | 2 | no | — |
| 40 | #550 | — | 2 | 2 | no | — |
| 41 | C2×Q32 | — | 4 | 1 | no | — |
| 42 | C2×SD32 | — | 2 | 2 | no | — |
| 43 | C2×SD32 | — | 2 | 2 | no | — |
| 44 | C2×SD32 | — | 2 | 2 | no | — |
| 45 | C2×D16 | — | 2 | 2 | no | — |
| 46 | C2×D16 | — | 4 | 1 | no | — |
| 47 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 48 | #495 | — | 2 | 2 | no | — |
| 49 | #494 | C23 | 1 | 2 | yes | — |
| 50 | #493 | — | 2 | 2 | no | — |
| 51 | #493 | C23 | 1 | 2 | yes | — |
| 52 | #486 | — | 2 | 2 | no | — |
| 53 | #486 | — | 2 | 2 | no | — |
| 54 | #482 | — | 4 | 2 | no | — |
| 55 | #475 | — | 2 | 2 | no | — |
| 56 | #465 | — | 4 | 2 | no | — |
| 57 | #464 | — | 4 | 2 | no | — |
| 58 | #461 | — | 2 | 2 | no | — |
| 59 | #460 | — | 2 | 2 | no | — |
| 60 | #459 | — | 2 | 2 | no | — |
| 61 | #459 | — | 4 | 2 | no | — |
| 62 | #452 | — | 4 | 2 | no | — |
| 63 | C8×Q8 | C23 | 1 | 2 | yes | — |
| 64 | C4×Q16 | — | 2 | 2 | no | — |
| 65 | C4×Q16 | D4 | 1 | 1 | yes | characteristic |
| 66 | C4×D8 | — | 2 | 2 | no | — |
| 67 | C4×D8 | C23 | 1 | 2 | yes | — |
| 68 | C8×D4 | — | 2 | 2 | no | — |
| 69 | #416 | — | 4 | 2 | no | — |
| 70 | #366 | D4 | 1 | 1 | yes | characteristic |
| 71 | C4⋊C16 | — | 2 | 2 | no | — |
| 72 | C4⋊C16 | C23 | 1 | 2 | yes | — |
| 73 | #358 | — | 2 | 2 | no | — |
| 74 | #357 | — | 2 | 2 | no | — |
| 75 | #357 | D4 | 1 | 2 | yes | — |
| 76 | #346 | — | 2 | 2 | no | — |
| 77 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 78 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 79 | #335 | D4 | 1 | 1 | yes | characteristic |
| 80 | #335 | C23 | 1 | 2 | yes | — |
| 81 | #332 | — | 2 | 2 | no | — |
| 82 | #331 | — | 2 | 2 | no | — |
| 83 | Q8⋊C8 | D4 | 1 | 2 | yes | — |
| 84 | D4⋊C8 | — | 2 | 2 | no | — |
| 85 | C82 | C23 | 1 | 1 | yes | characteristic |
| 86 | C2×#41 | — | 4 | 2 | no | — |
| 87 | C2×Q16 | — | 2 | 2 | no | — |
| 88 | C2×Q16 | — | 2 | 2 | no | — |
| 89 | C2×Q16 | D8 | 1 | 1 | yes | characteristic |
| 90 | C2×SD16 | — | 4 | 2 | no | — |
| 91 | C2×SD16 | — | 4 | 2 | no | — |
| 92 | C2×D8 | — | 2 | 2 | no | — |
| 93 | C2×D8 | — | 2 | 2 | no | — |
| 94 | C2×D8 | — | 4 | 2 | no | — |
| 95 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 96 | C22×C8 | — | 4 | 2 | no | — |
| 97 | C4⋊Q8 | — | 2 | 2 | no | — |
| 98 | #127 | — | 2 | 2 | no | — |
| 99 | #127 | — | 4 | 2 | no | — |
| 100 | #126 | — | 8 | 2 | no | — |
| 101 | #124 | — | 4 | 2 | no | — |
| 102 | C22⋊Q8 | — | 4 | 2 | no | — |
| 103 | #121 | — | 4 | 2 | no | — |
| 104 | #121 | — | 8 | 2 | no | — |
| 105 | C4×Q8 | — | 4 | 2 | no | — |
| 106 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 107 | C4×D4 | — | 2 | 2 | no | — |
| 108 | C4×D4 | — | 4 | 2 | no | — |
| 109 | #117 | — | 4 | 2 | no | — |
| 110 | Q32 | — | 8 | 2 | no | — |
| 111 | SD32 | — | 4 | 4 | no | — |
| 112 | SD32 | — | 8 | 2 | no | — |
| 113 | SD32 | — | 8 | 2 | no | — |
| 114 | D16 | — | 2 | 2 | no | — |
| 115 | D16 | — | 8 | 2 | no | — |
| 116 | D16 | — | 8 | 2 | no | — |
| 117 | C2×C16 | — | 2 | 1 | no | — |
| 118 | C2×C16 | — | 2 | 2 | no | — |
| 119 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 120 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 121 | #107 | D8 | 1 | 1 | yes | characteristic |
| 122 | #107 | C2×D4 | 1 | 2 | yes | — |
| 123 | #106 | — | 4 | 2 | no | — |
| 124 | C4⋊C8 | — | 2 | 2 | no | — |
| 125 | C4⋊C8 | — | 2 | 2 | no | — |
| 126 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 127 | #103 | — | 4 | 2 | no | — |
| 128 | #103 | — | 4 | 2 | no | — |
| 129 | #103 | D8 | 1 | 2 | yes | — |
| 130 | #102 | — | 2 | 2 | no | — |
| 131 | #102 | — | 4 | 2 | no | — |
| 132 | #102 | — | 4 | 2 | no | — |
| 133 | #102 | — | 4 | 2 | no | — |
| 134 | #102 | — | 4 | 2 | no | — |
| 135 | #102 | — | 8 | 2 | no | — |
| 136 | C22⋊C8 | — | 4 | 2 | no | — |
| 137 | C4×C8 | — | 2 | 2 | no | — |
| 138 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 139 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 140 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 141 | #41 | — | 8 | 4 | no | — |
| 142 | #41 | — | 16 | 2 | no | — |
| 143 | C2×Q8 | — | 4 | 2 | no | — |
| 144 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 145 | C2×D4 | — | 2 | 2 | no | — |
| 146 | C2×D4 | — | 4 | 2 | no | — |
| 147 | C2×D4 | — | 4 | 2 | no | — |
| 148 | C2×D4 | — | 4 | 2 | no | — |
| 149 | C2×D4 | — | 8 | 2 | no | — |
| 150 | C2×D4 | — | 8 | 2 | no | — |
| 151 | C22×C4 | — | 4 | 2 | no | — |
| 152 | C22×C4 | — | 8 | 2 | no | — |
| 153 | Q16 | — | 4 | 1 | no | — |
| 154 | Q16 | — | 4 | 2 | no | — |
| 155 | Q16 | — | 4 | 2 | no | — |
| 156 | Q16 | — | 8 | 2 | no | — |
| 157 | SD16 | — | 16 | 2 | no | — |
| 158 | SD16 | — | 16 | 2 | no | — |
| 159 | D8 | — | 2 | 2 | no | — |
| 160 | D8 | — | 2 | 2 | no | — |
| 161 | D8 | — | 4 | 2 | no | — |
| 162 | D8 | — | 4 | 2 | no | — |
| 163 | D8 | — | 8 | 2 | no | — |
| 164 | D8 | — | 16 | 2 | no | — |
| 165 | C2×C8 | — | 2 | 2 | no | — |
| 166 | C2×C8 | — | 2 | 1 | no | — |
| 167 | C2×C8 | — | 4 | 2 | no | — |
| 168 | C2×C8 | — | 16 | 2 | no | — |
| 169 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 173 | C4⋊C4 | — | 2 | 2 | no | — |
| 174 | C4⋊C4 | — | 2 | 2 | no | — |
| 175 | C4⋊C4 | — | 4 | 2 | no | — |
| 176 | C4⋊C4 | — | 4 | 2 | no | — |
| 177 | C4⋊C4 | — | 8 | 2 | no | — |
| 178 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 179 | C22⋊C4 | — | 4 | 2 | no | — |
| 180 | C22⋊C4 | — | 8 | 2 | no | — |
| 181 | C22⋊C4 | — | 8 | 2 | no | — |
| 182 | C22⋊C4 | — | 8 | 2 | no | — |
| 183 | C42 | — | 2 | 2 | no | — |
| 184 | C42 | — | 8 | 2 | no | — |
| 185 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 186 | C16 | — | 2 | 2 | no | — |
| 187 | C16 | — | 2 | 1 | no | — |
| 188 | C16 | — | 4 | 2 | no | — |
| 189 | C16 | #135 | 1 | 2 | yes | — |
| 190 | C23 | — | 4 | 2 | no | — |
| 191 | C23 | — | 8 | 2 | no | — |
| 192 | Q8 | — | 2 | 2 | no | — |
| 193 | Q8 | — | 2 | 2 | no | — |
| 194 | Q8 | — | 8 | 2 | no | — |
| 195 | Q8 | — | 8 | 2 | no | — |
| 196 | D4 | — | 4 | 2 | no | — |
| 197 | D4 | — | 4 | 2 | no | — |
| 198 | D4 | — | 8 | 2 | no | — |
| 199 | D4 | — | 8 | 2 | no | — |
| 200 | D4 | — | 8 | 2 | no | — |
| 201 | D4 | — | 8 | 2 | no | — |
| 202 | D4 | — | 16 | 2 | no | — |
| 203 | D4 | — | 16 | 4 | no | — |
| 204 | D4 | — | 32 | 2 | no | — |
| 205 | C2×C4 | — | 2 | 2 | no | — |
| 206 | C2×C4 | — | 2 | 2 | no | — |
| 207 | C2×C4 | — | 4 | 2 | no | — |
| 208 | C2×C4 | — | 8 | 2 | no | — |
| 209 | C2×C4 | — | 8 | 2 | no | — |
| 210 | C2×C4 | — | 8 | 2 | no | — |
| 211 | C2×C4 | — | 16 | 4 | no | — |
| 212 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 215 | C8 | — | 2 | 1 | no | — |
| 216 | C8 | — | 2 | 1 | no | — |
| 217 | C8 | — | 2 | 1 | no | — |
| 218 | C8 | — | 4 | 1 | no | — |
| 219 | C8 | — | 4 | 2 | no | — |
| 220 | C8 | — | 8 | 2 | no | — |
| 221 | C8 | C2×#135 | 1 | 1 | yes | characteristic |
| 222 | C8 | #576 | 1 | 1 | yes | characteristic |
| 223 | C22 | — | 8 | 2 | no | — |
| 224 | C22 | — | 8 | 2 | no | — |
| 225 | C22 | — | 8 | 2 | no | — |
| 226 | C22 | — | 16 | 2 | no | — |
| 227 | C22 | — | 16 | 2 | no | — |
| 228 | C22 | — | 16 | 2 | no | — |
| 229 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 230 | C4 | — | 2 | 1 | no | — |
| 231 | C4 | — | 4 | 2 | no | — |
| 232 | C4 | — | 4 | 2 | no | — |
| 233 | C4 | — | 8 | 2 | no | — |
| 234 | C4 | — | 16 | 2 | no | — |
| 235 | C4 | #3287 | 1 | 1 | yes | characteristic |
| 236 | C4 | #3292 | 1 | 1 | yes | characteristic |
| 237 | C4 | #3414 | 1 | 1 | yes | characteristic |
| 238 | C4 | #3415 | 1 | 1 | yes | characteristic |
| 239 | C2 | — | 16 | 2 | no | — |
| 240 | C2 | — | 32 | 2 | no | — |
| 241 | C2 | #19975 | 1 | 1 | yes | characteristic |
| 242 | C2 | #33839 | 1 | 1 | yes | characteristic |
| 243 | C2 | #33857 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33964 | C2 | semidirect |
| 2. #19995 | C2 | semidirect |
| 3. #12518 | C2 | semidirect |
| 4. #12444 | C2 | semidirect |
| 5. #12428 | C2 | semidirect |
| 6. #12335 | C2 | semidirect |
| 7. #12328 | C2 | semidirect |
| 8. #12054 | C2 | semidirect |
| 9. C8×D16 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33964 | 2 |
| #19995 | 2 |
| #12518 | 2 |
| #12444 | 2 |
| #12428 | 2 |
| #12335 | 2 |
| #12328 | 2 |
| #12054 | 2 |
| C8×D16 | 2 |