Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
430 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 114 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 | #33933 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #19909 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #12503 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #12485 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #12480 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12411 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12328 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12081 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #12054 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3391 | C22 | 1 | 2 | yes | — |
| 11 | #3380 | C22 | 1 | 2 | yes | — |
| 12 | #3333 | — | 2 | 2 | no | — |
| 13 | #2252 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2246 | C22 | 1 | 1 | yes | characteristic |
| 15 | #2244 | — | 2 | 2 | no | — |
| 16 | #2240 | C22 | 1 | 2 | yes | — |
| 17 | #2228 | — | 2 | 2 | no | — |
| 18 | #2226 | C22 | 1 | 2 | yes | — |
| 19 | #2206 | — | 2 | 2 | no | — |
| 20 | #2206 | C22 | 1 | 2 | yes | — |
| 21 | #2183 | C22 | 1 | 2 | yes | — |
| 22 | #2177 | — | 2 | 2 | no | — |
| 23 | C4×D16 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1706 | C22 | 1 | 2 | yes | — |
| 25 | #1697 | — | 2 | 2 | no | — |
| 26 | #1676 | C22 | 1 | 2 | yes | — |
| 27 | #1668 | C22 | 1 | 2 | yes | — |
| 28 | #1631 | — | 2 | 2 | no | — |
| 29 | #1622 | — | 2 | 2 | no | — |
| 30 | #1589 | C22 | 1 | 2 | yes | — |
| 31 | C8×Q16 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1371 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1370 | C22 | 1 | 1 | yes | characteristic |
| 34 | #1365 | 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 | #554 | — | 2 | 2 | no | — |
| 40 | #550 | — | 2 | 2 | no | — |
| 41 | C2×D16 | — | 2 | 2 | no | — |
| 42 | C2×D16 | — | 4 | 1 | no | — |
| 43 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 44 | #500 | C23 | 1 | 2 | yes | — |
| 45 | #499 | — | 2 | 2 | no | — |
| 46 | #493 | — | 2 | 2 | no | — |
| 47 | #493 | C23 | 1 | 2 | yes | — |
| 48 | #492 | — | 2 | 2 | no | — |
| 49 | #486 | — | 2 | 2 | no | — |
| 50 | #480 | — | 4 | 2 | no | — |
| 51 | #478 | — | 4 | 2 | no | — |
| 52 | #477 | — | 2 | 2 | no | — |
| 53 | #476 | — | 2 | 2 | no | — |
| 54 | #468 | — | 4 | 2 | no | — |
| 55 | #460 | — | 2 | 2 | no | — |
| 56 | #459 | — | 2 | 2 | no | — |
| 57 | #459 | — | 4 | 2 | no | — |
| 58 | #452 | — | 4 | 2 | no | — |
| 59 | C8×Q8 | C23 | 1 | 2 | yes | — |
| 60 | C4×Q16 | D4 | 1 | 1 | yes | characteristic |
| 61 | C4×D8 | — | 2 | 2 | no | — |
| 62 | C4×D8 | — | 2 | 2 | no | — |
| 63 | C4×D8 | C23 | 1 | 2 | yes | — |
| 64 | #436 | — | 2 | 2 | no | — |
| 65 | #417 | — | 4 | 2 | no | — |
| 66 | #367 | — | 2 | 2 | no | — |
| 67 | #366 | — | 4 | 1 | no | — |
| 68 | #366 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4⋊C16 | — | 2 | 2 | no | — |
| 70 | C4⋊C16 | C23 | 1 | 2 | yes | — |
| 71 | #357 | — | 2 | 2 | no | — |
| 72 | #357 | — | 2 | 2 | no | — |
| 73 | #357 | — | 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 | #333 | — | 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 | D8 | 1 | 1 | yes | characteristic |
| 88 | C2×SD16 | — | 4 | 2 | no | — |
| 89 | C2×SD16 | — | 8 | 2 | no | — |
| 90 | C2×D8 | — | 2 | 2 | no | — |
| 91 | C2×D8 | — | 2 | 2 | no | — |
| 92 | C2×D8 | — | 4 | 2 | no | — |
| 93 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 94 | C2×M16 | — | 4 | 2 | no | — |
| 95 | C4⋊Q8 | — | 2 | 2 | no | — |
| 96 | #127 | — | 2 | 2 | no | — |
| 97 | #127 | — | 4 | 2 | no | — |
| 98 | #125 | — | 4 | 2 | no | — |
| 99 | C22⋊Q8 | — | 4 | 2 | no | — |
| 100 | C22⋊Q8 | — | 8 | 2 | no | — |
| 101 | #121 | — | 4 | 2 | no | — |
| 102 | #121 | — | 8 | 2 | no | — |
| 103 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 104 | C4×D4 | — | 2 | 2 | no | — |
| 105 | C4×D4 | — | 4 | 2 | no | — |
| 106 | C4×D4 | — | 4 | 2 | no | — |
| 107 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 108 | D16 | — | 4 | 1 | no | — |
| 109 | D16 | — | 4 | 4 | no | — |
| 110 | D16 | — | 8 | 2 | no | — |
| 111 | C2×C16 | — | 2 | 1 | no | — |
| 112 | C2×C16 | — | 2 | 2 | no | — |
| 113 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 115 | #107 | — | 2 | 2 | no | — |
| 116 | #107 | — | 2 | 2 | no | — |
| 117 | #107 | — | 4 | 2 | no | — |
| 118 | #107 | D8 | 1 | 1 | yes | characteristic |
| 119 | #107 | C2×D4 | 1 | 2 | yes | — |
| 120 | #106 | — | 4 | 2 | no | — |
| 121 | C4⋊C8 | — | 2 | 2 | no | — |
| 122 | C4⋊C8 | — | 2 | 2 | no | — |
| 123 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 124 | #103 | — | 4 | 2 | no | — |
| 125 | #103 | D8 | 1 | 2 | yes | — |
| 126 | #102 | — | 2 | 2 | no | — |
| 127 | #102 | — | 4 | 2 | no | — |
| 128 | #102 | — | 4 | 2 | no | — |
| 129 | #102 | — | 4 | 2 | no | — |
| 130 | #102 | — | 4 | 2 | no | — |
| 131 | #102 | — | 4 | 2 | no | — |
| 132 | C22⋊C8 | — | 4 | 2 | no | — |
| 133 | C4×C8 | — | 2 | 2 | no | — |
| 134 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 135 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 136 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 137 | #41 | — | 8 | 4 | no | — |
| 138 | #41 | — | 16 | 2 | no | — |
| 139 | C2×Q8 | — | 4 | 2 | no | — |
| 140 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 141 | C2×D4 | — | 2 | 2 | no | — |
| 142 | C2×D4 | — | 4 | 2 | no | — |
| 143 | C2×D4 | — | 4 | 2 | no | — |
| 144 | C2×D4 | — | 4 | 2 | no | — |
| 145 | C2×D4 | — | 8 | 2 | no | — |
| 146 | C2×D4 | — | 8 | 2 | no | — |
| 147 | C22×C4 | — | 4 | 2 | no | — |
| 148 | C22×C4 | — | 8 | 2 | no | — |
| 149 | C22×C4 | — | 8 | 2 | no | — |
| 150 | Q16 | — | 4 | 1 | no | — |
| 151 | SD16 | — | 16 | 4 | no | — |
| 152 | SD16 | — | 16 | 2 | no | — |
| 153 | D8 | — | 2 | 2 | no | — |
| 154 | D8 | — | 2 | 2 | no | — |
| 155 | D8 | — | 4 | 2 | no | — |
| 156 | D8 | — | 4 | 2 | no | — |
| 157 | D8 | — | 8 | 2 | no | — |
| 158 | D8 | — | 16 | 2 | no | — |
| 159 | M16 | — | 16 | 2 | no | — |
| 160 | C2×C8 | — | 2 | 1 | no | — |
| 161 | C2×C8 | — | 2 | 2 | no | — |
| 162 | C2×C8 | — | 4 | 2 | no | — |
| 163 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 164 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 165 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 166 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 167 | C4⋊C4 | — | 2 | 2 | no | — |
| 168 | C4⋊C4 | — | 2 | 2 | no | — |
| 169 | C4⋊C4 | — | 4 | 2 | no | — |
| 170 | C4⋊C4 | — | 4 | 2 | no | — |
| 171 | C4⋊C4 | — | 8 | 2 | no | — |
| 172 | C4⋊C4 | — | 8 | 2 | no | — |
| 173 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 174 | C22⋊C4 | — | 4 | 2 | no | — |
| 175 | C22⋊C4 | — | 8 | 2 | no | — |
| 176 | C22⋊C4 | — | 8 | 2 | no | — |
| 177 | C42 | — | 2 | 2 | no | — |
| 178 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 179 | C16 | — | 2 | 2 | no | — |
| 180 | C16 | — | 2 | 1 | no | — |
| 181 | C16 | — | 2 | 1 | no | — |
| 182 | C16 | — | 4 | 2 | no | — |
| 183 | C23 | — | 4 | 2 | no | — |
| 184 | C23 | — | 8 | 2 | no | — |
| 185 | Q8 | — | 2 | 2 | no | — |
| 186 | Q8 | — | 2 | 2 | no | — |
| 187 | Q8 | — | 16 | 2 | no | — |
| 188 | D4 | — | 4 | 2 | no | — |
| 189 | D4 | — | 4 | 2 | no | — |
| 190 | D4 | — | 8 | 2 | no | — |
| 191 | D4 | — | 8 | 2 | no | — |
| 192 | D4 | — | 8 | 2 | no | — |
| 193 | D4 | — | 8 | 2 | no | — |
| 194 | D4 | — | 16 | 2 | no | — |
| 195 | D4 | — | 16 | 4 | no | — |
| 196 | D4 | — | 32 | 2 | no | — |
| 197 | C2×C4 | — | 2 | 2 | no | — |
| 198 | C2×C4 | — | 2 | 2 | no | — |
| 199 | C2×C4 | — | 4 | 2 | no | — |
| 200 | C2×C4 | — | 8 | 2 | no | — |
| 201 | C2×C4 | — | 8 | 2 | no | — |
| 202 | C2×C4 | — | 8 | 2 | no | — |
| 203 | C2×C4 | — | 16 | 4 | no | — |
| 204 | C2×C4 | — | 32 | 2 | no | — |
| 205 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 207 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 208 | C8 | — | 2 | 2 | no | — |
| 209 | C8 | — | 2 | 1 | no | — |
| 210 | C8 | — | 2 | 1 | no | — |
| 211 | C8 | — | 2 | 1 | no | — |
| 212 | C8 | — | 4 | 2 | no | — |
| 213 | C8 | — | 8 | 2 | no | — |
| 214 | C8 | #575 | 1 | 1 | yes | characteristic |
| 215 | C8 | #578 | 1 | 1 | yes | characteristic |
| 216 | C22 | — | 8 | 2 | no | — |
| 217 | C22 | — | 8 | 2 | no | — |
| 218 | C22 | — | 8 | 2 | no | — |
| 219 | C22 | — | 16 | 2 | no | — |
| 220 | C22 | — | 16 | 2 | no | — |
| 221 | C22 | — | 16 | 2 | no | — |
| 222 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 223 | C4 | — | 2 | 1 | no | — |
| 224 | C4 | — | 4 | 2 | no | — |
| 225 | C4 | — | 4 | 2 | no | — |
| 226 | C4 | — | 8 | 2 | no | — |
| 227 | C4 | — | 16 | 2 | no | — |
| 228 | C4 | #3277 | 1 | 1 | yes | characteristic |
| 229 | C4 | #3291 | 1 | 1 | yes | characteristic |
| 230 | C4 | #3414 | 1 | 1 | yes | characteristic |
| 231 | C4 | #3415 | 1 | 1 | yes | characteristic |
| 232 | C2 | — | 16 | 2 | no | — |
| 233 | C2 | — | 32 | 2 | no | — |
| 234 | C2 | #19888 | 1 | 1 | yes | characteristic |
| 235 | C2 | #33839 | 1 | 1 | yes | characteristic |
| 236 | C2 | #33858 | 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. #33933 | C2 | semidirect |
| 2. #19909 | C2 | semidirect |
| 3. #12503 | C2 | semidirect |
| 4. #12485 | C2 | semidirect |
| 5. #12480 | C2 | semidirect |
| 6. #12411 | C2 | semidirect |
| 7. #12328 | C2 | semidirect |
| 8. #12081 | C2 | semidirect |
| 9. #12054 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33933 | 2 |
| #19909 | 2 |
| #12503 | 2 |
| #12485 | 2 |
| #12480 | 2 |
| #12411 | 2 |
| #12328 | 2 |
| #12081 | 2 |
| #12054 | 2 |