Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1312 conjugacy classes of subgroups.
The subgroup lattice has 236 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62360 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #62359 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33959 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | Q8×SD32 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #33929 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #33926 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #33812 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #33736 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | C2×Q8×C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3502 | — | 2 | 1 | no | — |
| 11 | #3497 | — | 2 | 1 | no | — |
| 12 | #3395 | C22 | 1 | 4 | yes | — |
| 13 | #3391 | C22 | 1 | 4 | yes | — |
| 14 | Q8×Q16 | C22 | 1 | 4 | yes | — |
| 15 | Q8×D8 | C22 | 1 | 4 | yes | — |
| 16 | #3363 | — | 2 | 4 | no | — |
| 17 | #3358 | — | 2 | 4 | no | — |
| 18 | #3249 | — | 2 | 3 | no | — |
| 19 | #3245 | — | 2 | 3 | no | — |
| 20 | #3170 | C22 | 1 | 6 | yes | — |
| 21 | #3169 | C22 | 1 | 6 | yes | — |
| 22 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 23 | #2254 | C22 | 1 | 6 | yes | — |
| 24 | #2248 | C22 | 1 | 6 | yes | — |
| 25 | #2241 | C22 | 1 | 6 | yes | — |
| 26 | #2240 | C22 | 1 | 6 | yes | — |
| 27 | #2226 | C22 | 1 | 12 | yes | — |
| 28 | #2225 | C22 | 1 | 12 | yes | — |
| 29 | #2217 | C22 | 1 | 12 | yes | — |
| 30 | #2217 | C22 | 1 | 4 | yes | — |
| 31 | #2208 | C22 | 1 | 12 | yes | — |
| 32 | #2207 | C22 | 1 | 12 | yes | — |
| 33 | Q8×C16 | C22 | 1 | 4 | yes | — |
| 34 | Q8×C16 | C22 | 1 | 4 | yes | — |
| 35 | C4×SD32 | C22 | 1 | 12 | yes | — |
| 36 | C4×SD32 | C22 | 1 | 12 | yes | — |
| 37 | C2×C4⋊C16 | C22 | 1 | 3 | yes | — |
| 38 | C2×C4×C16 | C22 | 1 | 3 | yes | — |
| 39 | C2×2−1+4 | — | 4 | 1 | no | — |
| 40 | Q82 | — | 2 | 4 | no | — |
| 41 | #557 | — | 2 | 4 | no | — |
| 42 | #557 | — | 4 | 4 | no | — |
| 43 | #550 | — | 2 | 4 | no | — |
| 44 | D4×Q8 | — | 2 | 4 | no | — |
| 45 | #548 | — | 4 | 4 | no | — |
| 46 | #544 | — | 4 | 3 | no | — |
| 47 | #537 | — | 4 | 3 | no | — |
| 48 | #533 | — | 2 | 6 | no | — |
| 49 | #532 | — | 2 | 6 | no | — |
| 50 | #519 | — | 4 | 1 | no | — |
| 51 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 52 | C2×SD32 | C23 | 1 | 4 | yes | — |
| 53 | C2×SD32 | C23 | 1 | 12 | yes | — |
| 54 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 55 | C22×C16 | C23 | 1 | 3 | yes | — |
| 56 | #500 | C23 | 1 | 6 | yes | — |
| 57 | #500 | C23 | 1 | 6 | yes | — |
| 58 | #494 | C23 | 1 | 6 | yes | — |
| 59 | #493 | C23 | 1 | 6 | yes | — |
| 60 | #487 | — | 2 | 6 | no | — |
| 61 | #486 | — | 2 | 6 | no | — |
| 62 | #477 | — | 2 | 12 | no | — |
| 63 | #474 | — | 2 | 12 | no | — |
| 64 | #467 | — | 2 | 6 | no | — |
| 65 | #467 | C23 | 1 | 4 | yes | — |
| 66 | #466 | — | 2 | 6 | no | — |
| 67 | #466 | C23 | 1 | 4 | yes | — |
| 68 | #462 | — | 2 | 12 | no | — |
| 69 | #459 | — | 2 | 12 | no | — |
| 70 | C8×Q8 | D4 | 1 | 4 | yes | — |
| 71 | C8×Q8 | C23 | 1 | 4 | yes | — |
| 72 | C4×Q16 | — | 2 | 6 | no | — |
| 73 | C4×Q16 | C23 | 1 | 12 | yes | — |
| 74 | C4×D8 | — | 2 | 6 | no | — |
| 75 | C4×D8 | C23 | 1 | 12 | yes | — |
| 76 | C2×C4⋊C8 | D4 | 1 | 3 | yes | — |
| 77 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 78 | #367 | C23 | 1 | 4 | yes | — |
| 79 | #367 | C23 | 1 | 12 | yes | — |
| 80 | C4⋊C16 | C23 | 1 | 12 | yes | — |
| 81 | #358 | C23 | 1 | 4 | yes | — |
| 82 | #358 | C23 | 1 | 12 | yes | — |
| 83 | #357 | C23 | 1 | 4 | yes | — |
| 84 | #357 | C23 | 1 | 12 | yes | — |
| 85 | C4×C16 | C23 | 1 | 6 | yes | — |
| 86 | C4×C16 | C23 | 1 | 6 | yes | — |
| 87 | 2−1+4 | — | 8 | 8 | no | — |
| 88 | C2×#41 | — | 4 | 4 | no | — |
| 89 | C2×#41 | — | 4 | 6 | no | — |
| 90 | C22×Q8 | — | 4 | 4 | no | — |
| 91 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 92 | C2×Q16 | — | 2 | 6 | no | — |
| 93 | C2×Q16 | C24 | 1 | 4 | yes | — |
| 94 | C2×D8 | — | 2 | 6 | no | — |
| 95 | C2×D8 | C24 | 1 | 4 | yes | — |
| 96 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 97 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 98 | C4⋊Q8 | — | 2 | 6 | no | — |
| 99 | C4⋊Q8 | — | 2 | 6 | no | — |
| 100 | C4⋊Q8 | — | 2 | 6 | no | — |
| 101 | C4⋊Q8 | — | 4 | 12 | no | — |
| 102 | #127 | — | 2 | 6 | no | — |
| 103 | #125 | — | 4 | 12 | no | — |
| 104 | #125 | — | 4 | 6 | no | — |
| 105 | #124 | — | 4 | 6 | no | — |
| 106 | C22⋊Q8 | — | 2 | 4 | no | — |
| 107 | C22⋊Q8 | — | 4 | 6 | no | — |
| 108 | C22⋊Q8 | — | 4 | 12 | no | — |
| 109 | #121 | — | 2 | 4 | no | — |
| 110 | #121 | — | 4 | 6 | no | — |
| 111 | #121 | — | 4 | 12 | no | — |
| 112 | C4×Q8 | — | 2 | 12 | no | — |
| 113 | C4×Q8 | — | 4 | 4 | no | — |
| 114 | C4×Q8 | — | 4 | 4 | no | — |
| 115 | C4×Q8 | — | 4 | 6 | no | — |
| 116 | C4×Q8 | D8 | 1 | 4 | yes | — |
| 117 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 118 | C4×D4 | — | 2 | 12 | no | — |
| 119 | C4×D4 | — | 4 | 6 | no | — |
| 120 | #117 | — | 4 | 6 | no | — |
| 121 | C2×C4⋊C4 | D8 | 1 | 3 | yes | — |
| 122 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 123 | SD32 | — | 2 | 8 | no | — |
| 124 | SD32 | — | 4 | 12 | no | — |
| 125 | C2×C16 | — | 2 | 2 | no | — |
| 126 | C2×C16 | — | 4 | 3 | no | — |
| 127 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 128 | C2×C16 | C24 | 1 | 6 | yes | — |
| 129 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 130 | #107 | — | 2 | 6 | no | — |
| 131 | #107 | — | 2 | 6 | no | — |
| 132 | #107 | C24 | 1 | 4 | yes | — |
| 133 | #107 | C24 | 1 | 4 | yes | — |
| 134 | C4⋊C8 | C2×D4 | 1 | 12 | yes | — |
| 135 | #103 | — | 2 | 4 | no | — |
| 136 | #103 | — | 2 | 12 | no | — |
| 137 | #102 | — | 2 | 4 | no | — |
| 138 | #102 | — | 2 | 12 | no | — |
| 139 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 140 | C4×C8 | C24 | 1 | 6 | yes | — |
| 141 | #41 | — | 8 | 16 | no | — |
| 142 | #41 | — | 8 | 24 | no | — |
| 143 | C2×Q8 | — | 2 | 4 | no | — |
| 144 | C2×Q8 | — | 4 | 6 | no | — |
| 145 | C2×Q8 | — | 4 | 6 | no | — |
| 146 | C2×Q8 | — | 8 | 16 | no | — |
| 147 | C2×Q8 | SD32 | 1 | 8 | yes | complemented |
| 148 | C2×Q8 | C2×D8 | 1 | 4 | yes | — |
| 149 | C2×D4 | — | 2 | 4 | no | — |
| 150 | C2×D4 | — | 4 | 4 | no | — |
| 151 | C2×D4 | — | 4 | 6 | no | — |
| 152 | C2×D4 | — | 4 | 6 | no | — |
| 153 | C22×C4 | — | 4 | 12 | no | — |
| 154 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 155 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 156 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 157 | Q16 | — | 2 | 4 | no | — |
| 158 | Q16 | — | 2 | 4 | no | — |
| 159 | Q16 | — | 8 | 6 | no | — |
| 160 | D8 | — | 2 | 4 | no | — |
| 161 | D8 | — | 2 | 4 | no | — |
| 162 | D8 | — | 8 | 6 | no | — |
| 163 | C2×C8 | — | 2 | 1 | no | — |
| 164 | C2×C8 | — | 2 | 1 | no | — |
| 165 | C2×C8 | — | 4 | 3 | no | — |
| 166 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 167 | C2×C8 | C22×D4 | 1 | 6 | yes | — |
| 168 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 169 | C4⋊C4 | — | 2 | 4 | no | — |
| 170 | C4⋊C4 | — | 2 | 4 | no | — |
| 171 | C4⋊C4 | — | 4 | 6 | no | — |
| 172 | C4⋊C4 | — | 4 | 6 | no | — |
| 173 | C4⋊C4 | — | 4 | 6 | no | — |
| 174 | C4⋊C4 | — | 4 | 12 | no | — |
| 175 | C4⋊C4 | — | 4 | 4 | no | — |
| 176 | C4⋊C4 | — | 4 | 6 | no | — |
| 177 | C4⋊C4 | C2×D8 | 1 | 12 | yes | — |
| 178 | C22⋊C4 | — | 4 | 4 | no | — |
| 179 | C22⋊C4 | — | 4 | 12 | no | — |
| 180 | C22⋊C4 | — | 4 | 4 | no | — |
| 181 | C42 | — | 4 | 12 | no | — |
| 182 | C42 | C2×D8 | 1 | 6 | yes | — |
| 183 | C42 | C22×D4 | 1 | 6 | yes | — |
| 184 | C16 | — | 2 | 1 | no | — |
| 185 | C16 | — | 2 | 6 | no | — |
| 186 | C16 | 2−1+4 | 1 | 2 | yes | complemented |
| 187 | C23 | — | 4 | 4 | no | — |
| 188 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 4 | 4 | no | — |
| 190 | Q8 | — | 4 | 4 | no | — |
| 191 | Q8 | — | 8 | 12 | no | — |
| 192 | Q8 | — | 16 | 6 | no | — |
| 193 | Q8 | C2×SD32 | 1 | 16 | yes | — |
| 194 | D4 | — | 4 | 4 | no | — |
| 195 | D4 | — | 4 | 4 | no | — |
| 196 | D4 | — | 8 | 8 | no | — |
| 197 | D4 | — | 8 | 12 | no | — |
| 198 | D4 | — | 16 | 6 | no | — |
| 199 | C2×C4 | — | 2 | 1 | no | — |
| 200 | C2×C4 | — | 2 | 1 | no | — |
| 201 | C2×C4 | — | 4 | 3 | no | — |
| 202 | C2×C4 | — | 4 | 4 | no | — |
| 203 | C2×C4 | — | 4 | 4 | no | — |
| 204 | C2×C4 | — | 4 | 4 | no | — |
| 205 | C2×C4 | — | 8 | 24 | no | — |
| 206 | C2×C4 | C2×SD32 | 1 | 12 | yes | — |
| 207 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 209 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 210 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 211 | C8 | — | 2 | 6 | no | — |
| 212 | C8 | — | 2 | 1 | no | — |
| 213 | C8 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 214 | C8 | #585 | 1 | 1 | yes | characteristic |
| 215 | C22 | — | 2 | 1 | no | — |
| 216 | C22 | — | 2 | 1 | no | — |
| 217 | C22 | — | 8 | 4 | no | — |
| 218 | C22 | — | 8 | 4 | no | — |
| 219 | C22 | — | 8 | 4 | no | — |
| 220 | C22 | C22×SD32 | 1 | 2 | yes | — |
| 221 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 222 | C4 | — | 2 | 1 | no | — |
| 223 | C4 | — | 2 | 6 | no | — |
| 224 | C4 | — | 8 | 4 | no | — |
| 225 | C4 | — | 8 | 4 | no | — |
| 226 | C4 | — | 8 | 4 | no | — |
| 227 | C4 | C22×SD32 | 1 | 12 | yes | — |
| 228 | C4 | #3497 | 1 | 1 | yes | characteristic |
| 229 | C4 | #3584 | 1 | 1 | yes | characteristic |
| 230 | C2 | — | 2 | 2 | no | — |
| 231 | C2 | — | 16 | 4 | no | — |
| 232 | C2 | #62359 | 1 | 1 | yes | characteristic |
| 233 | C2 | C23×SD32 | 1 | 1 | yes | characteristic |
| 234 | C2 | #62635 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #62359 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #62360 | C2 | semidirect |
| 3. #33959 | C2 | semidirect |
| 4. Q8×SD32 | C2 | semidirect |
| 5. #33929 | C2 | semidirect |
| 6. #33926 | C2 | semidirect |
| 7. #33812 | C2 | semidirect |
| 8. #33736 | C2 | semidirect |
| 9. C2×Q8×C16 | C2 | semidirect |
| 147. C2×Q8 | SD32 | semidirect |
| 186. C16 | 2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62360 | 2 |
| #62359 | 2 |
| #33959 | 2 |
| Q8×SD32 | 2 |
| #33929 | 2 |
| #33926 | 2 |
| #33812 | 2 |
| #33736 | 2 |
| C2×Q8×C16 | 2 |