Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
850 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 850 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C4×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #9640 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #7567 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×H(3,𝔽4) | C4 | 1 | 1 | yes | characteristic |
| 5 | #2379 | — | 2 | 3 | no | — |
| 6 | #2379 | C4 | 1 | 3 | yes | — |
| 7 | #2375 | — | 2 | 3 | no | — |
| 8 | C4×#124 | — | 2 | 3 | no | — |
| 9 | C4×#124 | C22 | 1 | 3 | yes | — |
| 10 | C4×#120 | — | 2 | 3 | no | — |
| 11 | #1842 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | #1744 | C22 | 1 | 3 | yes | complemented |
| 13 | C2×Q8⋊C8 | — | 2 | 3 | no | — |
| 14 | C2×D4⋊C8 | — | 2 | 3 | no | — |
| 15 | #1272 | — | 2 | 3 | no | — |
| 16 | #1270 | — | 2 | 3 | no | — |
| 17 | C24×C4 | — | 2 | 1 | no | — |
| 18 | H(3,𝔽4) | — | 2 | 6 | no | — |
| 19 | H(3,𝔽4) | C8 | 1 | 4 | yes | complemented |
| 20 | C2×#124 | — | 2 | 3 | no | — |
| 21 | C2×#124 | C2×C4 | 1 | 3 | yes | — |
| 22 | C2×#120 | — | 2 | 3 | no | — |
| 23 | C2×C4×Q8 | D4 | 1 | 3 | yes | — |
| 24 | C2×C4×D4 | — | 2 | 3 | no | — |
| 25 | C2×C4×D4 | D4 | 1 | 3 | yes | — |
| 26 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 27 | C2×C4×C8 | — | 2 | 3 | no | — |
| 28 | #391 | — | 2 | 3 | no | — |
| 29 | #391 | D4 | 1 | 3 | yes | — |
| 30 | #390 | — | 2 | 3 | no | — |
| 31 | #390 | D4 | 1 | 3 | yes | — |
| 32 | #388 | — | 2 | 6 | no | — |
| 33 | #388 | — | 2 | 6 | no | — |
| 34 | #388 | — | 2 | 6 | no | — |
| 35 | #386 | — | 2 | 6 | no | — |
| 36 | #386 | — | 2 | 3 | no | — |
| 37 | #385 | — | 2 | 3 | no | — |
| 38 | #385 | — | 2 | 6 | no | — |
| 39 | #382 | — | 2 | 3 | no | — |
| 40 | #382 | C2×C4 | 1 | 3 | yes | — |
| 41 | #379 | — | 2 | 3 | no | — |
| 42 | C4×C22⋊C4 | — | 2 | 3 | no | — |
| 43 | C4×C22⋊C4 | — | 2 | 6 | no | — |
| 44 | C43 | — | 2 | 1 | no | — |
| 45 | C43 | C23 | 1 | 1 | yes | characteristic |
| 46 | #336 | — | 2 | 6 | no | — |
| 47 | Q8⋊C8 | — | 4 | 6 | no | — |
| 48 | D4⋊C8 | — | 4 | 6 | no | — |
| 49 | C25 | — | 2 | 1 | no | — |
| 50 | C22×Q8 | C22⋊C4 | 1 | 3 | yes | — |
| 51 | C22×D4 | — | 2 | 3 | no | — |
| 52 | C22×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 53 | C23×C4 | — | 2 | 3 | no | — |
| 54 | C23×C4 | — | 2 | 3 | no | — |
| 55 | C23×C4 | — | 8 | 3 | no | — |
| 56 | C23×C4 | — | 8 | 3 | no | — |
| 57 | C22×C8 | — | 4 | 3 | no | — |
| 58 | C22×C8 | — | 4 | 1 | no | — |
| 59 | #124 | — | 2 | 12 | no | — |
| 60 | #124 | — | 2 | 12 | no | — |
| 61 | #124 | — | 2 | 6 | no | — |
| 62 | #124 | C2×C8 | 1 | 6 | yes | — |
| 63 | #124 | M16 | 1 | 6 | yes | — |
| 64 | #120 | — | 2 | 12 | no | — |
| 65 | #120 | — | 2 | 12 | no | — |
| 66 | C4×Q8 | — | 4 | 3 | no | — |
| 67 | C4×Q8 | — | 4 | 3 | no | — |
| 68 | C4×D4 | — | 4 | 3 | no | — |
| 69 | C4×D4 | — | 4 | 3 | no | — |
| 70 | C4×D4 | — | 8 | 3 | no | — |
| 71 | C4×D4 | — | 8 | 3 | no | — |
| 72 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 73 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 74 | C2×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 75 | C2×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 76 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 77 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 78 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 79 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 80 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 81 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 82 | C2×C42 | — | 2 | 1 | no | — |
| 83 | C2×C42 | — | 2 | 3 | no | — |
| 84 | C2×C42 | — | 2 | 3 | no | — |
| 85 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 87 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 88 | C4⋊C8 | — | 4 | 3 | no | — |
| 89 | C4⋊C8 | — | 4 | 3 | no | — |
| 90 | C4×C8 | — | 2 | 6 | no | — |
| 91 | C4×C8 | — | 4 | 3 | no | — |
| 92 | #95 | — | 2 | 6 | no | — |
| 93 | #95 | — | 2 | 6 | no | — |
| 94 | #95 | — | 2 | 6 | no | — |
| 95 | C24 | — | 2 | 4 | no | — |
| 96 | C24 | — | 2 | 3 | no | — |
| 97 | C24 | — | 8 | 3 | no | — |
| 98 | C24 | — | 8 | 3 | no | — |
| 99 | C2×Q8 | — | 2 | 3 | no | — |
| 100 | C2×Q8 | — | 4 | 3 | no | — |
| 101 | C2×Q8 | — | 4 | 3 | no | — |
| 102 | C2×Q8 | C22⋊C8 | 1 | 6 | yes | — |
| 103 | C2×D4 | — | 2 | 12 | no | — |
| 104 | C2×D4 | — | 2 | 3 | no | — |
| 105 | C2×D4 | — | 4 | 3 | no | — |
| 106 | C2×D4 | — | 4 | 3 | no | — |
| 107 | C2×D4 | — | 8 | 3 | no | — |
| 108 | C2×D4 | — | 8 | 3 | no | — |
| 109 | C2×D4 | C22⋊C8 | 1 | 6 | yes | — |
| 110 | C22×C4 | — | 2 | 3 | no | — |
| 111 | C22×C4 | — | 2 | 3 | no | — |
| 112 | C22×C4 | — | 2 | 3 | no | — |
| 113 | C22×C4 | — | 4 | 3 | no | — |
| 114 | C22×C4 | — | 4 | 6 | no | — |
| 115 | C22×C4 | — | 4 | 3 | no | — |
| 116 | C22×C4 | — | 4 | 6 | no | — |
| 117 | C22×C4 | — | 8 | 6 | no | — |
| 118 | C22×C4 | — | 8 | 3 | no | — |
| 119 | C22×C4 | — | 8 | 4 | no | — |
| 120 | C22×C4 | — | 8 | 12 | no | — |
| 121 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 122 | C22×C4 | #120 | 1 | 3 | yes | — |
| 123 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 124 | C2×C8 | — | 4 | 6 | no | — |
| 125 | C2×C8 | — | 8 | 6 | no | — |
| 126 | C2×C8 | — | 8 | 3 | no | — |
| 127 | C4⋊C4 | — | 4 | 3 | no | — |
| 128 | C4⋊C4 | — | 4 | 3 | no | — |
| 129 | C4⋊C4 | — | 8 | 3 | no | — |
| 130 | C4⋊C4 | — | 8 | 3 | no | — |
| 131 | C22⋊C4 | — | 2 | 12 | no | — |
| 132 | C22⋊C4 | — | 2 | 12 | no | — |
| 133 | C22⋊C4 | — | 2 | 12 | no | — |
| 134 | C22⋊C4 | — | 8 | 6 | no | — |
| 135 | C22⋊C4 | — | 8 | 3 | no | — |
| 136 | C22⋊C4 | — | 8 | 3 | no | — |
| 137 | C22⋊C4 | — | 8 | 6 | no | — |
| 138 | C42 | — | 2 | 4 | no | — |
| 139 | C42 | — | 2 | 3 | no | — |
| 140 | C42 | — | 2 | 3 | no | — |
| 141 | C42 | — | 4 | 3 | no | — |
| 142 | C42 | — | 4 | 3 | no | — |
| 143 | C42 | — | 4 | 3 | no | — |
| 144 | C42 | — | 8 | 3 | no | — |
| 145 | C42 | C22×C8 | 1 | 1 | yes | characteristic |
| 146 | C42 | C2×M16 | 1 | 3 | yes | — |
| 147 | C23 | — | 2 | 6 | no | — |
| 148 | C23 | — | 4 | 3 | no | — |
| 149 | C23 | — | 4 | 6 | no | — |
| 150 | C23 | — | 4 | 3 | no | — |
| 151 | C23 | — | 4 | 6 | no | — |
| 152 | C23 | — | 8 | 12 | no | — |
| 153 | C23 | — | 8 | 4 | no | — |
| 154 | C23 | — | 8 | 12 | no | — |
| 155 | C23 | #379 | 1 | 1 | yes | characteristic |
| 156 | Q8 | — | 4 | 6 | no | — |
| 157 | Q8 | — | 8 | 3 | no | — |
| 158 | D4 | — | 4 | 6 | no | — |
| 159 | D4 | — | 8 | 3 | no | — |
| 160 | D4 | — | 8 | 12 | no | — |
| 161 | C2×C4 | — | 2 | 3 | no | — |
| 162 | C2×C4 | — | 2 | 3 | no | — |
| 163 | C2×C4 | — | 2 | 3 | no | — |
| 164 | C2×C4 | — | 2 | 6 | no | — |
| 165 | C2×C4 | — | 2 | 6 | no | — |
| 166 | C2×C4 | — | 4 | 3 | no | — |
| 167 | C2×C4 | — | 4 | 3 | no | — |
| 168 | C2×C4 | — | 4 | 3 | no | — |
| 169 | C2×C4 | — | 4 | 3 | no | — |
| 170 | C2×C4 | — | 4 | 6 | no | — |
| 171 | C2×C4 | — | 4 | 6 | no | — |
| 172 | C2×C4 | — | 8 | 12 | no | — |
| 173 | C2×C4 | — | 8 | 3 | no | — |
| 174 | C2×C4 | — | 8 | 3 | no | — |
| 175 | C2×C4 | — | 8 | 6 | no | — |
| 176 | C2×C4 | C2×C22⋊C8 | 1 | 3 | yes | — |
| 177 | C2×C4 | #407 | 1 | 3 | yes | — |
| 178 | C2×C4 | #453 | 1 | 3 | yes | — |
| 179 | C2×C4 | #455 | 1 | 3 | yes | — |
| 180 | C8 | — | 8 | 6 | no | — |
| 181 | C8 | — | 16 | 1 | no | — |
| 182 | C22 | — | 4 | 6 | no | — |
| 183 | C22 | — | 4 | 12 | no | — |
| 184 | C22 | — | 8 | 16 | no | — |
| 185 | C22 | — | 8 | 3 | no | — |
| 186 | C22 | — | 8 | 6 | no | — |
| 187 | C22 | — | 8 | 3 | no | — |
| 188 | C22 | #1779 | 1 | 1 | yes | characteristic |
| 189 | C22 | #1792 | 1 | 3 | yes | — |
| 190 | C22 | #1794 | 1 | 3 | yes | — |
| 191 | C4 | — | 2 | 3 | no | — |
| 192 | C4 | — | 2 | 3 | no | — |
| 193 | C4 | — | 4 | 3 | no | — |
| 194 | C4 | — | 4 | 6 | no | — |
| 195 | C4 | — | 8 | 3 | no | — |
| 196 | C4 | — | 8 | 3 | no | — |
| 197 | C4 | #1655 | 1 | 1 | yes | characteristic |
| 198 | C4 | #1659 | 1 | 3 | yes | — |
| 199 | C2 | — | 8 | 6 | no | — |
| 200 | C2 | #8318 | 1 | 3 | yes | — |
| 201 | C2 | #8504 | 1 | 3 | yes | — |
| 202 | C2 | #8506 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #379 |
| prime radical | C2×H(3,𝔽4) | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #9640 | C2 | semidirect |
| 3. #7567 | C2 | semidirect |
| 11. #1842 | C22 | semidirect |
| 12. #1744 | C22 | semidirect |
| 19. H(3,𝔽4) | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×H(3,𝔽4) | 2 |
| #9640 | 2 |
| #7567 | 2 |