Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
419 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 51 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 | D8×#272 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11720838 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #63653⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11720253 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11720251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | D3×#826 | — | 3 | 2 | no | maximal |
| 7 | D4×#272 | — | 2 | 1 | no | — |
| 8 | #88655 | — | 2 | 1 | no | — |
| 9 | D8×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #88545 | — | 2 | 1 | no | — |
| 11 | #88529 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C8×#272 | C22 | 1 | 1 | yes | characteristic |
| 13 | #3745⋊C3 | C22 | 1 | 2 | yes | complemented |
| 14 | #88432 | — | 2 | 1 | no | — |
| 15 | (C3⋊(C3⋊C16))⋊C3 | C22 | 1 | 2 | yes | complemented |
| 16 | C3×D3×D8 | — | 3 | 2 | no | — |
| 17 | C3×D3×D8 | — | 6 | 1 | no | — |
| 18 | D3×D24 | — | 3 | 2 | no | — |
| 19 | #63653 | D3 | 1 | 2 | yes | complemented |
| 20 | C3×#826 | — | 3 | 2 | no | — |
| 21 | #63546 | — | 3 | 2 | no | — |
| 22 | #63545 | — | 3 | 2 | no | — |
| 23 | D3×C3⋊C16 | — | 3 | 2 | no | — |
| 24 | C22×#272 | — | 4 | 1 | no | — |
| 25 | C2×((C3⋊D3)⋊C3)⋊C2 | — | 4 | 1 | no | — |
| 26 | D4×H(3,𝔽3) | — | 2 | 1 | no | — |
| 27 | #6235 | — | 4 | 1 | no | — |
| 28 | C4⋊#272 | — | 2 | 1 | no | — |
| 29 | C4×#272 | D4 | 1 | 1 | yes | characteristic |
| 30 | C4⋊((C3⋊D3)⋊C3) | — | 2 | 2 | no | — |
| 31 | #6198 | — | 4 | 1 | no | — |
| 32 | C8×H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 33 | #6180 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×#826 | — | 9 | 1 | no | — |
| 35 | D3×D16 | — | 9 | 2 | no | — |
| 36 | C3×D3×D4 | — | 6 | 2 | no | — |
| 37 | C3×D3×D4 | — | 6 | 2 | no | — |
| 38 | D3×D12 | — | 6 | 2 | no | — |
| 39 | C32×D8 | — | 2 | 1 | no | — |
| 40 | C32×D8 | C2×D3 | 1 | 2 | yes | complemented |
| 41 | #3745 | C2×D3 | 1 | 2 | yes | complemented |
| 42 | C3×#213 | — | 6 | 2 | no | — |
| 43 | C3×#213 | — | 6 | 2 | no | — |
| 44 | C3×D24 | — | 3 | 2 | no | — |
| 45 | C3×D24 | — | 3 | 2 | no | — |
| 46 | C3×D24 | — | 6 | 1 | no | — |
| 47 | C3×D3×C8 | — | 3 | 2 | no | — |
| 48 | C3×D3×C8 | — | 6 | 1 | no | — |
| 49 | #3714 | — | 6 | 2 | no | — |
| 50 | C3⋊(C3⋊C16) | C2×D3 | 1 | 2 | yes | complemented |
| 51 | C3×C3⋊C16 | — | 3 | 2 | no | — |
| 52 | C22×H(3,𝔽3) | — | 4 | 1 | no | — |
| 53 | C2×#272 | — | 4 | 1 | no | — |
| 54 | C2×#272 | — | 8 | 2 | no | — |
| 55 | C2×#272 | D8 | 1 | 1 | yes | characteristic |
| 56 | C2×(C3⋊D3)⋊C3 | — | 4 | 2 | no | — |
| 57 | ((C3⋊D3)⋊C3)⋊C2 | — | 8 | 2 | no | — |
| 58 | C4×H(3,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | #1084 | D8 | 1 | 1 | yes | characteristic |
| 60 | C2×C3×D8 | — | 9 | 1 | no | — |
| 61 | D3×D8 | — | 3 | 2 | no | — |
| 62 | D3×D8 | — | 6 | 1 | no | — |
| 63 | D3×D8 | — | 9 | 2 | no | — |
| 64 | C2×D24 | — | 9 | 1 | no | — |
| 65 | C3×D16 | — | 9 | 2 | no | — |
| 66 | #826 | — | 3 | 2 | no | — |
| 67 | #826 | — | 9 | 2 | no | — |
| 68 | #826 | — | 9 | 2 | no | — |
| 69 | #826 | — | 9 | 2 | no | — |
| 70 | C2×C3⋊C16 | — | 9 | 1 | no | — |
| 71 | D48 | — | 9 | 2 | no | — |
| 72 | D3×C16 | — | 9 | 2 | no | — |
| 73 | C22×C3×D3 | — | 12 | 2 | no | — |
| 74 | C22×C3×D3 | — | 12 | 2 | no | — |
| 75 | C2×D32 | — | 12 | 2 | no | — |
| 76 | C32×D4 | — | 2 | 2 | no | — |
| 77 | C32×D4 | — | 2 | 2 | no | — |
| 78 | C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 79 | C3×#67 | — | 12 | 2 | no | — |
| 80 | C3×#67 | — | 12 | 2 | no | — |
| 81 | C3×D12 | — | 6 | 2 | no | — |
| 82 | C3×D12 | — | 6 | 2 | no | — |
| 83 | C3×D12 | — | 6 | 2 | no | — |
| 84 | C3×C4×D3 | — | 3 | 2 | no | — |
| 85 | C3×C4×D3 | — | 6 | 1 | no | — |
| 86 | #626 | — | 12 | 2 | no | — |
| 87 | C32×C8 | — | 2 | 1 | no | — |
| 88 | C32×C8 | C22×D3 | 1 | 2 | yes | — |
| 89 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 90 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 91 | C2×D16 | — | 27 | 1 | no | Sylow |
| 92 | C2×H(3,𝔽3) | — | 8 | 1 | no | — |
| 93 | C2×H(3,𝔽3) | C2×D8 | 1 | 1 | yes | characteristic |
| 94 | #272 | — | 8 | 1 | no | — |
| 95 | #272 | D16 | 1 | 2 | yes | complemented |
| 96 | (C3⋊D3)⋊C3 | — | 8 | 2 | no | — |
| 97 | C2×C3×D4 | — | 18 | 1 | no | — |
| 98 | D3×D4 | — | 6 | 2 | no | — |
| 99 | D3×D4 | — | 6 | 2 | no | — |
| 100 | D3×D4 | — | 18 | 2 | no | — |
| 101 | C2×D12 | — | 18 | 1 | no | — |
| 102 | C3×D8 | — | 3 | 2 | no | — |
| 103 | C3×D8 | — | 6 | 1 | no | — |
| 104 | C3×D8 | — | 9 | 2 | no | — |
| 105 | C3×D8 | — | 9 | 1 | no | — |
| 106 | C3×D8 | — | 18 | 1 | no | — |
| 107 | C3×D8 | D32 | 1 | 1 | yes | characteristic |
| 108 | C2×C3×C8 | — | 9 | 1 | no | — |
| 109 | #213 | — | 6 | 2 | no | — |
| 110 | #213 | — | 6 | 2 | no | — |
| 111 | #213 | — | 18 | 2 | no | — |
| 112 | D24 | — | 3 | 2 | no | — |
| 113 | D24 | — | 3 | 2 | no | — |
| 114 | D24 | — | 6 | 1 | no | — |
| 115 | D24 | — | 9 | 2 | no | — |
| 116 | D24 | — | 18 | 1 | no | — |
| 117 | D3×C8 | — | 3 | 2 | no | — |
| 118 | D3×C8 | — | 6 | 1 | no | — |
| 119 | C3×C16 | — | 9 | 2 | no | — |
| 120 | C3⋊C16 | — | 3 | 2 | no | — |
| 121 | C3⋊C16 | — | 9 | 2 | no | — |
| 122 | C22×C32 | — | 4 | 2 | no | — |
| 123 | C22×C32 | — | 4 | 2 | no | — |
| 124 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 125 | C2×C3×D3 | — | 3 | 2 | no | — |
| 126 | C2×C3×D3 | — | 6 | 1 | no | — |
| 127 | C2×C3×D3 | — | 12 | 2 | no | — |
| 128 | C2×C3×D3 | — | 12 | 2 | no | — |
| 129 | C2×C3×D3 | — | 12 | 2 | no | — |
| 130 | C2×C3×D3 | — | 24 | 4 | no | — |
| 131 | C2×C3×D3 | — | 24 | 4 | no | — |
| 132 | D32 | — | 24 | 4 | no | — |
| 133 | C32×C4 | — | 2 | 1 | no | — |
| 134 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 135 | C3×Dic3 | — | 3 | 2 | no | — |
| 136 | C3×Dic3 | — | 6 | 1 | no | — |
| 137 | C2×D8 | — | 9 | 1 | no | — |
| 138 | C2×D8 | — | 27 | 1 | no | — |
| 139 | D16 | — | 9 | 2 | no | — |
| 140 | D16 | — | 27 | 2 | no | — |
| 141 | C2×C16 | — | 27 | 1 | no | — |
| 142 | H(3,𝔽3) | C2×D16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 143 | C23×C3 | — | 36 | 1 | no | — |
| 144 | C22×D3 | — | 12 | 2 | no | — |
| 145 | C22×D3 | — | 12 | 2 | no | — |
| 146 | C22×D3 | — | 36 | 2 | no | — |
| 147 | C22×D3 | — | 36 | 1 | no | — |
| 148 | C3×D4 | — | 2 | 1 | no | — |
| 149 | C3×D4 | — | 6 | 2 | no | — |
| 150 | C3×D4 | — | 6 | 2 | no | — |
| 151 | C3×D4 | — | 18 | 2 | no | — |
| 152 | C3×D4 | — | 18 | 1 | no | — |
| 153 | C3×D4 | — | 36 | 1 | no | — |
| 154 | C2×C3×C4 | — | 9 | 1 | no | — |
| 155 | #67 | — | 12 | 2 | no | — |
| 156 | #67 | — | 12 | 2 | no | — |
| 157 | #67 | — | 36 | 2 | no | — |
| 158 | D12 | — | 6 | 2 | no | — |
| 159 | D12 | — | 6 | 2 | no | — |
| 160 | D12 | — | 6 | 2 | no | — |
| 161 | D12 | — | 18 | 2 | no | — |
| 162 | D12 | — | 36 | 1 | no | — |
| 163 | C4×D3 | — | 3 | 2 | no | — |
| 164 | C4×D3 | — | 6 | 1 | no | — |
| 165 | C3×C8 | — | 3 | 2 | no | — |
| 166 | C3×C8 | — | 6 | 1 | no | — |
| 167 | C3×C8 | — | 9 | 1 | no | — |
| 168 | C3×C8 | C2×D32 | 1 | 1 | yes | characteristic |
| 169 | C3⋊C8 | — | 3 | 2 | no | — |
| 170 | C3⋊C8 | — | 6 | 1 | no | — |
| 171 | C2×C32 | — | 2 | 1 | no | — |
| 172 | C2×C32 | — | 8 | 2 | no | — |
| 173 | C2×C32 | — | 8 | 2 | no | — |
| 174 | C2×C32 | D3×D8 | 1 | 2 | yes | — |
| 175 | C3⋊D3 | — | 8 | 2 | no | — |
| 176 | C3×D3 | — | 3 | 4 | no | — |
| 177 | C3×D3 | — | 6 | 2 | no | — |
| 178 | C3×D3 | — | 24 | 2 | no | — |
| 179 | C3×D3 | — | 24 | 2 | no | — |
| 180 | C3×D3 | — | 24 | 2 | no | — |
| 181 | C2×D4 | — | 18 | 1 | no | — |
| 182 | C2×D4 | — | 54 | 1 | no | — |
| 183 | D8 | — | 9 | 2 | no | — |
| 184 | D8 | — | 9 | 1 | no | — |
| 185 | D8 | — | 18 | 1 | no | — |
| 186 | D8 | — | 54 | 1 | no | — |
| 187 | D8 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 188 | C2×C8 | — | 9 | 1 | no | — |
| 189 | C16 | — | 9 | 2 | no | — |
| 190 | C22×C3 | — | 4 | 1 | no | — |
| 191 | C22×C3 | — | 9 | 1 | no | — |
| 192 | C22×C3 | — | 12 | 2 | no | — |
| 193 | C22×C3 | — | 12 | 2 | no | — |
| 194 | C22×C3 | — | 36 | 2 | no | — |
| 195 | C22×C3 | — | 36 | 1 | no | — |
| 196 | C22×C3 | — | 72 | 2 | no | — |
| 197 | C2×D3 | — | 3 | 2 | no | — |
| 198 | C2×D3 | — | 6 | 1 | no | — |
| 199 | C2×D3 | — | 12 | 2 | no | — |
| 200 | C2×D3 | — | 12 | 2 | no | — |
| 201 | C2×D3 | — | 12 | 2 | no | — |
| 202 | C2×D3 | — | 24 | 4 | no | — |
| 203 | C2×D3 | — | 24 | 4 | no | — |
| 204 | C2×D3 | — | 36 | 2 | no | — |
| 205 | C2×D3 | — | 72 | 4 | no | — |
| 206 | C2×D3 | — | 72 | 2 | no | — |
| 207 | C3×C4 | — | 3 | 2 | no | — |
| 208 | C3×C4 | — | 6 | 1 | no | — |
| 209 | C3×C4 | — | 9 | 1 | no | — |
| 210 | C3×C4 | #3811 | 1 | 1 | yes | characteristic |
| 211 | Dic3 | — | 3 | 2 | no | — |
| 212 | Dic3 | — | 6 | 1 | no | — |
| 213 | C32 | — | 2 | 1 | no | — |
| 214 | C32 | D3×D16 | 1 | 2 | yes | complemented |
| 215 | C23 | — | 36 | 1 | no | — |
| 216 | C23 | — | 108 | 1 | no | — |
| 217 | D4 | — | 2 | 1 | no | — |
| 218 | D4 | — | 18 | 1 | no | — |
| 219 | D4 | — | 18 | 2 | no | — |
| 220 | D4 | — | 36 | 1 | no | — |
| 221 | D4 | — | 108 | 1 | no | — |
| 222 | C2×C4 | — | 9 | 1 | no | — |
| 223 | C8 | — | 9 | 1 | no | — |
| 224 | C8 | C2×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 225 | C2×C3 | — | 3 | 2 | no | — |
| 226 | C2×C3 | — | 6 | 1 | no | — |
| 227 | C2×C3 | — | 8 | 1 | no | — |
| 228 | C2×C3 | — | 9 | 2 | no | — |
| 229 | C2×C3 | — | 24 | 2 | no | — |
| 230 | C2×C3 | — | 24 | 2 | no | — |
| 231 | C2×C3 | — | 72 | 2 | no | — |
| 232 | C2×C3 | — | 72 | 1 | no | — |
| 233 | C2×C3 | #63926 | 1 | 1 | yes | characteristic |
| 234 | D3 | — | 3 | 4 | no | — |
| 235 | D3 | — | 6 | 2 | no | — |
| 236 | D3 | — | 24 | 2 | no | — |
| 237 | D3 | — | 24 | 2 | no | — |
| 238 | D3 | — | 24 | 2 | no | — |
| 239 | D3 | — | 72 | 2 | no | — |
| 240 | C22 | — | 4 | 1 | no | — |
| 241 | C22 | — | 9 | 1 | no | — |
| 242 | C22 | — | 36 | 1 | no | — |
| 243 | C22 | — | 36 | 2 | no | — |
| 244 | C22 | — | 72 | 2 | no | — |
| 245 | C22 | — | 216 | 2 | no | — |
| 246 | C4 | — | 9 | 1 | no | — |
| 247 | C4 | #88677 | 1 | 1 | yes | characteristic |
| 248 | C3 | — | 3 | 2 | no | — |
| 249 | C3 | — | 6 | 1 | no | — |
| 250 | C3 | #10591025 | 1 | 1 | yes | characteristic |
| 251 | C2 | — | 8 | 1 | no | — |
| 252 | C2 | — | 9 | 2 | no | — |
| 253 | C2 | — | 72 | 1 | no | — |
| 254 | C2 | — | 72 | 2 | no | — |
| 255 | C2 | #11721190 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D8×#272 | C2 | semidirect |
| 2. #11720838 | C2 | semidirect |
| 3. #63653⋊C3 | C2 | semidirect |
| 4. #11720253 | C2 | semidirect |
| 5. #11720251 | C2 | semidirect |
| 9. D8×H(3,𝔽3) | C22 | semidirect |
| 11. #88529 | C22 | semidirect |
| 13. #3745⋊C3 | C22 | semidirect |
| 15. (C3⋊(C3⋊C16))⋊C3 | C22 | semidirect |
| 19. #63653 | D3 | semidirect |
| 40. C32×D8 | C2×D3 | semidirect |
| 41. #3745 | C2×D3 | semidirect |
| 50. C3⋊(C3⋊C16) | C2×D3 | semidirect |
| 95. #272 | D16 | semidirect |
| 142. H(3,𝔽3) | C2×D16 | semidirect |
| 187. D8 | ((C3⋊D3)⋊C3)⋊C2 | semidirect |
| 214. C32 | D3×D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D8×#272 | 2 |
| #11720838 | 2 |
| #63653⋊C3 | 2 |
| #11720253 | 2 |
| #11720251 | 2 |
| D3×#826 | 3 |