Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
579 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 579 conjugacy classes of subgroups — too many to draw. Only its 79 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 | C2×#88651 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | M16×#272 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11721065 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #63651⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11720321 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | D3×#824 | — | 3 | 2 | no | maximal |
| 7 | C2×C4×#272 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#634⋊C3 | C22 | 1 | 2 | yes | — |
| 9 | C2×#6199 | C4 | 1 | 2 | yes | — |
| 10 | C2×#6196 | C4 | 1 | 2 | yes | — |
| 11 | #88651 | — | 2 | 1 | no | — |
| 12 | #88651 | — | 2 | 1 | no | — |
| 13 | #88651 | — | 2 | 1 | no | — |
| 14 | #88651 | — | 2 | 1 | no | — |
| 15 | M16×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 16 | #88538 | C22 | 1 | 1 | yes | characteristic |
| 17 | #88527 | — | 2 | 1 | no | — |
| 18 | C8×#272 | — | 2 | 1 | no | — |
| 19 | #3748⋊C3 | C22 | 1 | 2 | yes | — |
| 20 | #88435 | — | 2 | 1 | no | — |
| 21 | #88434 | — | 2 | 1 | no | — |
| 22 | C2×D3×Dic6 | — | 3 | 2 | no | — |
| 23 | C3×D3×M16 | — | 3 | 2 | no | — |
| 24 | C3×D3×M16 | — | 6 | 1 | no | — |
| 25 | D3×#208 | — | 3 | 2 | no | — |
| 26 | #63651 | D3 | 1 | 2 | yes | complemented |
| 27 | C3×#824 | — | 3 | 2 | no | — |
| 28 | #63561 | — | 3 | 2 | no | — |
| 29 | #63560 | — | 3 | 2 | no | — |
| 30 | C22×#272 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C2×C4×H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×#1084 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C4×#272 | D4 | 1 | 1 | yes | characteristic |
| 34 | C4×#272 | D4 | 1 | 1 | yes | characteristic |
| 35 | C4×#272 | D4 | 1 | 1 | yes | characteristic |
| 36 | C4×#272 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×(C3⋊Dic3)⋊C3 | C2×C4 | 1 | 4 | yes | — |
| 38 | #634⋊C3 | — | 2 | 2 | no | — |
| 39 | #634⋊C3 | — | 2 | 2 | no | — |
| 40 | #6199 | — | 2 | 2 | no | — |
| 41 | #6199 | — | 2 | 2 | no | — |
| 42 | #6196 | — | 2 | 2 | no | — |
| 43 | #6196 | — | 2 | 2 | no | — |
| 44 | C8×H(3,𝔽3) | — | 2 | 1 | no | — |
| 45 | #6180 | — | 2 | 1 | no | — |
| 46 | (C3⋊(C3⋊C8))⋊C3 | — | 2 | 2 | no | — |
| 47 | C2×#824 | — | 9 | 1 | no | — |
| 48 | D3×#101 | — | 9 | 2 | no | — |
| 49 | C2×#634 | C2×D3 | 1 | 2 | yes | — |
| 50 | C2×C3×C4×D3 | — | 3 | 2 | no | — |
| 51 | C2×C3×C4×D3 | — | 6 | 1 | no | — |
| 52 | C2×C3×Dic6 | — | 3 | 2 | no | — |
| 53 | C2×#627 | — | 3 | 4 | no | — |
| 54 | C2×D3×Dic3 | — | 3 | 4 | no | — |
| 55 | D3×Dic6 | — | 6 | 2 | no | — |
| 56 | D3×Dic6 | — | 6 | 2 | no | — |
| 57 | D3×Dic6 | — | 6 | 2 | no | — |
| 58 | D3×Dic6 | — | 6 | 2 | no | — |
| 59 | C32×M16 | — | 2 | 1 | no | — |
| 60 | C32×M16 | C2×D3 | 1 | 2 | yes | — |
| 61 | #3748 | C2×D3 | 1 | 2 | yes | — |
| 62 | C3×#208 | — | 3 | 2 | no | — |
| 63 | C3×#208 | — | 3 | 2 | no | — |
| 64 | C3×#208 | — | 6 | 1 | no | — |
| 65 | C3×ΓL(1,𝔽25) | — | 6 | 2 | no | — |
| 66 | C3×ΓL(1,𝔽25) | — | 6 | 2 | no | — |
| 67 | C3×D3×C8 | — | 6 | 2 | no | — |
| 68 | C3×D3×C8 | — | 6 | 2 | no | — |
| 69 | #3711 | — | 6 | 2 | no | — |
| 70 | D3×C3⋊C8 | — | 6 | 2 | no | — |
| 71 | C22×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | C2×#272 | — | 2 | 2 | no | — |
| 73 | C2×#272 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 74 | C2×#272 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 75 | C4×H(3,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C4×H(3,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | #1084 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 78 | #1084 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 79 | (C3⋊Dic3)⋊C3 | — | 2 | 4 | no | — |
| 80 | C2×D3×Q8 | — | 9 | 2 | no | — |
| 81 | C22×Dic6 | — | 9 | 1 | no | — |
| 82 | C2×C3×M16 | — | 9 | 1 | no | — |
| 83 | C2×#208 | — | 9 | 1 | no | — |
| 84 | D3×M16 | — | 3 | 2 | no | — |
| 85 | D3×M16 | — | 6 | 1 | no | — |
| 86 | D3×M16 | — | 9 | 2 | no | — |
| 87 | C3×#101 | — | 9 | 2 | no | — |
| 88 | #836 | — | 9 | 2 | no | — |
| 89 | #824 | — | 3 | 2 | no | — |
| 90 | #824 | — | 9 | 2 | no | — |
| 91 | #824 | — | 9 | 2 | no | — |
| 92 | #824 | — | 9 | 2 | no | — |
| 93 | C22×C3×D3 | — | 3 | 2 | no | — |
| 94 | C22×C3×D3 | — | 6 | 1 | no | — |
| 95 | C2×C32×C4 | — | 2 | 1 | no | — |
| 96 | C2×C32×C4 | C22×D3 | 1 | 2 | yes | — |
| 97 | C2×C3⋊Dic3 | C4×D3 | 1 | 4 | yes | — |
| 98 | #634 | — | 2 | 2 | no | — |
| 99 | #634 | — | 2 | 2 | no | — |
| 100 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 101 | C2×C3×Dic3 | — | 3 | 4 | no | — |
| 102 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 103 | C3×C4×D3 | — | 3 | 2 | no | — |
| 104 | C3×C4×D3 | — | 3 | 2 | no | — |
| 105 | C3×C4×D3 | — | 3 | 2 | no | — |
| 106 | C3×C4×D3 | — | 3 | 2 | no | — |
| 107 | C3×C4×D3 | — | 6 | 1 | no | — |
| 108 | C3×C4×D3 | — | 6 | 1 | no | — |
| 109 | C3×C4×D3 | — | 6 | 1 | no | — |
| 110 | C3×C4×D3 | — | 6 | 1 | no | — |
| 111 | C3×Dic6 | — | 6 | 2 | no | — |
| 112 | C3×Dic6 | — | 6 | 2 | no | — |
| 113 | #627 | — | 6 | 4 | no | — |
| 114 | #627 | — | 6 | 4 | no | — |
| 115 | D3×Dic3 | — | 6 | 4 | no | — |
| 116 | D3×Dic3 | — | 6 | 4 | no | — |
| 117 | C32×C8 | — | 2 | 2 | no | — |
| 118 | C32×C8 | — | 2 | 2 | no | — |
| 119 | C3⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 120 | C3×C3⋊C8 | — | 6 | 2 | no | — |
| 121 | C3×C3⋊C8 | — | 6 | 2 | no | — |
| 122 | C3×C3⋊C8 | — | 6 | 2 | no | — |
| 123 | C2×#101 | — | 27 | 1 | no | Sylow |
| 124 | C2×H(3,𝔽3) | — | 2 | 1 | no | — |
| 125 | C2×H(3,𝔽3) | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 126 | #272 | — | 2 | 1 | no | — |
| 127 | #272 | #101 | 1 | 2 | yes | complemented |
| 128 | C2×C3×Q8 | — | 9 | 2 | no | — |
| 129 | C22×C3×C4 | — | 9 | 1 | no | — |
| 130 | C22×Dic3 | — | 9 | 2 | no | — |
| 131 | D3×Q8 | — | 18 | 2 | no | — |
| 132 | D3×Q8 | — | 18 | 2 | no | — |
| 133 | D3×Q8 | — | 18 | 2 | no | — |
| 134 | D3×Q8 | — | 18 | 2 | no | — |
| 135 | C2×C4×D3 | — | 3 | 2 | no | — |
| 136 | C2×C4×D3 | — | 6 | 1 | no | — |
| 137 | C2×C4×D3 | — | 9 | 4 | no | — |
| 138 | C2×Dic6 | — | 3 | 2 | no | — |
| 139 | C2×Dic6 | — | 9 | 2 | no | — |
| 140 | C2×Dic6 | — | 9 | 2 | no | — |
| 141 | C2×Dic6 | — | 9 | 4 | no | — |
| 142 | C2×Dic6 | — | 18 | 1 | no | — |
| 143 | C2×Dic6 | — | 18 | 1 | no | — |
| 144 | C2×Dic6 | — | 18 | 1 | no | — |
| 145 | C2×Dic6 | — | 18 | 1 | no | — |
| 146 | C3×M16 | — | 3 | 2 | no | — |
| 147 | C3×M16 | — | 6 | 1 | no | — |
| 148 | C3×M16 | — | 9 | 2 | no | — |
| 149 | C3×M16 | — | 9 | 1 | no | — |
| 150 | C3×M16 | — | 18 | 1 | no | — |
| 151 | C3×M16 | D32 | 1 | 1 | yes | characteristic |
| 152 | C2×C3×C8 | — | 18 | 1 | no | — |
| 153 | #208 | — | 3 | 2 | no | — |
| 154 | #208 | — | 3 | 2 | no | — |
| 155 | #208 | — | 6 | 1 | no | — |
| 156 | #208 | — | 9 | 2 | no | — |
| 157 | #208 | — | 18 | 1 | no | — |
| 158 | C2×C3⋊C8 | — | 18 | 1 | no | — |
| 159 | ΓL(1,𝔽25) | — | 6 | 2 | no | — |
| 160 | ΓL(1,𝔽25) | — | 6 | 2 | no | — |
| 161 | ΓL(1,𝔽25) | — | 18 | 2 | no | — |
| 162 | D3×C8 | — | 6 | 2 | no | — |
| 163 | D3×C8 | — | 6 | 2 | no | — |
| 164 | D3×C8 | — | 18 | 2 | no | — |
| 165 | C22×C32 | — | 2 | 1 | no | — |
| 166 | C22×C32 | C2×C4×D3 | 1 | 2 | yes | — |
| 167 | C2×C3×D3 | — | 3 | 2 | no | — |
| 168 | C2×C3×D3 | — | 3 | 2 | no | — |
| 169 | C2×C3×D3 | — | 6 | 1 | no | — |
| 170 | C2×C3×D3 | — | 6 | 1 | no | — |
| 171 | C2×C3×D3 | — | 6 | 4 | no | — |
| 172 | C2×C3×D3 | — | 12 | 2 | no | — |
| 173 | C32×C4 | — | 2 | 1 | no | — |
| 174 | C32×C4 | — | 2 | 1 | no | — |
| 175 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 176 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 177 | C3⋊Dic3 | — | 2 | 4 | no | — |
| 178 | C3×Dic3 | — | 3 | 2 | no | — |
| 179 | C3×Dic3 | — | 3 | 2 | no | — |
| 180 | C3×Dic3 | — | 6 | 1 | no | — |
| 181 | C3×Dic3 | — | 6 | 1 | no | — |
| 182 | C3×Dic3 | — | 6 | 4 | no | — |
| 183 | C22×Q8 | — | 27 | 1 | no | — |
| 184 | C2×M16 | — | 9 | 1 | no | — |
| 185 | C2×M16 | — | 27 | 1 | no | — |
| 186 | #101 | — | 9 | 2 | no | — |
| 187 | #101 | — | 27 | 2 | no | — |
| 188 | H(3,𝔽3) | C2×#101 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 189 | C23×C3 | — | 9 | 1 | no | — |
| 190 | C22×D3 | — | 3 | 2 | no | — |
| 191 | C22×D3 | — | 6 | 1 | no | — |
| 192 | C3×Q8 | — | 18 | 2 | no | — |
| 193 | C3×Q8 | — | 18 | 2 | no | — |
| 194 | C2×C3×C4 | — | 3 | 2 | no | — |
| 195 | C2×C3×C4 | — | 6 | 1 | no | — |
| 196 | C2×C3×C4 | — | 9 | 1 | no | — |
| 197 | C2×C3×C4 | — | 9 | 1 | no | — |
| 198 | C2×C3×C4 | — | 9 | 1 | no | — |
| 199 | C2×C3×C4 | — | 9 | 1 | no | — |
| 200 | C2×C3×C4 | — | 9 | 1 | no | — |
| 201 | C2×C3×C4 | — | 9 | 4 | no | — |
| 202 | C2×C3×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 203 | C2×Dic3 | — | 3 | 2 | no | — |
| 204 | C2×Dic3 | — | 3 | 4 | no | — |
| 205 | C2×Dic3 | — | 6 | 1 | no | — |
| 206 | C2×Dic3 | — | 9 | 4 | no | — |
| 207 | C2×Dic3 | — | 18 | 2 | no | — |
| 208 | C2×Dic3 | — | 18 | 2 | no | — |
| 209 | C4×D3 | — | 3 | 2 | no | — |
| 210 | C4×D3 | — | 3 | 2 | no | — |
| 211 | C4×D3 | — | 3 | 2 | no | — |
| 212 | C4×D3 | — | 3 | 2 | no | — |
| 213 | C4×D3 | — | 6 | 1 | no | — |
| 214 | C4×D3 | — | 6 | 1 | no | — |
| 215 | C4×D3 | — | 6 | 1 | no | — |
| 216 | C4×D3 | — | 6 | 1 | no | — |
| 217 | C4×D3 | — | 18 | 4 | no | — |
| 218 | C4×D3 | — | 18 | 4 | no | — |
| 219 | Dic6 | — | 6 | 2 | no | — |
| 220 | Dic6 | — | 6 | 2 | no | — |
| 221 | Dic6 | — | 18 | 2 | no | — |
| 222 | Dic6 | — | 18 | 2 | no | — |
| 223 | Dic6 | — | 18 | 4 | no | — |
| 224 | Dic6 | — | 18 | 4 | no | — |
| 225 | Dic6 | — | 18 | 2 | no | — |
| 226 | Dic6 | — | 18 | 2 | no | — |
| 227 | C3×C8 | — | 2 | 1 | no | — |
| 228 | C3×C8 | — | 6 | 2 | no | — |
| 229 | C3×C8 | — | 6 | 2 | no | — |
| 230 | C3×C8 | — | 18 | 2 | no | — |
| 231 | C3×C8 | — | 18 | 1 | no | — |
| 232 | C3⋊C8 | — | 6 | 2 | no | — |
| 233 | C3⋊C8 | — | 6 | 2 | no | — |
| 234 | C3⋊C8 | — | 6 | 2 | no | — |
| 235 | C3⋊C8 | — | 18 | 2 | no | — |
| 236 | C2×C32 | — | 2 | 2 | no | — |
| 237 | C2×C32 | — | 2 | 1 | no | — |
| 238 | C2×C32 | — | 4 | 1 | no | — |
| 239 | C2×C32 | D3×C22⋊C4 | 1 | 2 | yes | — |
| 240 | C3×D3 | — | 3 | 4 | no | — |
| 241 | C3×D3 | — | 6 | 2 | no | — |
| 242 | C3×D3 | — | 6 | 2 | no | — |
| 243 | C3×D3 | — | 12 | 1 | no | — |
| 244 | C2×Q8 | — | 9 | 2 | no | — |
| 245 | C2×Q8 | — | 27 | 2 | no | — |
| 246 | C2×Q8 | — | 54 | 1 | no | — |
| 247 | C2×Q8 | — | 54 | 1 | no | — |
| 248 | C2×Q8 | — | 54 | 1 | no | — |
| 249 | C2×Q8 | — | 54 | 1 | no | — |
| 250 | C22×C4 | — | 9 | 1 | no | — |
| 251 | C22×C4 | — | 27 | 2 | no | — |
| 252 | M16 | — | 9 | 2 | no | — |
| 253 | M16 | — | 9 | 1 | no | — |
| 254 | M16 | — | 18 | 1 | no | — |
| 255 | M16 | — | 54 | 1 | no | — |
| 256 | M16 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | — | 18 | 1 | no | — |
| 258 | C2×C8 | — | 54 | 1 | no | — |
| 259 | C22×C3 | — | 3 | 2 | no | — |
| 260 | C22×C3 | — | 6 | 1 | no | — |
| 261 | C22×C3 | — | 9 | 1 | no | — |
| 262 | C22×C3 | — | 9 | 1 | no | — |
| 263 | C22×C3 | — | 18 | 2 | no | — |
| 264 | C22×C3 | C4×D32 | 1 | 1 | yes | characteristic |
| 265 | C2×D3 | — | 3 | 2 | no | — |
| 266 | C2×D3 | — | 3 | 2 | no | — |
| 267 | C2×D3 | — | 6 | 1 | no | — |
| 268 | C2×D3 | — | 6 | 1 | no | — |
| 269 | C2×D3 | — | 6 | 4 | no | — |
| 270 | C2×D3 | — | 12 | 2 | no | — |
| 271 | C3×C4 | — | 3 | 2 | no | — |
| 272 | C3×C4 | — | 3 | 2 | no | — |
| 273 | C3×C4 | — | 6 | 1 | no | — |
| 274 | C3×C4 | — | 6 | 1 | no | — |
| 275 | C3×C4 | — | 9 | 1 | no | — |
| 276 | C3×C4 | — | 9 | 1 | no | — |
| 277 | C3×C4 | — | 18 | 4 | no | — |
| 278 | C3×C4 | #3802 | 1 | 1 | yes | characteristic |
| 279 | C3×C4 | #3811 | 1 | 1 | yes | characteristic |
| 280 | Dic3 | — | 3 | 2 | no | — |
| 281 | Dic3 | — | 3 | 2 | no | — |
| 282 | Dic3 | — | 6 | 1 | no | — |
| 283 | Dic3 | — | 6 | 1 | no | — |
| 284 | Dic3 | — | 6 | 4 | no | — |
| 285 | Dic3 | — | 18 | 4 | no | — |
| 286 | C32 | — | 2 | 1 | no | — |
| 287 | C32 | D3×#101 | 1 | 2 | yes | complemented |
| 288 | C23 | — | 9 | 1 | no | — |
| 289 | Q8 | — | 18 | 2 | no | — |
| 290 | Q8 | — | 18 | 2 | no | — |
| 291 | Q8 | — | 54 | 2 | no | — |
| 292 | Q8 | — | 54 | 2 | no | — |
| 293 | C2×C4 | — | 9 | 1 | no | — |
| 294 | C2×C4 | — | 9 | 1 | no | — |
| 295 | C2×C4 | — | 9 | 1 | no | — |
| 296 | C2×C4 | — | 9 | 1 | no | — |
| 297 | C2×C4 | — | 9 | 1 | no | — |
| 298 | C2×C4 | — | 9 | 4 | no | — |
| 299 | C2×C4 | — | 54 | 2 | no | — |
| 300 | C2×C4 | — | 54 | 2 | no | — |
| 301 | C2×C4 | C2×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 302 | C8 | — | 2 | 1 | no | — |
| 303 | C8 | — | 18 | 2 | no | — |
| 304 | C8 | — | 18 | 1 | no | — |
| 305 | C2×C3 | — | 2 | 1 | no | — |
| 306 | C2×C3 | — | 3 | 2 | no | — |
| 307 | C2×C3 | — | 6 | 2 | no | — |
| 308 | C2×C3 | — | 6 | 1 | no | — |
| 309 | C2×C3 | — | 9 | 2 | no | — |
| 310 | C2×C3 | — | 12 | 1 | no | — |
| 311 | C2×C3 | — | 18 | 1 | no | — |
| 312 | C2×C3 | #63921 | 1 | 1 | yes | characteristic |
| 313 | D3 | — | 3 | 4 | no | — |
| 314 | D3 | — | 6 | 2 | no | — |
| 315 | D3 | — | 6 | 2 | no | — |
| 316 | D3 | — | 12 | 1 | no | — |
| 317 | C22 | — | 9 | 1 | no | — |
| 318 | C22 | — | 9 | 1 | no | — |
| 319 | C22 | — | 18 | 2 | no | — |
| 320 | C22 | C4×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 321 | C4 | — | 9 | 1 | no | — |
| 322 | C4 | — | 9 | 1 | no | — |
| 323 | C4 | — | 18 | 4 | no | — |
| 324 | C4 | #88655 | 1 | 1 | yes | characteristic |
| 325 | C4 | #88677 | 1 | 1 | yes | characteristic |
| 326 | C3 | — | 3 | 2 | no | — |
| 327 | C3 | — | 6 | 1 | no | — |
| 328 | C3 | #10591003 | 1 | 1 | yes | characteristic |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 9 | 2 | no | — |
| 331 | C2 | — | 18 | 1 | no | — |
| 332 | C2 | #11721161 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #63651⋊C3 | C2 | semidirect |
| 5. #11720321 | C2 | semidirect |
| 26. #63651 | D3 | semidirect |
| 127. #272 | #101 | semidirect |
| 188. H(3,𝔽3) | C2×#101 | semidirect |
| 287. C32 | D3×#101 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88651 | 2 |
| M16×#272 | 2 |
| #11721065 | 2 |
| #63651⋊C3 | 2 |
| #11720321 | 2 |
| D3×#824 | 3 |