Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
930 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 930 conjugacy classes of subgroups — too many to draw. Only its 169 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 | #11722639 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #11721869 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#88433 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #11721060 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #11721058 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #10595450 | — | 3 | 2 | no | maximal |
| 7 | #41×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | Q8×(C3⋊D3)⋊C3 | — | 2 | 2 | no | — |
| 9 | #3830⋊C3 | — | 2 | 6 | no | — |
| 10 | #3828⋊C3 | C22 | 1 | 6 | yes | — |
| 11 | C2×#634⋊C3 | C22 | 1 | 6 | yes | — |
| 12 | #88538 | C22 | 1 | 3 | yes | — |
| 13 | C2×#6180 | C22 | 1 | 3 | yes | — |
| 14 | #88433 | C22 | 1 | 6 | yes | — |
| 15 | #88433 | C22 | 1 | 3 | yes | — |
| 16 | #88432 | C22 | 1 | 1 | yes | characteristic |
| 17 | #88431 | C22 | 1 | 6 | yes | — |
| 18 | #64367 | D3 | 1 | 2 | yes | complemented |
| 19 | C3×#1010 | — | 3 | 2 | no | — |
| 20 | C3×#948 | — | 3 | 2 | no | — |
| 21 | C3×#948 | — | 6 | 1 | no | — |
| 22 | C2×#3719 | — | 3 | 6 | no | — |
| 23 | #63833 | — | 3 | 6 | no | — |
| 24 | #63831 | — | 3 | 6 | no | — |
| 25 | #63829 | — | 3 | 6 | no | — |
| 26 | Q8×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 27 | D4×H(3,𝔽3) | D4 | 1 | 3 | yes | — |
| 28 | C2×C4×H(3,𝔽3) | C23 | 1 | 3 | yes | — |
| 29 | #638⋊C3 | — | 2 | 6 | no | — |
| 30 | C2×(C3⋊Dic3)⋊C3 | — | 2 | 6 | no | — |
| 31 | C4⋊((C3⋊D3)⋊C3) | C23 | 1 | 2 | yes | — |
| 32 | C4×(C3⋊D3)⋊C3 | — | 2 | 6 | no | — |
| 33 | #634⋊C3 | — | 2 | 6 | no | — |
| 34 | #634⋊C3 | C23 | 1 | 6 | yes | — |
| 35 | #6180 | C23 | 1 | 3 | yes | — |
| 36 | #6180 | C23 | 1 | 1 | yes | characteristic |
| 37 | #5838 | — | 9 | 2 | no | — |
| 38 | #5753 | — | 9 | 1 | no | — |
| 39 | C32×#41 | — | 2 | 1 | no | — |
| 40 | C32×#41 | C2×D3 | 1 | 2 | yes | — |
| 41 | Q8×C3⋊D3 | — | 2 | 2 | no | — |
| 42 | #3830 | — | 2 | 6 | no | — |
| 43 | #3828 | C2×D3 | 1 | 6 | yes | — |
| 44 | C2×#634 | C2×D3 | 1 | 6 | yes | — |
| 45 | C3×D3×Q8 | — | 6 | 2 | no | — |
| 46 | C3×#237 | — | 6 | 6 | no | — |
| 47 | C3×#235 | — | 3 | 6 | no | — |
| 48 | C2×C3×Dic6 | — | 3 | 6 | no | — |
| 49 | C3×#208 | — | 3 | 6 | no | — |
| 50 | C3×#208 | — | 6 | 3 | no | — |
| 51 | C2×C3×C3⋊C8 | — | 3 | 6 | no | — |
| 52 | C2×C3×C3⋊C8 | — | 6 | 3 | no | — |
| 53 | #3719 | — | 3 | 6 | no | — |
| 54 | #3719 | — | 3 | 12 | no | — |
| 55 | #3717 | — | 3 | 6 | no | — |
| 56 | #3716 | — | 3 | 6 | no | — |
| 57 | #3714 | — | 3 | 2 | no | — |
| 58 | C22×H(3,𝔽3) | C2×D4 | 1 | 3 | yes | — |
| 59 | C2×(C3⋊D3)⋊C3 | — | 2 | 2 | no | — |
| 60 | C4×H(3,𝔽3) | C2×D4 | 1 | 3 | yes | — |
| 61 | C4×H(3,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 62 | (C3⋊Dic3)⋊C3 | — | 2 | 6 | no | — |
| 63 | C3×2−1+4 | — | 9 | 2 | no | — |
| 64 | #1010 | — | 3 | 2 | no | — |
| 65 | #1010 | — | 9 | 2 | no | — |
| 66 | C3×#131 | — | 9 | 1 | no | — |
| 67 | #951 | — | 9 | 6 | no | — |
| 68 | #950 | — | 9 | 6 | no | — |
| 69 | #948 | — | 3 | 2 | no | — |
| 70 | #948 | — | 6 | 1 | no | — |
| 71 | C2×#216 | — | 9 | 6 | no | — |
| 72 | #942 | — | 9 | 6 | no | — |
| 73 | #909 | — | 9 | 6 | no | — |
| 74 | C2×Dic12 | — | 9 | 3 | no | — |
| 75 | #904 | — | 9 | 3 | no | — |
| 76 | C32×Q8 | — | 2 | 1 | no | — |
| 77 | C32×Q8 | #67 | 1 | 2 | yes | — |
| 78 | C32×D4 | — | 2 | 3 | no | — |
| 79 | C32×D4 | #67 | 1 | 6 | yes | — |
| 80 | C2×C32×C4 | — | 2 | 3 | no | — |
| 81 | C2×C32×C4 | C22×D3 | 1 | 6 | yes | — |
| 82 | #638 | — | 2 | 6 | no | — |
| 83 | C2×C3⋊Dic3 | — | 2 | 6 | no | — |
| 84 | C4⋊(C3⋊D3) | C22×D3 | 1 | 2 | yes | — |
| 85 | C4×C3⋊D3 | — | 2 | 6 | no | — |
| 86 | #634 | — | 2 | 6 | no | — |
| 87 | #634 | C22×D3 | 1 | 6 | yes | — |
| 88 | C3×#67 | — | 6 | 6 | no | — |
| 89 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 90 | C3×D12 | — | 3 | 2 | no | — |
| 91 | C3×C4×D3 | — | 6 | 6 | no | — |
| 92 | C3×Dic6 | — | 3 | 6 | no | — |
| 93 | C3×Dic6 | — | 6 | 6 | no | — |
| 94 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 95 | C3×C3⋊C8 | — | 3 | 6 | no | — |
| 96 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 97 | C3×C3⋊C8 | — | 6 | 3 | no | — |
| 98 | #578 | — | 27 | 1 | no | Sylow |
| 99 | C2×H(3,𝔽3) | — | 2 | 3 | no | — |
| 100 | C2×H(3,𝔽3) | C22×D4 | 1 | 1 | yes | characteristic |
| 101 | (C3⋊D3)⋊C3 | — | 4 | 2 | no | — |
| 102 | C3×#41 | — | 3 | 2 | no | — |
| 103 | C3×#41 | — | 6 | 1 | no | — |
| 104 | C3×#41 | — | 9 | 6 | no | — |
| 105 | C3×#41 | — | 18 | 6 | no | — |
| 106 | C3×#41 | D32 | 1 | 1 | yes | characteristic |
| 107 | C2×C3×Q8 | — | 9 | 6 | no | — |
| 108 | C2×C3×Q8 | — | 18 | 2 | no | — |
| 109 | D3×Q8 | — | 6 | 2 | no | — |
| 110 | D3×Q8 | — | 18 | 2 | no | — |
| 111 | #237 | — | 6 | 6 | no | — |
| 112 | #237 | — | 18 | 6 | no | — |
| 113 | #235 | — | 3 | 6 | no | — |
| 114 | #235 | — | 9 | 6 | no | — |
| 115 | C2×Dic6 | — | 3 | 6 | no | — |
| 116 | C2×Dic6 | — | 9 | 6 | no | — |
| 117 | C3×M16 | — | 9 | 3 | no | — |
| 118 | C2×C3×C8 | — | 9 | 3 | no | — |
| 119 | #216 | — | 9 | 12 | no | — |
| 120 | #216 | — | 9 | 6 | no | — |
| 121 | #215 | — | 9 | 6 | no | — |
| 122 | #214 | — | 9 | 6 | no | — |
| 123 | #213 | — | 9 | 2 | no | — |
| 124 | #208 | — | 3 | 6 | no | — |
| 125 | #208 | — | 6 | 3 | no | — |
| 126 | C2×C3⋊C8 | — | 3 | 6 | no | — |
| 127 | C2×C3⋊C8 | — | 6 | 3 | no | — |
| 128 | Dic12 | — | 9 | 3 | no | — |
| 129 | Dic12 | — | 9 | 6 | no | — |
| 130 | D24 | — | 9 | 1 | no | — |
| 131 | #204 | — | 9 | 6 | no | — |
| 132 | C22×C32 | — | 2 | 3 | no | — |
| 133 | C22×C32 | C2×#67 | 1 | 6 | yes | — |
| 134 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 135 | C2×C3×D3 | — | 6 | 2 | no | — |
| 136 | C32×C4 | — | 2 | 1 | no | — |
| 137 | C32×C4 | — | 2 | 3 | no | — |
| 138 | C32×C4 | C2×#67 | 1 | 6 | yes | — |
| 139 | C32×C4 | C23×D3 | 1 | 2 | yes | — |
| 140 | C3⋊Dic3 | — | 2 | 6 | no | — |
| 141 | C3×Dic3 | — | 6 | 6 | no | — |
| 142 | 2−1+4 | — | 9 | 2 | no | — |
| 143 | #137 | — | 27 | 6 | no | — |
| 144 | #135 | — | 27 | 3 | no | — |
| 145 | C2×Q16 | — | 27 | 3 | no | — |
| 146 | #131 | — | 9 | 1 | no | — |
| 147 | H(3,𝔽3) | #578 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 148 | C3×Q8 | — | 3 | 2 | no | — |
| 149 | C3×Q8 | — | 6 | 1 | no | — |
| 150 | C3×Q8 | — | 9 | 6 | no | — |
| 151 | C3×Q8 | — | 18 | 6 | no | — |
| 152 | C3×Q8 | #625 | 1 | 1 | yes | characteristic |
| 153 | C3×D4 | — | 3 | 6 | no | — |
| 154 | C3×D4 | — | 6 | 3 | no | — |
| 155 | C3×D4 | — | 9 | 2 | no | — |
| 156 | C3×D4 | — | 18 | 6 | no | — |
| 157 | C3×D4 | #625 | 1 | 3 | yes | — |
| 158 | C2×C3×C4 | — | 3 | 6 | no | — |
| 159 | C2×C3×C4 | — | 6 | 3 | no | — |
| 160 | C2×C3×C4 | — | 18 | 6 | no | — |
| 161 | C2×C3×C4 | — | 18 | 6 | no | — |
| 162 | C2×C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 163 | #67 | — | 6 | 6 | no | — |
| 164 | #67 | — | 18 | 6 | no | — |
| 165 | C2×Dic3 | — | 6 | 6 | no | — |
| 166 | C2×Dic3 | — | 18 | 6 | no | — |
| 167 | D12 | — | 3 | 2 | no | — |
| 168 | D12 | — | 9 | 2 | no | — |
| 169 | C4×D3 | — | 6 | 6 | no | — |
| 170 | C4×D3 | — | 18 | 6 | no | — |
| 171 | Dic6 | — | 3 | 6 | no | — |
| 172 | Dic6 | — | 6 | 6 | no | — |
| 173 | Dic6 | — | 9 | 6 | no | — |
| 174 | Dic6 | — | 18 | 6 | no | — |
| 175 | C3×C8 | — | 9 | 3 | no | — |
| 176 | C3×C8 | — | 9 | 1 | no | — |
| 177 | C3⋊C8 | — | 3 | 2 | no | — |
| 178 | C3⋊C8 | — | 3 | 6 | no | — |
| 179 | C3⋊C8 | — | 6 | 1 | no | — |
| 180 | C3⋊C8 | — | 6 | 3 | no | — |
| 181 | C2×C32 | — | 2 | 6 | no | — |
| 182 | C2×C32 | — | 2 | 1 | no | — |
| 183 | C2×C32 | — | 4 | 3 | no | — |
| 184 | C2×C32 | C22×#67 | 1 | 2 | yes | — |
| 185 | C3⋊D3 | — | 4 | 2 | no | — |
| 186 | C3×D3 | — | 12 | 2 | no | — |
| 187 | #41 | — | 9 | 6 | no | — |
| 188 | #41 | — | 18 | 6 | no | — |
| 189 | #41 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 190 | C2×Q8 | — | 9 | 6 | no | — |
| 191 | C2×Q8 | — | 18 | 2 | no | — |
| 192 | Q16 | — | 27 | 3 | no | — |
| 193 | Q16 | — | 27 | 6 | no | — |
| 194 | SD16 | — | 27 | 6 | no | — |
| 195 | D8 | — | 27 | 1 | no | — |
| 196 | M16 | — | 9 | 3 | no | — |
| 197 | C2×C8 | — | 9 | 3 | no | — |
| 198 | C22×C3 | — | 3 | 6 | no | — |
| 199 | C22×C3 | — | 6 | 3 | no | — |
| 200 | C22×C3 | — | 18 | 2 | no | — |
| 201 | C22×C3 | C2×#625 | 1 | 3 | yes | — |
| 202 | C2×D3 | — | 6 | 2 | no | — |
| 203 | C2×D3 | — | 18 | 2 | no | — |
| 204 | C3×C4 | — | 3 | 2 | no | — |
| 205 | C3×C4 | — | 3 | 6 | no | — |
| 206 | C3×C4 | — | 6 | 1 | no | — |
| 207 | C3×C4 | — | 6 | 3 | no | — |
| 208 | C3×C4 | — | 18 | 6 | no | — |
| 209 | C3×C4 | C2×#625 | 1 | 3 | yes | — |
| 210 | C3×C4 | C22×D32 | 1 | 1 | yes | characteristic |
| 211 | Dic3 | — | 6 | 6 | no | — |
| 212 | Dic3 | — | 18 | 6 | no | — |
| 213 | C32 | — | 2 | 1 | no | — |
| 214 | C32 | #5838 | 1 | 2 | yes | complemented |
| 215 | Q8 | — | 9 | 6 | no | — |
| 216 | Q8 | — | 18 | 6 | no | — |
| 217 | Q8 | #6198 | 1 | 1 | yes | characteristic |
| 218 | D4 | — | 9 | 2 | no | — |
| 219 | D4 | — | 18 | 6 | no | — |
| 220 | D4 | #6198 | 1 | 3 | yes | — |
| 221 | C2×C4 | — | 18 | 6 | no | — |
| 222 | C2×C4 | — | 18 | 6 | no | — |
| 223 | C2×C4 | C2×((C3⋊D3)⋊C3)⋊C2 | 1 | 3 | yes | — |
| 224 | C8 | — | 9 | 1 | no | — |
| 225 | C8 | — | 9 | 3 | no | — |
| 226 | C2×C3 | — | 2 | 3 | no | — |
| 227 | C2×C3 | — | 3 | 2 | no | — |
| 228 | C2×C3 | — | 6 | 6 | no | — |
| 229 | C2×C3 | — | 6 | 1 | no | — |
| 230 | C2×C3 | — | 12 | 3 | no | — |
| 231 | C2×C3 | — | 36 | 2 | no | — |
| 232 | C2×C3 | C22×#625 | 1 | 1 | yes | characteristic |
| 233 | D3 | — | 12 | 2 | no | — |
| 234 | D3 | — | 36 | 2 | no | — |
| 235 | C22 | — | 18 | 2 | no | — |
| 236 | C22 | C2×#6198 | 1 | 3 | yes | — |
| 237 | C4 | — | 18 | 6 | no | — |
| 238 | C4 | C2×#6198 | 1 | 3 | yes | — |
| 239 | C4 | C22×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 240 | C3 | — | 3 | 2 | no | — |
| 241 | C3 | — | 6 | 1 | no | — |
| 242 | C3 | #10595445 | 1 | 1 | yes | characteristic |
| 243 | C2 | — | 2 | 3 | no | — |
| 244 | C2 | — | 36 | 2 | no | — |
| 245 | C2 | C22×#6198 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722639 | C2 | semidirect |
| 2. #11721869 | C2 | semidirect |
| 3. C2×#88433 | C2 | semidirect |
| 4. #11721060 | C2 | semidirect |
| 5. #11721058 | C2 | semidirect |
| 18. #64367 | D3 | semidirect |
| 147. H(3,𝔽3) | #578 | semidirect |
| 214. C32 | #5838 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722639 | 2 |
| #11721869 | 2 |
| C2×#88433 | 2 |
| #11721060 | 2 |
| #11721058 | 2 |
| #10595450 | 3 |