Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
988 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 988 conjugacy classes of subgroups — too many to draw. Only its 37 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 | #11722587 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×#3832 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊#5948 | — | 3 | 1 | no | maximal |
| 4 | (C3⋊Dic3)⋊A4 | — | 4 | 1 | no | maximal |
| 5 | #88622 | — | 4 | 4 | no | maximal |
| 6 | C22×C32×Q8 | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | C2×#3832 | — | 3 | 4 | no | — |
| 8 | C2×#3832 | — | 3 | 4 | no | — |
| 9 | C22×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 10 | C22×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 11 | C3×C22⋊SL(2,𝔽3) | — | 2 | 1 | no | — |
| 12 | C3×C22⋊SL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×C22⋊H(3,𝔽3) | — | 4 | 1 | no | — |
| 14 | Q8⋊H(3,𝔽3) | — | 4 | 4 | no | — |
| 15 | C22×#239 | — | 3 | 1 | no | — |
| 16 | C22×#239 | — | 9 | 1 | no | — |
| 17 | #5948 | — | 9 | 1 | no | — |
| 18 | C23×C3⋊D3 | — | 3 | 1 | no | — |
| 19 | C2×C32×Q8 | — | 3 | 4 | no | — |
| 20 | C22×C32×C4 | — | 3 | 1 | no | — |
| 21 | C22×C3⋊Dic3 | A4 | 1 | 1 | yes | characteristic |
| 22 | #3832 | — | 3 | 12 | no | — |
| 23 | #3832 | — | 3 | 4 | no | — |
| 24 | #3832 | — | 3 | 4 | no | — |
| 25 | #3832 | A4 | 1 | 4 | yes | complemented |
| 26 | C2×C4⋊(C3⋊D3) | — | 3 | 12 | no | — |
| 27 | C2×C4×C3⋊D3 | — | 3 | 6 | no | — |
| 28 | C2×C4×C3⋊D3 | — | 3 | 6 | no | — |
| 29 | Dic3×A4 | — | 12 | 1 | no | — |
| 30 | C3⋊(#41⋊C3) | — | 12 | 4 | no | — |
| 31 | C22⋊H(3,𝔽3) | — | 4 | 1 | no | — |
| 32 | (C3⋊Dic3)⋊C3 | — | 16 | 1 | no | — |
| 33 | C22×C3×Q8 | — | 3 | 1 | no | — |
| 34 | C22×C3×Q8 | C3×D3 | 1 | 1 | yes | characteristic |
| 35 | C2×#239 | — | 9 | 4 | no | — |
| 36 | C2×#239 | — | 9 | 12 | no | — |
| 37 | C2×#239 | — | 9 | 12 | no | — |
| 38 | C2×#239 | — | 9 | 4 | no | — |
| 39 | C22×D12 | — | 9 | 1 | no | — |
| 40 | C22×D12 | — | 9 | 3 | no | — |
| 41 | C22×C4×D3 | — | 9 | 1 | no | — |
| 42 | C22×C4×D3 | — | 9 | 3 | no | — |
| 43 | C22⋊SL(2,𝔽3) | — | 3 | 1 | no | — |
| 44 | C22⋊SL(2,𝔽3) | — | 6 | 1 | no | — |
| 45 | C23×C32 | C2×A4 | 1 | 1 | yes | characteristic |
| 46 | C22×C3⋊D3 | — | 3 | 6 | no | — |
| 47 | C22×C3⋊D3 | — | 6 | 3 | no | — |
| 48 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 49 | C2×C3×A4 | — | 4 | 1 | no | — |
| 50 | C2×C3×A4 | — | 8 | 1 | no | — |
| 51 | C32×Q8 | — | 3 | 4 | no | — |
| 52 | C32×Q8 | C2×A4 | 1 | 4 | yes | — |
| 53 | C2×C32×C4 | — | 3 | 6 | no | — |
| 54 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 55 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 56 | C4⋊(C3⋊D3) | — | 3 | 4 | no | — |
| 57 | C4⋊(C3⋊D3) | — | 3 | 12 | no | — |
| 58 | C4×C3⋊D3 | — | 3 | 12 | no | — |
| 59 | C4×C3⋊D3 | — | 3 | 4 | no | — |
| 60 | C3×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 61 | C3×SL(2,𝔽3) | — | 8 | 4 | no | — |
| 62 | C22×#41 | — | 9 | 1 | no | Sylow |
| 63 | C2×H(3,𝔽3) | — | 16 | 1 | no | — |
| 64 | C23×D3 | — | 9 | 3 | no | — |
| 65 | C23×D3 | — | 9 | 1 | no | — |
| 66 | C2×C3×Q8 | — | 3 | 12 | no | — |
| 67 | C2×C3×Q8 | — | 3 | 4 | no | — |
| 68 | C22×C3×C4 | — | 3 | 3 | no | — |
| 69 | C22×C3×C4 | — | 3 | 1 | no | — |
| 70 | C22×Dic3 | — | 3 | 1 | no | — |
| 71 | C22×Dic3 | — | 9 | 1 | no | — |
| 72 | #239 | — | 3 | 4 | no | — |
| 73 | #239 | — | 9 | 4 | no | — |
| 74 | #239 | — | 9 | 4 | no | — |
| 75 | #239 | — | 9 | 12 | no | — |
| 76 | #239 | — | 9 | 12 | no | — |
| 77 | #239 | — | 9 | 36 | no | — |
| 78 | #239 | — | 9 | 12 | no | — |
| 79 | #239 | — | 9 | 4 | no | — |
| 80 | C2×D12 | — | 9 | 12 | no | — |
| 81 | C2×D12 | — | 9 | 36 | no | — |
| 82 | C2×C4×D3 | — | 9 | 6 | no | — |
| 83 | C2×C4×D3 | — | 9 | 18 | no | — |
| 84 | C2×C4×D3 | — | 9 | 18 | no | — |
| 85 | C2×C4×D3 | — | 9 | 6 | no | — |
| 86 | #41⋊C3 | — | 36 | 4 | no | — |
| 87 | C4×A4 | — | 36 | 1 | no | — |
| 88 | C22×C32 | — | 3 | 1 | no | — |
| 89 | C22×C32 | — | 3 | 1 | no | — |
| 90 | C22×C32 | #41⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 91 | C2×C3⋊D3 | — | 3 | 4 | no | — |
| 92 | C2×C3⋊D3 | — | 6 | 6 | no | — |
| 93 | C2×C3⋊D3 | — | 6 | 6 | no | — |
| 94 | C3×A4 | — | 4 | 1 | no | — |
| 95 | C3×A4 | — | 8 | 1 | no | — |
| 96 | C32×C4 | — | 3 | 4 | no | — |
| 97 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 98 | C3⋊Dic3 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 99 | C3×Dic3 | — | 48 | 1 | no | — |
| 100 | C2×#41 | — | 27 | 4 | no | — |
| 101 | C2×#41 | — | 27 | 4 | no | — |
| 102 | C22×Q8 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 103 | C22×D4 | — | 27 | 1 | no | — |
| 104 | C23×C4 | — | 27 | 1 | no | — |
| 105 | H(3,𝔽3) | — | 16 | 1 | no | Sylow |
| 106 | C23×C3 | — | 3 | 1 | no | — |
| 107 | C23×C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 108 | C22×D3 | — | 9 | 18 | no | — |
| 109 | C22×D3 | — | 9 | 6 | no | — |
| 110 | C22×D3 | — | 18 | 3 | no | — |
| 111 | C22×D3 | — | 18 | 9 | no | — |
| 112 | C22×D3 | — | 18 | 3 | no | — |
| 113 | C22×D3 | — | 18 | 1 | no | — |
| 114 | C2×A4 | — | 12 | 1 | no | — |
| 115 | C2×A4 | — | 24 | 1 | no | — |
| 116 | C3×Q8 | — | 3 | 12 | no | — |
| 117 | C3×Q8 | — | 3 | 4 | no | — |
| 118 | C3×Q8 | — | 3 | 4 | no | — |
| 119 | C3×Q8 | D3×A4 | 1 | 4 | yes | — |
| 120 | C2×C3×C4 | — | 3 | 18 | no | — |
| 121 | C2×C3×C4 | — | 3 | 6 | no | — |
| 122 | C2×Dic3 | — | 9 | 1 | no | — |
| 123 | C2×Dic3 | — | 9 | 3 | no | — |
| 124 | C2×Dic3 | — | 9 | 3 | no | — |
| 125 | C2×Dic3 | — | 9 | 1 | no | — |
| 126 | D12 | — | 9 | 4 | no | — |
| 127 | D12 | — | 9 | 12 | no | — |
| 128 | D12 | — | 9 | 36 | no | — |
| 129 | D12 | — | 9 | 12 | no | — |
| 130 | C4×D3 | — | 9 | 4 | no | — |
| 131 | C4×D3 | — | 9 | 12 | no | — |
| 132 | C4×D3 | — | 9 | 36 | no | — |
| 133 | C4×D3 | — | 9 | 12 | no | — |
| 134 | SL(2,𝔽3) | — | 12 | 4 | no | — |
| 135 | SL(2,𝔽3) | — | 24 | 4 | no | — |
| 136 | C2×C32 | — | 3 | 1 | no | — |
| 137 | C2×C32 | — | 3 | 1 | no | — |
| 138 | C2×C32 | — | 16 | 1 | no | — |
| 139 | C2×C32 | — | 32 | 1 | no | — |
| 140 | C2×C32 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 141 | C3⋊D3 | — | 6 | 4 | no | — |
| 142 | C24 | — | 27 | 1 | no | — |
| 143 | #41 | — | 9 | 4 | no | — |
| 144 | #41 | — | 27 | 4 | no | — |
| 145 | #41 | — | 27 | 12 | no | — |
| 146 | #41 | — | 27 | 4 | no | — |
| 147 | C2×Q8 | — | 3 | 4 | no | — |
| 148 | C2×D4 | — | 27 | 12 | no | — |
| 149 | C22×C4 | — | 3 | 1 | no | — |
| 150 | C22×C4 | — | 9 | 1 | no | — |
| 151 | C22×C4 | — | 27 | 6 | no | — |
| 152 | C22×C4 | — | 27 | 6 | no | — |
| 153 | C22×C3 | — | 3 | 1 | no | — |
| 154 | C22×C3 | — | 3 | 3 | no | — |
| 155 | C22×C3 | — | 3 | 3 | no | — |
| 156 | C22×C3 | — | 3 | 1 | no | — |
| 157 | C22×C3 | — | 3 | 1 | no | — |
| 158 | C22×C3 | C3⋊(#41⋊C3) | 1 | 1 | yes | characteristic |
| 159 | C2×D3 | — | 9 | 12 | no | — |
| 160 | C2×D3 | — | 9 | 4 | no | — |
| 161 | C2×D3 | — | 18 | 18 | no | — |
| 162 | C2×D3 | — | 18 | 18 | no | — |
| 163 | C2×D3 | — | 18 | 6 | no | — |
| 164 | C2×D3 | — | 18 | 6 | no | — |
| 165 | A4 | — | 12 | 1 | no | — |
| 166 | A4 | — | 24 | 1 | no | — |
| 167 | C3×C4 | — | 3 | 12 | no | — |
| 168 | C3×C4 | — | 3 | 4 | no | — |
| 169 | C3×C4 | — | 144 | 1 | no | — |
| 170 | Dic3 | — | 3 | 1 | no | — |
| 171 | Dic3 | — | 9 | 1 | no | — |
| 172 | Dic3 | — | 9 | 3 | no | — |
| 173 | Dic3 | — | 9 | 1 | no | — |
| 174 | C32 | — | 16 | 1 | no | — |
| 175 | C32 | — | 32 | 1 | no | — |
| 176 | C32 | #5948 | 1 | 1 | yes | characteristic, complemented |
| 177 | C23 | — | 27 | 6 | no | — |
| 178 | C23 | — | 54 | 3 | no | — |
| 179 | C23 | — | 54 | 1 | no | — |
| 180 | C23 | (C3⋊D3)⋊A4 | 1 | 1 | yes | characteristic |
| 181 | Q8 | — | 3 | 4 | no | — |
| 182 | Q8 | (C3⋊D3)⋊A4 | 1 | 4 | yes | — |
| 183 | D4 | — | 27 | 4 | no | — |
| 184 | D4 | — | 27 | 12 | no | — |
| 185 | C2×C4 | — | 3 | 6 | no | — |
| 186 | C2×C4 | — | 27 | 4 | no | — |
| 187 | C2×C4 | — | 27 | 12 | no | — |
| 188 | C2×C4 | — | 27 | 1 | no | — |
| 189 | C2×C4 | — | 27 | 1 | no | — |
| 190 | C2×C3 | — | 3 | 3 | no | — |
| 191 | C2×C3 | — | 3 | 3 | no | — |
| 192 | C2×C3 | — | 3 | 1 | no | — |
| 193 | C2×C3 | — | 3 | 1 | no | — |
| 194 | C2×C3 | — | 3 | 1 | no | — |
| 195 | C2×C3 | — | 48 | 1 | no | — |
| 196 | C2×C3 | — | 96 | 1 | no | — |
| 197 | C2×C3 | D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 198 | D3 | — | 18 | 12 | no | — |
| 199 | D3 | — | 18 | 4 | no | — |
| 200 | C22 | — | 3 | 1 | no | — |
| 201 | C22 | — | 3 | 1 | no | — |
| 202 | C22 | — | 27 | 4 | no | — |
| 203 | C22 | — | 54 | 6 | no | — |
| 204 | C22 | — | 54 | 6 | no | — |
| 205 | C22 | #88622 | 1 | 1 | yes | characteristic, complemented |
| 206 | C4 | — | 3 | 4 | no | — |
| 207 | C4 | — | 9 | 1 | no | — |
| 208 | C4 | — | 27 | 1 | no | — |
| 209 | C3 | — | 3 | 1 | no | — |
| 210 | C3 | — | 48 | 1 | no | — |
| 211 | C3 | — | 96 | 1 | no | — |
| 212 | C3 | C3⋊#5948 | 1 | 1 | yes | characteristic |
| 213 | C2 | — | 3 | 1 | no | — |
| 214 | C2 | — | 3 | 1 | no | — |
| 215 | C2 | — | 54 | 4 | no | — |
| 216 | C2 | (C3⋊D3)⋊(C22⋊A4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊D3)⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×C32×Q8 | C2×C3 |
| Frattini | C2×C3 | D3×C22⋊A4 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ C22×C32×Q8 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722587 | C2 | semidirect |
| 2. C22×#3832 | C3 | semidirect |
| 12. C3×C22⋊SL(2,𝔽3) | D3 | semidirect |
| 25. #3832 | A4 | semidirect |
| 90. C22×C32 | #41⋊C3 | semidirect |
| 176. C32 | #5948 | semidirect |
| 205. C22 | #88622 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722587 | 2 |
| C22×#3832 | 3 |
| C3⋊#5948 | 3 |
| (C3⋊Dic3)⋊A4 | 4 |
| #88622 | 4 |