Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1400 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 1400 conjugacy classes of subgroups — too many to draw. Only its 187 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 | C32×#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11724470 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11724469 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10596587 | — | 3 | 12 | no | maximal |
| 5 | #10596307 | — | 3 | 1 | no | maximal |
| 6 | Q8×C3⋊(C3⋊D3) | — | 4 | 1 | no | maximal |
| 7 | C32×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 8 | C3⋊(C3⋊GL(2,𝔽3)) | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3⋊(C3⋊#226) | C22 | 1 | 3 | yes | complemented |
| 10 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #64367 | — | 3 | 1 | no | — |
| 12 | C3×#994 | D3 | 1 | 12 | yes | complemented |
| 13 | #64266 | — | 3 | 36 | no | — |
| 14 | #64265 | — | 3 | 36 | no | — |
| 15 | #64160 | — | 3 | 3 | no | — |
| 16 | #64159 | — | 3 | 3 | no | — |
| 17 | #64157 | — | 3 | 1 | no | — |
| 18 | C2×#3754 | — | 3 | 3 | no | — |
| 19 | #64141 | — | 3 | 3 | no | — |
| 20 | C33×Q8 | — | 4 | 1 | no | — |
| 21 | C4×C3⋊(C3⋊D3) | — | 4 | 3 | no | — |
| 22 | #6308 | — | 4 | 3 | no | — |
| 23 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 24 | #5924 | — | 9 | 9 | no | — |
| 25 | #5836 | — | 9 | 4 | no | — |
| 26 | C32×#41 | — | 3 | 1 | no | — |
| 27 | C32×#41 | C2×D3 | 1 | 3 | yes | — |
| 28 | C2×C32×D4 | — | 3 | 3 | no | — |
| 29 | Q8×C3⋊D3 | — | 6 | 1 | no | — |
| 30 | Q8×C3⋊D3 | — | 12 | 12 | no | — |
| 31 | #3830 | — | 6 | 3 | no | — |
| 32 | #3828 | — | 3 | 3 | no | — |
| 33 | C2×#634 | — | 3 | 3 | no | — |
| 34 | C3×#41⋊C3 | C2×D3 | 1 | 36 | yes | — |
| 35 | C3⋊GL(2,𝔽3) | — | 3 | 12 | no | — |
| 36 | C3⋊#226 | — | 3 | 36 | no | — |
| 37 | #3756 | — | 3 | 3 | no | — |
| 38 | #3755 | — | 3 | 1 | no | — |
| 39 | #3754 | — | 3 | 6 | no | — |
| 40 | #3754 | — | 3 | 3 | no | — |
| 41 | #3753 | — | 3 | 3 | no | — |
| 42 | #3748 | — | 3 | 3 | no | — |
| 43 | C2×C3⋊(C3⋊C8) | — | 3 | 3 | no | — |
| 44 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 45 | C33×C4 | — | 4 | 3 | no | — |
| 46 | C3⋊(C3⋊Dic3) | — | 4 | 3 | no | — |
| 47 | C3×H(5,𝔽2) | C3⋊D3 | 1 | 4 | yes | complemented |
| 48 | #1010 | — | 9 | 4 | no | — |
| 49 | #994 | C3⋊D3 | 1 | 9 | yes | complemented |
| 50 | #985 | — | 9 | 27 | no | — |
| 51 | #984 | — | 9 | 27 | no | — |
| 52 | #951 | — | 9 | 12 | no | — |
| 53 | #950 | — | 9 | 12 | no | — |
| 54 | #948 | — | 9 | 4 | no | — |
| 55 | C2×#214 | — | 9 | 12 | no | — |
| 56 | #932 | — | 9 | 12 | no | — |
| 57 | C23×C32 | — | 6 | 1 | no | — |
| 58 | C32×Q8 | — | 4 | 12 | no | — |
| 59 | C32×Q8 | S4 | 1 | 1 | yes | characteristic |
| 60 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 61 | C32×D4 | — | 3 | 3 | no | — |
| 62 | C32×D4 | — | 3 | 3 | no | — |
| 63 | C2×C32×C4 | — | 3 | 3 | no | — |
| 64 | #638 | — | 6 | 3 | no | — |
| 65 | C2×C3⋊Dic3 | — | 6 | 3 | no | — |
| 66 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 67 | C4×C3⋊D3 | — | 6 | 3 | no | — |
| 68 | C4×C3⋊D3 | — | 12 | 36 | no | — |
| 69 | #634 | — | 3 | 3 | no | — |
| 70 | #634 | — | 6 | 3 | no | — |
| 71 | #634 | — | 12 | 36 | no | — |
| 72 | C3×SL(2,𝔽3) | C22×D3 | 1 | 12 | yes | — |
| 73 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 74 | C3⋊(C3⋊C8) | — | 3 | 3 | no | — |
| 75 | #577 | — | 27 | 1 | no | Sylow |
| 76 | C2×C33 | — | 4 | 1 | no | — |
| 77 | C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 78 | C3×#41 | — | 3 | 4 | no | — |
| 79 | C3×#41 | C2×C3⋊D3 | 1 | 12 | yes | — |
| 80 | C2×C3×D4 | — | 3 | 12 | no | — |
| 81 | D3×Q8 | — | 18 | 4 | no | — |
| 82 | D3×Q8 | — | 36 | 9 | no | — |
| 83 | #237 | — | 18 | 12 | no | — |
| 84 | #235 | — | 9 | 12 | no | — |
| 85 | C2×Dic6 | — | 9 | 12 | no | — |
| 86 | #41⋊C3 | C2×C3⋊D3 | 1 | 27 | yes | — |
| 87 | GL(2,𝔽3) | — | 9 | 9 | no | — |
| 88 | #226 | — | 9 | 27 | no | — |
| 89 | #216 | — | 9 | 12 | no | — |
| 90 | #215 | — | 9 | 4 | no | — |
| 91 | #214 | — | 9 | 24 | no | — |
| 92 | #214 | — | 9 | 12 | no | — |
| 93 | #213 | — | 9 | 12 | no | — |
| 94 | #208 | — | 9 | 12 | no | — |
| 95 | C2×C3⋊C8 | — | 9 | 12 | no | — |
| 96 | C22×C32 | — | 3 | 3 | no | — |
| 97 | C22×C32 | — | 24 | 1 | no | — |
| 98 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 99 | C2×C3⋊D3 | — | 12 | 12 | no | — |
| 100 | C32×C4 | — | 3 | 1 | no | — |
| 101 | C32×C4 | — | 4 | 36 | no | — |
| 102 | C32×C4 | C2×S4 | 1 | 3 | yes | — |
| 103 | C3⋊Dic3 | — | 6 | 3 | no | — |
| 104 | C3⋊Dic3 | — | 12 | 36 | no | — |
| 105 | 2−1+4 | — | 27 | 1 | no | — |
| 106 | H(5,𝔽2) | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 107 | #137 | — | 27 | 3 | no | — |
| 108 | AGL(1,ℤ/8) | — | 27 | 3 | no | — |
| 109 | #135 | — | 27 | 3 | no | — |
| 110 | C2×SD16 | — | 27 | 3 | no | — |
| 111 | #131 | — | 27 | 1 | no | — |
| 112 | C33 | — | 4 | 1 | no | Sylow |
| 113 | C23×C3 | — | 6 | 4 | no | — |
| 114 | C3×Q8 | — | 4 | 9 | no | — |
| 115 | C3×Q8 | C3⋊S4 | 1 | 4 | yes | — |
| 116 | C3×Q8 | C22×C3⋊D3 | 1 | 4 | yes | — |
| 117 | C3×D4 | — | 3 | 12 | no | — |
| 118 | C3×D4 | — | 3 | 12 | no | — |
| 119 | C2×C3×C4 | — | 3 | 12 | no | — |
| 120 | #67 | — | 18 | 12 | no | — |
| 121 | C2×Dic3 | — | 18 | 12 | no | — |
| 122 | D12 | — | 9 | 4 | no | — |
| 123 | C4×D3 | — | 18 | 12 | no | — |
| 124 | C4×D3 | — | 36 | 27 | no | — |
| 125 | Dic6 | — | 9 | 12 | no | — |
| 126 | Dic6 | — | 18 | 12 | no | — |
| 127 | Dic6 | — | 36 | 27 | no | — |
| 128 | SL(2,𝔽3) | C22×C3⋊D3 | 1 | 9 | yes | — |
| 129 | C3⋊C8 | — | 9 | 4 | no | — |
| 130 | C3⋊C8 | — | 9 | 12 | no | — |
| 131 | C2×C32 | — | 4 | 12 | no | — |
| 132 | C2×C32 | — | 6 | 3 | no | — |
| 133 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 134 | C3⋊D3 | — | 12 | 24 | no | — |
| 135 | C3⋊D3 | — | 12 | 1 | no | — |
| 136 | #41 | — | 3 | 1 | no | — |
| 137 | #41 | — | 27 | 3 | no | — |
| 138 | #41 | — | 54 | 3 | no | — |
| 139 | #41 | C2×C3⋊(C3⋊D3) | 1 | 3 | yes | — |
| 140 | C2×Q8 | — | 27 | 3 | no | — |
| 141 | C2×Q8 | — | 54 | 1 | no | — |
| 142 | C2×D4 | — | 3 | 3 | no | — |
| 143 | Q16 | — | 27 | 3 | no | — |
| 144 | SD16 | — | 27 | 3 | no | — |
| 145 | SD16 | — | 27 | 6 | no | — |
| 146 | SD16 | — | 27 | 1 | no | — |
| 147 | D8 | — | 27 | 3 | no | — |
| 148 | M16 | — | 27 | 3 | no | — |
| 149 | C2×C8 | — | 27 | 3 | no | — |
| 150 | C22×C3 | — | 3 | 12 | no | — |
| 151 | C22×C3 | — | 24 | 4 | no | — |
| 152 | C2×D3 | — | 18 | 4 | no | — |
| 153 | C2×D3 | — | 36 | 9 | no | — |
| 154 | C3×C4 | — | 3 | 4 | no | — |
| 155 | C3×C4 | — | 4 | 27 | no | — |
| 156 | C3×C4 | C2×C3⋊S4 | 1 | 12 | yes | — |
| 157 | Dic3 | — | 18 | 12 | no | — |
| 158 | Dic3 | — | 36 | 27 | no | — |
| 159 | C32 | — | 4 | 12 | no | — |
| 160 | C32 | #5924 | 1 | 1 | yes | characteristic, complemented |
| 161 | C23 | — | 6 | 1 | no | — |
| 162 | Q8 | — | 27 | 3 | no | — |
| 163 | Q8 | — | 54 | 3 | no | — |
| 164 | Q8 | C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic |
| 165 | Q8 | C22×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 166 | D4 | — | 3 | 3 | no | — |
| 167 | D4 | — | 3 | 3 | no | — |
| 168 | D4 | — | 27 | 1 | no | — |
| 169 | D4 | — | 54 | 3 | no | — |
| 170 | C2×C4 | — | 3 | 3 | no | — |
| 171 | C2×C4 | — | 54 | 3 | no | — |
| 172 | C2×C4 | — | 54 | 3 | no | — |
| 173 | C8 | — | 27 | 1 | no | — |
| 174 | C8 | — | 27 | 3 | no | — |
| 175 | C2×C3 | — | 4 | 9 | no | — |
| 176 | C2×C3 | — | 6 | 12 | no | — |
| 177 | C2×C3 | C22×C3⋊S4 | 1 | 4 | yes | — |
| 178 | D3 | — | 36 | 18 | no | — |
| 179 | D3 | — | 36 | 4 | no | — |
| 180 | C22 | — | 3 | 3 | no | — |
| 181 | C22 | — | 24 | 1 | no | — |
| 182 | C22 | — | 54 | 1 | no | — |
| 183 | C4 | — | 3 | 1 | no | — |
| 184 | C4 | — | 54 | 3 | no | — |
| 185 | C4 | C2×C3⋊(C3⋊S4) | 1 | 3 | yes | — |
| 186 | C3 | — | 4 | 9 | no | — |
| 187 | C3 | #10596587 | 1 | 4 | yes | complemented |
| 188 | C2 | — | 6 | 3 | no | — |
| 189 | C2 | — | 108 | 1 | no | — |
| 190 | C2 | C22×C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊(C3⋊S4) |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | Q8 | C3⋊(C3⋊S4) |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#994 | C2 | semidirect |
| 2. #11724470 | C2 | semidirect |
| 3. #11724469 | C2 | semidirect |
| 8. C3⋊(C3⋊GL(2,𝔽3)) | C22 | semidirect |
| 9. C3⋊(C3⋊#226) | C22 | semidirect |
| 10. C32×H(5,𝔽2) | D3 | semidirect |
| 12. C3×#994 | D3 | semidirect |
| 47. C3×H(5,𝔽2) | C3⋊D3 | semidirect |
| 49. #994 | C3⋊D3 | semidirect |
| 106. H(5,𝔽2) | C3⋊(C3⋊D3) | semidirect |
| 160. C32 | #5924 | semidirect |
| 187. C3 | #10596587 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#994 | 2 |
| #11724470 | 2 |
| #11724469 | 2 |
| #10596587 | 3 |
| #10596307 | 3 |
| Q8×C3⋊(C3⋊D3) | 4 |