Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
476 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 55 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 | #11724470 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊(C3⋊(#131⋊C3)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593222 | — | 3 | 4 | no | maximal |
| 5 | #10592522 | — | 3 | 1 | no | maximal |
| 6 | C32×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊(C3⋊GL(2,𝔽3)) | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3⋊(C3⋊#226) | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | #88788 | — | 4 | 1 | no | maximal |
| 10 | #64266 | — | 3 | 4 | no | — |
| 11 | #64266 | — | 6 | 4 | no | — |
| 12 | #64159 | — | 3 | 1 | no | — |
| 13 | #64157 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #64116 | — | 3 | 1 | no | — |
| 15 | C3⋊(#131⋊C3) | — | 3 | 4 | no | — |
| 16 | C3×GU(2,𝔽3) | D3 | 1 | 4 | yes | complemented |
| 17 | C32×#104 | — | 3 | 1 | no | — |
| 18 | #63649 | — | 3 | 1 | no | — |
| 19 | #63632 | — | 3 | 1 | no | — |
| 20 | #63630 | — | 3 | 1 | no | — |
| 21 | C4×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 22 | C32×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C3×C3⋊(C3⋊C8) | — | 4 | 1 | no | — |
| 24 | C32×C3⋊C8 | — | 4 | 1 | no | — |
| 25 | #5405 | — | 9 | 1 | no | — |
| 26 | #4823 | — | 9 | 4 | no | — |
| 27 | C32×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | #3828 | — | 3 | 1 | no | — |
| 29 | C3×#41⋊C3 | — | 2 | 4 | no | — |
| 30 | C3×#41⋊C3 | C2×D3 | 1 | 4 | yes | — |
| 31 | C3⋊GL(2,𝔽3) | — | 3 | 4 | no | — |
| 32 | C3⋊GL(2,𝔽3) | — | 6 | 4 | no | — |
| 33 | C3⋊#226 | — | 3 | 4 | no | — |
| 34 | C3⋊#226 | — | 6 | 4 | no | — |
| 35 | C32×M16 | — | 3 | 1 | no | — |
| 36 | C32×C42 | — | 3 | 1 | no | — |
| 37 | #3756 | — | 3 | 1 | no | — |
| 38 | #3755 | — | 3 | 1 | no | — |
| 39 | #3754 | — | 3 | 1 | no | — |
| 40 | #3753 | — | 3 | 1 | no | — |
| 41 | #3748 | — | 3 | 1 | no | — |
| 42 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 43 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 44 | #3743 | — | 6 | 1 | no | — |
| 45 | C8×C3⋊D3 | — | 6 | 1 | no | — |
| 46 | #3712 | — | 12 | 4 | no | — |
| 47 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 48 | C33×C4 | — | 4 | 1 | no | — |
| 49 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 50 | #985 | — | 9 | 1 | no | — |
| 51 | #985 | — | 18 | 4 | no | — |
| 52 | #950 | — | 9 | 4 | no | — |
| 53 | #948 | — | 3 | 4 | no | — |
| 54 | #907 | — | 9 | 4 | no | — |
| 55 | #131⋊C3 | — | 9 | 1 | no | — |
| 56 | GU(2,𝔽3) | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 57 | C3×#104 | — | 3 | 4 | no | — |
| 58 | #822 | — | 9 | 4 | no | — |
| 59 | #805 | — | 9 | 4 | no | — |
| 60 | #803 | — | 9 | 4 | no | — |
| 61 | C32×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 62 | C32×D4 | — | 3 | 1 | no | — |
| 63 | C2×C32×C4 | — | 3 | 1 | no | — |
| 64 | C2×C32×C4 | — | 6 | 1 | no | — |
| 65 | #638 | — | 6 | 1 | no | — |
| 66 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 67 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 68 | C4×C3⋊D3 | — | 12 | 4 | no | — |
| 69 | C4×C3⋊D3 | — | 24 | 4 | no | — |
| 70 | #634 | — | 3 | 1 | no | — |
| 71 | C3×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 72 | C3×SL(2,𝔽3) | C4×D3 | 1 | 4 | yes | — |
| 73 | C32×C8 | — | 6 | 1 | no | — |
| 74 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 75 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 76 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 77 | C3⋊(C3⋊C8) | S4 | 1 | 1 | yes | characteristic |
| 78 | C3×C3⋊C8 | — | 4 | 4 | no | — |
| 79 | C3×C3⋊C8 | — | 12 | 4 | no | — |
| 80 | #444 | — | 27 | 1 | no | Sylow |
| 81 | C2×C33 | — | 4 | 1 | no | — |
| 82 | C3⋊(C3⋊D3) | — | 8 | 1 | no | — |
| 83 | C3×#41 | D32 | 1 | 4 | yes | — |
| 84 | #235 | — | 9 | 4 | no | — |
| 85 | #41⋊C3 | — | 2 | 4 | no | — |
| 86 | #41⋊C3 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 87 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 88 | GL(2,𝔽3) | — | 18 | 4 | no | — |
| 89 | #226 | — | 9 | 1 | no | — |
| 90 | #226 | — | 18 | 4 | no | — |
| 91 | C3×M16 | — | 3 | 4 | no | — |
| 92 | C3×C42 | — | 3 | 4 | no | — |
| 93 | #216 | — | 9 | 4 | no | — |
| 94 | #215 | — | 9 | 4 | no | — |
| 95 | #214 | — | 9 | 4 | no | — |
| 96 | #213 | — | 9 | 4 | no | — |
| 97 | #208 | — | 9 | 4 | no | — |
| 98 | C2×C3⋊C8 | — | 9 | 4 | no | — |
| 99 | C2×C3⋊C8 | — | 9 | 4 | no | — |
| 100 | ΓL(1,𝔽25) | — | 18 | 4 | no | — |
| 101 | ΓL(1,𝔽25) | — | 36 | 1 | no | — |
| 102 | D3×C8 | — | 18 | 4 | no | — |
| 103 | C22×C32 | — | 3 | 1 | no | — |
| 104 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 105 | C2×C3⋊D3 | — | 12 | 4 | no | — |
| 106 | C2×C3⋊D3 | — | 24 | 4 | no | — |
| 107 | C32×C4 | — | 3 | 1 | no | — |
| 108 | C32×C4 | — | 4 | 4 | no | — |
| 109 | C32×C4 | — | 8 | 4 | no | — |
| 110 | C32×C4 | — | 12 | 1 | no | — |
| 111 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 112 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 113 | C3⋊Dic3 | — | 12 | 4 | no | — |
| 114 | C3⋊Dic3 | — | 24 | 4 | no | — |
| 115 | #135 | — | 27 | 1 | no | — |
| 116 | #131 | — | 9 | 1 | no | — |
| 117 | #131 | — | 27 | 1 | no | — |
| 118 | #108 | — | 27 | 1 | no | — |
| 119 | #104 | — | 3 | 1 | no | — |
| 120 | #104 | — | 27 | 1 | no | — |
| 121 | #97 | — | 27 | 1 | no | — |
| 122 | C33 | — | 4 | 1 | no | Sylow |
| 123 | C3×Q8 | #624 | 1 | 4 | yes | — |
| 124 | C3×D4 | — | 3 | 4 | no | — |
| 125 | C2×C3×C4 | — | 3 | 4 | no | — |
| 126 | C2×C3×C4 | — | 6 | 4 | no | — |
| 127 | #67 | — | 18 | 4 | no | — |
| 128 | D12 | — | 9 | 4 | no | — |
| 129 | C4×D3 | — | 18 | 4 | no | — |
| 130 | C4×D3 | — | 36 | 1 | no | — |
| 131 | C4×D3 | — | 72 | 4 | no | — |
| 132 | Dic6 | — | 9 | 4 | no | — |
| 133 | SL(2,𝔽3) | — | 2 | 4 | no | — |
| 134 | SL(2,𝔽3) | C4×C3⋊D3 | 1 | 1 | yes | characteristic |
| 135 | C3×C8 | — | 6 | 4 | no | — |
| 136 | C3×C8 | — | 36 | 1 | no | — |
| 137 | C3⋊C8 | — | 3 | 4 | no | — |
| 138 | C3⋊C8 | — | 4 | 1 | no | — |
| 139 | C3⋊C8 | — | 9 | 4 | no | — |
| 140 | C3⋊C8 | — | 9 | 4 | no | — |
| 141 | C3⋊C8 | — | 9 | 4 | no | — |
| 142 | C2×C32 | — | 4 | 4 | no | — |
| 143 | C2×C32 | — | 6 | 1 | no | — |
| 144 | C2×C32 | — | 8 | 4 | no | — |
| 145 | C2×C32 | C4×S4 | 1 | 1 | yes | characteristic |
| 146 | C3⋊D3 | — | 12 | 1 | no | — |
| 147 | C3⋊D3 | — | 24 | 4 | no | — |
| 148 | C3⋊D3 | — | 24 | 8 | no | — |
| 149 | #41 | — | 27 | 1 | no | — |
| 150 | #41 | D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 151 | Q16 | — | 27 | 1 | no | — |
| 152 | SD16 | — | 27 | 1 | no | — |
| 153 | SD16 | — | 27 | 1 | no | — |
| 154 | D8 | — | 27 | 1 | no | — |
| 155 | M16 | — | 3 | 1 | no | — |
| 156 | M16 | — | 27 | 1 | no | — |
| 157 | M16 | — | 54 | 1 | no | — |
| 158 | C2×C8 | — | 27 | 1 | no | — |
| 159 | C2×C8 | — | 27 | 1 | no | — |
| 160 | C2×C8 | — | 54 | 1 | no | — |
| 161 | C42 | — | 3 | 1 | no | — |
| 162 | C22×C3 | — | 3 | 4 | no | — |
| 163 | C2×D3 | — | 18 | 4 | no | — |
| 164 | C2×D3 | — | 36 | 1 | no | — |
| 165 | C2×D3 | — | 72 | 4 | no | — |
| 166 | C3×C4 | — | 3 | 4 | no | — |
| 167 | C3×C4 | — | 4 | 1 | no | — |
| 168 | C3×C4 | — | 8 | 4 | no | — |
| 169 | C3×C4 | — | 12 | 4 | no | — |
| 170 | C3×C4 | D3×S4 | 1 | 4 | yes | — |
| 171 | Dic3 | — | 18 | 4 | no | — |
| 172 | Dic3 | — | 36 | 1 | no | — |
| 173 | Dic3 | — | 72 | 4 | no | — |
| 174 | C32 | — | 4 | 4 | no | — |
| 175 | C32 | — | 8 | 4 | no | — |
| 176 | C32 | #5405 | 1 | 1 | yes | characteristic, complemented |
| 177 | Q8 | — | 27 | 1 | no | — |
| 178 | Q8 | #6289 | 1 | 1 | yes | characteristic |
| 179 | D4 | — | 3 | 1 | no | — |
| 180 | D4 | — | 27 | 1 | no | — |
| 181 | D4 | — | 54 | 1 | no | — |
| 182 | C2×C4 | — | 3 | 1 | no | — |
| 183 | C2×C4 | — | 6 | 1 | no | — |
| 184 | C2×C4 | — | 54 | 1 | no | — |
| 185 | C8 | — | 6 | 1 | no | — |
| 186 | C8 | — | 9 | 1 | no | — |
| 187 | C8 | — | 27 | 1 | no | — |
| 188 | C8 | — | 27 | 1 | no | — |
| 189 | C8 | — | 27 | 1 | no | — |
| 190 | C2×C3 | — | 4 | 1 | no | — |
| 191 | C2×C3 | — | 6 | 4 | no | — |
| 192 | C2×C3 | — | 8 | 4 | no | — |
| 193 | C2×C3 | (C3⋊S4)⋊C4 | 1 | 4 | yes | — |
| 194 | D3 | — | 36 | 4 | no | — |
| 195 | D3 | — | 72 | 1 | no | — |
| 196 | D3 | — | 72 | 8 | no | — |
| 197 | C22 | — | 3 | 1 | no | — |
| 198 | C22 | — | 54 | 1 | no | — |
| 199 | C4 | — | 3 | 1 | no | — |
| 200 | C4 | — | 12 | 1 | no | — |
| 201 | C4 | — | 54 | 1 | no | — |
| 202 | C4 | C3⋊D3×S4 | 1 | 1 | yes | characteristic |
| 203 | C3 | — | 4 | 1 | no | — |
| 204 | C3 | — | 8 | 4 | no | — |
| 205 | C3 | #10593222 | 1 | 4 | yes | complemented |
| 206 | C2 | — | 6 | 1 | no | — |
| 207 | C2 | — | 108 | 1 | no | — |
| 208 | C2 | #11724186 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | C3⋊D3×S4 |
| commutator, hypocenter | C32×SL(2,𝔽3) | C2×C4 |
| Fitting | C32×#41 | C2×D3 |
| socle | C2×C32 | C4×S4 |
| prime radical | #11724470 | C2 |
| cosocle | C32×#41⋊C3 | C22 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3⋊(C3⋊(#131⋊C3)) | C2 | semidirect |
| 3. C32×GU(2,𝔽3) | C2 | semidirect |
| 7. C3⋊(C3⋊GL(2,𝔽3)) | C4 | semidirect |
| 8. C3⋊(C3⋊#226) | C4 | semidirect |
| 13. #64157 | D3 | semidirect |
| 16. C3×GU(2,𝔽3) | D3 | semidirect |
| 56. GU(2,𝔽3) | C3⋊D3 | semidirect |
| 176. C32 | #5405 | semidirect |
| 205. C3 | #10593222 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724470 | 2 |
| C3⋊(C3⋊(#131⋊C3)) | 2 |
| C32×GU(2,𝔽3) | 2 |
| #10593222 | 3 |
| #10592522 | 3 |
| #88788 | 4 |