Finite Group DatabasebetaFinite Group DatabasebetaOrder 4374.
12748 conjugacy classes of subgroups; only the 2700 normal subgroups are shown.
This group has 12748 conjugacy classes of subgroups — too many to draw. Only its 2700 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⋊H(5,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C3×#92404 | — | 3 | 3 | no | maximal |
| 3 | C3×C3⋊(C3⋊#272) | — | 3 | 1 | no | maximal |
| 4 | #13091502 | — | 3 | 81 | no | maximal |
| 5 | #13091466 | — | 3 | 24 | no | maximal |
| 6 | C3⋊#92338 | — | 3 | 12 | no | maximal |
| 7 | C3×H(5,𝔽3) | D3 | 1 | 3 | yes | complemented |
| 8 | C33×H(3,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | H(3,𝔽3)2 | D3 | 1 | 81 | yes | complemented |
| 10 | #10624880 | D3 | 1 | 24 | yes | complemented |
| 11 | C3×C32⋊H(3,𝔽3) | D3 | 1 | 12 | yes | complemented |
| 12 | C32×C3⋊(C3⋊D3) | — | 9 | 4 | no | — |
| 13 | #92404 | — | 3 | 3 | no | — |
| 14 | C3⋊(C3⋊#272) | — | 3 | 1 | no | — |
| 15 | C3×C3⋊#272 | — | 9 | 1 | no | — |
| 16 | C3×C3⋊#272 | — | 9 | 36 | no | — |
| 17 | C3×C3⋊#272 | — | 9 | 108 | no | — |
| 18 | C3×C3⋊#272 | — | 9 | 8 | no | — |
| 19 | C3×C3⋊#272 | — | 9 | 27 | no | — |
| 20 | C3×C3⋊#272 | — | 9 | 81 | no | — |
| 21 | #92338 | — | 9 | 9 | no | — |
| 22 | #92338 | — | 9 | 72 | no | — |
| 23 | #92338 | — | 9 | 216 | no | — |
| 24 | #92338 | — | 9 | 648 | no | — |
| 25 | C35 | C3⋊D3 | 1 | 4 | yes | complemented |
| 26 | H(5,𝔽3) | — | 3 | 3 | no | — |
| 27 | H(5,𝔽3) | — | 6 | 12 | no | — |
| 28 | H(5,𝔽3) | — | 6 | 27 | no | — |
| 29 | C32×H(3,𝔽3) | — | 3 | 1 | no | — |
| 30 | C32×H(3,𝔽3) | — | 6 | 9 | no | — |
| 31 | C32×H(3,𝔽3) | — | 6 | 4 | no | — |
| 32 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 1 | yes | characteristic |
| 33 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 8 | yes | complemented |
| 34 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 27 | yes | complemented |
| 35 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 36 | yes | complemented |
| 36 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 81 | yes | complemented |
| 37 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 108 | yes | complemented |
| 38 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 9 | yes | complemented |
| 39 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 72 | yes | complemented |
| 40 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 216 | yes | complemented |
| 41 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 648 | yes | complemented |
| 42 | C3×C3⋊(C3⋊D3) | — | 9 | 4 | no | — |
| 43 | C3×C3⋊(C3⋊D3) | — | 9 | 12 | no | — |
| 44 | C32×C3⋊D3 | — | 27 | 8 | no | — |
| 45 | C32×C3⋊D3 | — | 27 | 1 | no | — |
| 46 | C32×C3⋊D3 | — | 27 | 36 | no | — |
| 47 | C32×C3⋊D3 | — | 27 | 4 | no | — |
| 48 | C32×C3⋊D3 | — | 27 | 108 | no | — |
| 49 | C3⋊#272 | — | 9 | 8 | no | — |
| 50 | C3⋊#272 | — | 9 | 27 | no | — |
| 51 | C3⋊#272 | — | 9 | 36 | no | — |
| 52 | C3⋊#272 | — | 9 | 81 | no | — |
| 53 | C3⋊#272 | — | 9 | 108 | no | — |
| 54 | C3⋊#272 | — | 9 | 1 | no | — |
| 55 | C3×#272 | — | 27 | 27 | no | — |
| 56 | C3×#272 | — | 27 | 81 | no | — |
| 57 | C3×#272 | — | 27 | 72 | no | — |
| 58 | C3×#272 | — | 27 | 9 | no | — |
| 59 | C3×#272 | — | 27 | 648 | no | — |
| 60 | C3×#272 | — | 27 | 216 | no | — |
| 61 | C34 | — | 3 | 4 | no | — |
| 62 | C34 | — | 6 | 4 | no | — |
| 63 | C34 | — | 6 | 12 | no | — |
| 64 | C34 | — | 9 | 12 | no | — |
| 65 | C34 | — | 18 | 12 | no | — |
| 66 | C34 | C3⋊(C3⋊D3) | 1 | 4 | yes | — |
| 67 | C34 | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 68 | C34 | C3⋊(C3⋊D3) | 1 | 36 | yes | — |
| 69 | C34 | C3⋊(C3⋊D3) | 1 | 8 | yes | — |
| 70 | C34 | C3⋊(C3⋊D3) | 1 | 108 | yes | complemented |
| 71 | C3×H(3,𝔽3) | — | 2 | 4 | no | — |
| 72 | C3×H(3,𝔽3) | — | 2 | 9 | no | — |
| 73 | C3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 74 | C3×H(3,𝔽3) | — | 3 | 36 | no | — |
| 75 | C3×H(3,𝔽3) | — | 3 | 27 | no | — |
| 76 | C3×H(3,𝔽3) | — | 3 | 81 | no | — |
| 77 | C3×H(3,𝔽3) | — | 6 | 108 | no | — |
| 78 | C3×H(3,𝔽3) | — | 6 | 324 | no | — |
| 79 | C3×H(3,𝔽3) | — | 6 | 24 | no | — |
| 80 | C3×H(3,𝔽3) | — | 6 | 36 | no | — |
| 81 | C3×H(3,𝔽3) | — | 6 | 72 | no | — |
| 82 | C3×H(3,𝔽3) | — | 6 | 8 | no | — |
| 83 | C3×H(3,𝔽3) | — | 6 | 36 | no | — |
| 84 | C3×H(3,𝔽3) | — | 9 | 108 | no | — |
| 85 | C3×H(3,𝔽3) | — | 18 | 108 | no | — |
| 86 | C3×H(3,𝔽3) | #272 | 1 | 1 | yes | characteristic, complemented |
| 87 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 9 | yes | — |
| 88 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 27 | yes | — |
| 89 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 72 | yes | — |
| 90 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 81 | yes | — |
| 91 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 216 | yes | complemented |
| 92 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 648 | yes | complemented |
| 93 | C3⋊(C3⋊D3) | — | 9 | 4 | no | — |
| 94 | C3×C3⋊D3 | — | 27 | 108 | no | — |
| 95 | C3×C3⋊D3 | — | 27 | 8 | no | — |
| 96 | C3×C3⋊D3 | — | 27 | 36 | no | — |
| 97 | C3×C3⋊D3 | — | 27 | 36 | no | — |
| 98 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 99 | C3×C3⋊D3 | — | 27 | 3 | no | — |
| 100 | C3×C3⋊D3 | — | 27 | 324 | no | — |
| 101 | C3×C3⋊D3 | — | 27 | 24 | no | — |
| 102 | C3×C3⋊D3 | — | 27 | 72 | no | — |
| 103 | C3×C3⋊D3 | — | 27 | 12 | no | — |
| 104 | C3×C3⋊D3 | — | 27 | 4 | no | — |
| 105 | C32×D3 | — | 81 | 24 | no | — |
| 106 | C32×D3 | — | 81 | 81 | no | — |
| 107 | C32×D3 | — | 81 | 1 | no | — |
| 108 | C32×D3 | — | 81 | 12 | no | — |
| 109 | C32×D3 | — | 81 | 3 | no | — |
| 110 | #272 | — | 27 | 72 | no | — |
| 111 | #272 | — | 27 | 27 | no | — |
| 112 | #272 | — | 27 | 216 | no | — |
| 113 | #272 | — | 27 | 81 | no | — |
| 114 | #272 | — | 27 | 648 | no | — |
| 115 | #272 | — | 27 | 9 | no | — |
| 116 | C33 | — | 2 | 4 | no | — |
| 117 | C33 | — | 2 | 12 | no | — |
| 118 | C33 | — | 3 | 1 | no | — |
| 119 | C33 | — | 3 | 3 | no | — |
| 120 | C33 | — | 3 | 36 | no | — |
| 121 | C33 | — | 3 | 8 | no | — |
| 122 | C33 | — | 3 | 36 | no | — |
| 123 | C33 | — | 3 | 108 | no | — |
| 124 | C33 | — | 6 | 1 | no | — |
| 125 | C33 | — | 6 | 3 | no | — |
| 126 | C33 | — | 6 | 36 | no | — |
| 127 | C33 | — | 6 | 8 | no | — |
| 128 | C33 | — | 6 | 36 | no | — |
| 129 | C33 | — | 6 | 108 | no | — |
| 130 | C33 | — | 9 | 12 | no | — |
| 131 | C33 | — | 9 | 72 | no | — |
| 132 | C33 | — | 9 | 24 | no | — |
| 133 | C33 | — | 9 | 324 | no | — |
| 134 | C33 | — | 9 | 4 | no | — |
| 135 | C33 | — | 18 | 4 | no | — |
| 136 | C33 | — | 18 | 12 | no | — |
| 137 | C33 | — | 18 | 4 | no | — |
| 138 | C33 | — | 18 | 36 | no | — |
| 139 | C33 | — | 18 | 24 | no | — |
| 140 | C33 | — | 18 | 36 | no | — |
| 141 | C33 | — | 18 | 8 | no | — |
| 142 | C33 | — | 18 | 36 | no | — |
| 143 | C33 | C3⋊#272 | 1 | 4 | yes | complemented |
| 144 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 1 | yes | characteristic |
| 145 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 12 | yes | — |
| 146 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 3 | yes | — |
| 147 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 24 | yes | — |
| 148 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 81 | yes | — |
| 149 | H(3,𝔽3) | — | 2 | 36 | no | — |
| 150 | H(3,𝔽3) | — | 3 | 72 | no | — |
| 151 | H(3,𝔽3) | — | 3 | 27 | no | — |
| 152 | H(3,𝔽3) | — | 3 | 216 | no | — |
| 153 | H(3,𝔽3) | — | 3 | 81 | no | — |
| 154 | H(3,𝔽3) | — | 6 | 72 | no | — |
| 155 | H(3,𝔽3) | — | 6 | 27 | no | — |
| 156 | H(3,𝔽3) | — | 6 | 216 | no | — |
| 157 | H(3,𝔽3) | — | 6 | 81 | no | — |
| 158 | H(3,𝔽3) | — | 9 | 648 | no | — |
| 159 | H(3,𝔽3) | C3⋊#272 | 1 | 9 | yes | complemented |
| 160 | C2×C32 | — | 243 | 1 | no | — |
| 161 | C3⋊D3 | — | 27 | 1 | no | — |
| 162 | C3⋊D3 | — | 27 | 4 | no | — |
| 163 | C3⋊D3 | — | 27 | 36 | no | — |
| 164 | C3⋊D3 | — | 27 | 8 | no | — |
| 165 | C3⋊D3 | — | 27 | 108 | no | — |
| 166 | C3×D3 | — | 81 | 81 | no | — |
| 167 | C3×D3 | — | 81 | 24 | no | — |
| 168 | C3×D3 | — | 81 | 243 | no | — |
| 169 | C3×D3 | — | 81 | 72 | no | — |
| 170 | C3×D3 | — | 81 | 3 | no | — |
| 171 | C3×D3 | — | 81 | 36 | no | — |
| 172 | C3×D3 | — | 81 | 3 | no | — |
| 173 | C3×D3 | — | 81 | 6 | no | — |
| 174 | C3×D3 | — | 81 | 12 | no | — |
| 175 | C3×D3 | — | 81 | 1 | no | — |
| 176 | C3×D3 | — | 81 | 3 | no | — |
| 177 | C32 | — | 2 | 12 | no | — |
| 178 | C32 | — | 2 | 1 | no | — |
| 179 | C32 | — | 2 | 3 | no | — |
| 180 | C32 | — | 3 | 1 | no | — |
| 181 | C32 | — | 3 | 3 | no | — |
| 182 | C32 | — | 3 | 3 | no | — |
| 183 | C32 | — | 3 | 6 | no | — |
| 184 | C32 | — | 3 | 36 | no | — |
| 185 | C32 | — | 3 | 24 | no | — |
| 186 | C32 | — | 3 | 81 | no | — |
| 187 | C32 | — | 3 | 243 | no | — |
| 188 | C32 | — | 3 | 72 | no | — |
| 189 | C32 | — | 6 | 3 | no | — |
| 190 | C32 | — | 6 | 3 | no | — |
| 191 | C32 | — | 6 | 1 | no | — |
| 192 | C32 | — | 6 | 3 | no | — |
| 193 | C32 | — | 6 | 3 | no | — |
| 194 | C32 | — | 9 | 4 | no | — |
| 195 | C32 | — | 9 | 36 | no | — |
| 196 | C32 | — | 9 | 8 | no | — |
| 197 | C32 | — | 18 | 4 | no | — |
| 198 | C32 | — | 18 | 12 | no | — |
| 199 | C32 | — | 18 | 36 | no | — |
| 200 | C32 | — | 18 | 36 | no | — |
| 201 | C32 | — | 18 | 8 | no | — |
| 202 | C32 | — | 18 | 72 | no | — |
| 203 | C32 | — | 18 | 24 | no | — |
| 204 | C32 | — | 27 | 108 | no | — |
| 205 | C32 | — | 54 | 108 | no | — |
| 206 | C32 | C3⋊(C3⋊#272) | 1 | 12 | yes | — |
| 207 | C32 | C3⋊(C3⋊#272) | 1 | 1 | yes | characteristic |
| 208 | C32 | #92404 | 1 | 3 | yes | complemented |
| 209 | C32 | C3⋊(C3⋊(C3⋊(C3⋊D3))) | 1 | 1 | yes | characteristic |
| 210 | C2×C3 | — | 243 | 1 | no | — |
| 211 | C2×C3 | — | 243 | 3 | no | — |
| 212 | D3 | — | 81 | 1 | no | — |
| 213 | D3 | — | 81 | 3 | no | — |
| 214 | D3 | — | 81 | 12 | no | — |
| 215 | D3 | — | 81 | 24 | no | — |
| 216 | D3 | — | 81 | 81 | no | — |
| 217 | C3 | — | 3 | 1 | no | — |
| 218 | C3 | — | 3 | 3 | no | — |
| 219 | C3 | — | 3 | 12 | no | — |
| 220 | C3 | — | 6 | 1 | no | — |
| 221 | C3 | — | 6 | 3 | no | — |
| 222 | C3 | — | 6 | 12 | no | — |
| 223 | C3 | — | 9 | 24 | no | — |
| 224 | C3 | — | 9 | 81 | no | — |
| 225 | C3 | C3⋊(C3⋊(C3⋊#272)) | 1 | 1 | yes | characteristic |
| 226 | C3 | C3⋊#92404 | 1 | 3 | yes | — |
| 227 | C2 | — | 243 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C32 | C3⋊(C3⋊(C3⋊(C3⋊D3))) |
| commutator, Fitting, cosocle, hypocenter | C32⋊H(5,𝔽3) | C2 |
| LCS | G ▷ C32⋊H(5,𝔽3) | |
| UCS | C1 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32⋊H(5,𝔽3) | C2 | semidirect |
| 7. C3×H(5,𝔽3) | D3 | semidirect |
| 8. C33×H(3,𝔽3) | D3 | semidirect |
| 9. H(3,𝔽3)2 | D3 | semidirect |
| 10. #10624880 | D3 | semidirect |
| 11. C3×C32⋊H(3,𝔽3) | D3 | semidirect |
| 25. C35 | C3⋊D3 | semidirect |
| 33. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 34. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 35. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 36. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 37. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 38. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 39. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 40. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 41. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 70. C34 | C3⋊(C3⋊D3) | semidirect |
| 86. C3×H(3,𝔽3) | #272 | semidirect |
| 91. C3×H(3,𝔽3) | C3⋊(C3⋊D3) | semidirect |
| 92. C3×H(3,𝔽3) | C3⋊(C3⋊D3) | semidirect |
| 143. C33 | C3⋊#272 | semidirect |
| 159. H(3,𝔽3) | C3⋊#272 | semidirect |
| 208. C32 | #92404 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32⋊H(5,𝔽3) | 2 |
| C3×#92404 | 3 |
| C3×C3⋊(C3⋊#272) | 3 |
| #13091502 | 3 |
| #13091466 | 3 |
| C3⋊#92338 | 3 |