Finite Group DatabasebetaFinite Group DatabasebetaOrder 4374.
17980 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 17980 conjugacy classes of subgroups — too many to draw. Only its 298 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×C32⋊H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C32×C3⋊(C3⋊(C3⋊D3)) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×C3⋊((C3⋊D3)⋊C3) | — | 3 | 4 | no | maximal |
| 4 | C3×#92301 | C3 | 1 | 12 | yes | maximal, direct product factor |
| 5 | C36 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C33×H(3,𝔽3) | — | 2 | 4 | no | — |
| 7 | C33×H(3,𝔽3) | D3 | 1 | 4 | yes | complemented |
| 8 | C3×C32⋊H(3,𝔽3) | — | 2 | 48 | no | — |
| 9 | C3×C32⋊H(3,𝔽3) | C2×C3 | 1 | 12 | yes | complemented |
| 10 | C3×C3⋊(C3⋊(C3⋊D3)) | C32 | 1 | 4 | yes | complemented |
| 11 | C32×C3⋊(C3⋊D3) | — | 3 | 4 | no | — |
| 12 | C32×C3⋊(C3⋊D3) | — | 9 | 12 | no | — |
| 13 | C33×C3⋊D3 | — | 9 | 1 | no | — |
| 14 | C3×C3⋊((C3⋊D3)⋊C3) | — | 3 | 48 | no | — |
| 15 | C32×(C3⋊D3)⋊C3 | — | 9 | 12 | no | — |
| 16 | #92301 | C32 | 1 | 9 | yes | direct product factor |
| 17 | C35 | — | 2 | 16 | no | — |
| 18 | C35 | — | 2 | 4 | no | — |
| 19 | C35 | — | 3 | 12 | no | — |
| 20 | C35 | — | 6 | 48 | no | — |
| 21 | C35 | C3×D3 | 1 | 4 | yes | — |
| 22 | C35 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | C35 | C2×C32 | 1 | 4 | yes | complemented |
| 24 | C32×H(3,𝔽3) | — | 2 | 96 | no | — |
| 25 | C32×H(3,𝔽3) | — | 2 | 48 | no | — |
| 26 | C32×H(3,𝔽3) | — | 2 | 48 | no | — |
| 27 | C32×H(3,𝔽3) | — | 3 | 12 | no | — |
| 28 | C32×H(3,𝔽3) | — | 6 | 48 | no | — |
| 29 | C32×H(3,𝔽3) | — | 6 | 96 | no | — |
| 30 | C32×H(3,𝔽3) | — | 6 | 12 | no | — |
| 31 | C32×H(3,𝔽3) | C3×D3 | 1 | 48 | yes | complemented |
| 32 | C32⋊H(3,𝔽3) | — | 2 | 216 | no | — |
| 33 | C32⋊H(3,𝔽3) | — | 2 | 144 | no | — |
| 34 | C32⋊H(3,𝔽3) | C2×C32 | 1 | 9 | yes | complemented |
| 35 | C3⋊(C3⋊(C3⋊D3)) | C33 | 1 | 1 | yes | characteristic, complemented |
| 36 | C3×C3⋊(C3⋊D3) | — | 3 | 16 | no | — |
| 37 | C3×C3⋊(C3⋊D3) | — | 9 | 48 | no | — |
| 38 | C32×C3⋊D3 | — | 9 | 1 | no | — |
| 39 | C32×C3⋊D3 | — | 9 | 12 | no | — |
| 40 | C32×C3⋊D3 | — | 9 | 12 | no | — |
| 41 | C32×C3⋊D3 | — | 27 | 12 | no | — |
| 42 | C32×C3⋊D3 | — | 27 | 27 | no | — |
| 43 | C33×D3 | — | 27 | 4 | no | — |
| 44 | C3⋊((C3⋊D3)⋊C3) | — | 3 | 36 | no | — |
| 45 | C3×(C3⋊D3)⋊C3 | — | 9 | 144 | no | — |
| 46 | C34 | — | 2 | 48 | no | — |
| 47 | C34 | — | 2 | 16 | no | — |
| 48 | C34 | — | 2 | 24 | no | — |
| 49 | C34 | — | 2 | 16 | no | — |
| 50 | C34 | — | 2 | 48 | no | — |
| 51 | C34 | — | 3 | 48 | no | — |
| 52 | C34 | — | 3 | 4 | no | — |
| 53 | C34 | — | 3 | 27 | no | — |
| 54 | C34 | — | 3 | 12 | no | — |
| 55 | C34 | — | 6 | 16 | no | — |
| 56 | C34 | — | 6 | 48 | no | — |
| 57 | C34 | — | 6 | 48 | no | — |
| 58 | C34 | — | 6 | 144 | no | — |
| 59 | C34 | — | 6 | 288 | no | — |
| 60 | C34 | — | 6 | 96 | no | — |
| 61 | C34 | — | 6 | 32 | no | — |
| 62 | C34 | — | 6 | 96 | no | — |
| 63 | C34 | — | 6 | 432 | no | — |
| 64 | C34 | — | 6 | 648 | no | — |
| 65 | C34 | — | 6 | 48 | no | — |
| 66 | C34 | — | 6 | 4 | no | — |
| 67 | C34 | — | 6 | 12 | no | — |
| 68 | C34 | (C3⋊D3)⋊C3 | 1 | 12 | yes | complemented |
| 69 | C34 | C32×D3 | 1 | 16 | yes | — |
| 70 | C34 | C3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 71 | C34 | C3×C3⋊D3 | 1 | 12 | yes | complemented |
| 72 | C34 | C2×C33 | 1 | 1 | yes | characteristic, complemented |
| 73 | C3×H(3,𝔽3) | — | 2 | 288 | no | — |
| 74 | C3×H(3,𝔽3) | — | 2 | 36 | no | — |
| 75 | C3×H(3,𝔽3) | — | 2 | 144 | no | — |
| 76 | C3×H(3,𝔽3) | — | 3 | 144 | no | — |
| 77 | C3×H(3,𝔽3) | — | 6 | 144 | no | — |
| 78 | C3×H(3,𝔽3) | — | 6 | 144 | no | — |
| 79 | C3×H(3,𝔽3) | — | 6 | 288 | no | — |
| 80 | C3×H(3,𝔽3) | — | 6 | 576 | no | — |
| 81 | C3×H(3,𝔽3) | — | 6 | 288 | no | — |
| 82 | C3×H(3,𝔽3) | — | 6 | 288 | no | — |
| 83 | C3×H(3,𝔽3) | — | 6 | 144 | no | — |
| 84 | C3×H(3,𝔽3) | C32×D3 | 1 | 36 | yes | complemented |
| 85 | C2×C33 | — | 81 | 1 | no | — |
| 86 | C3⋊(C3⋊D3) | — | 3 | 4 | no | — |
| 87 | C3⋊(C3⋊D3) | — | 9 | 12 | no | — |
| 88 | C3×C3⋊D3 | — | 9 | 4 | no | — |
| 89 | C3×C3⋊D3 | — | 9 | 9 | no | — |
| 90 | C3×C3⋊D3 | — | 9 | 48 | no | — |
| 91 | C3×C3⋊D3 | — | 27 | 48 | no | — |
| 92 | C3×C3⋊D3 | — | 27 | 108 | no | — |
| 93 | C32×D3 | — | 27 | 48 | no | — |
| 94 | C32×D3 | — | 27 | 4 | no | — |
| 95 | C32×D3 | — | 81 | 12 | no | — |
| 96 | (C3⋊D3)⋊C3 | — | 9 | 108 | no | — |
| 97 | C33 | — | 2 | 16 | no | — |
| 98 | C33 | — | 2 | 96 | no | — |
| 99 | C33 | — | 2 | 48 | no | — |
| 100 | C33 | — | 2 | 48 | no | — |
| 101 | C33 | — | 2 | 16 | no | — |
| 102 | C33 | — | 2 | 36 | no | — |
| 103 | C33 | — | 3 | 12 | no | — |
| 104 | C33 | — | 3 | 108 | no | — |
| 105 | C33 | — | 3 | 12 | no | — |
| 106 | C33 | — | 3 | 48 | no | — |
| 107 | C33 | — | 3 | 48 | no | — |
| 108 | C33 | — | 6 | 48 | no | — |
| 109 | C33 | — | 6 | 48 | no | — |
| 110 | C33 | — | 6 | 48 | no | — |
| 111 | C33 | — | 6 | 144 | no | — |
| 112 | C33 | — | 6 | 144 | no | — |
| 113 | C33 | — | 6 | 288 | no | — |
| 114 | C33 | — | 6 | 96 | no | — |
| 115 | C33 | — | 6 | 288 | no | — |
| 116 | C33 | — | 6 | 96 | no | — |
| 117 | C33 | — | 6 | 288 | no | — |
| 118 | C33 | — | 6 | 432 | no | — |
| 119 | C33 | — | 6 | 432 | no | — |
| 120 | C33 | — | 6 | 864 | no | — |
| 121 | C33 | — | 6 | 2592 | no | — |
| 122 | C33 | — | 6 | 48 | no | — |
| 123 | C33 | — | 6 | 48 | no | — |
| 124 | C33 | — | 6 | 48 | no | — |
| 125 | C33 | — | 6 | 144 | no | — |
| 126 | C33 | — | 9 | 1 | no | — |
| 127 | C33 | — | 18 | 16 | no | — |
| 128 | C33 | — | 18 | 24 | no | — |
| 129 | C33 | — | 18 | 192 | no | — |
| 130 | C33 | — | 18 | 32 | no | — |
| 131 | C33 | — | 18 | 96 | no | — |
| 132 | C33 | — | 18 | 4 | no | — |
| 133 | C33 | C3×(C3⋊D3)⋊C3 | 1 | 48 | yes | complemented |
| 134 | C33 | C3⋊((C3⋊D3)⋊C3) | 1 | 4 | yes | — |
| 135 | C33 | C33×D3 | 1 | 4 | yes | — |
| 136 | C33 | C32×C3⋊D3 | 1 | 4 | yes | — |
| 137 | C33 | C32×C3⋊D3 | 1 | 9 | yes | complemented |
| 138 | H(3,𝔽3) | — | 3 | 108 | no | — |
| 139 | H(3,𝔽3) | — | 6 | 432 | no | — |
| 140 | H(3,𝔽3) | — | 6 | 864 | no | — |
| 141 | H(3,𝔽3) | — | 6 | 108 | no | — |
| 142 | C2×C32 | — | 81 | 12 | no | — |
| 143 | C2×C32 | — | 81 | 1 | no | — |
| 144 | C3⋊D3 | — | 9 | 1 | no | — |
| 145 | C3⋊D3 | — | 9 | 12 | no | — |
| 146 | C3⋊D3 | — | 27 | 12 | no | — |
| 147 | C3⋊D3 | — | 27 | 27 | no | — |
| 148 | C3×D3 | — | 27 | 36 | no | — |
| 149 | C3×D3 | — | 27 | 16 | no | — |
| 150 | C3×D3 | — | 81 | 48 | no | — |
| 151 | C32 | — | 2 | 24 | no | — |
| 152 | C32 | — | 2 | 16 | no | — |
| 153 | C32 | — | 2 | 16 | no | — |
| 154 | C32 | — | 2 | 48 | no | — |
| 155 | C32 | — | 3 | 27 | no | — |
| 156 | C32 | — | 3 | 12 | no | — |
| 157 | C32 | — | 3 | 48 | no | — |
| 158 | C32 | — | 3 | 36 | no | — |
| 159 | C32 | — | 6 | 48 | no | — |
| 160 | C32 | — | 6 | 144 | no | — |
| 161 | C32 | — | 6 | 288 | no | — |
| 162 | C32 | — | 6 | 96 | no | — |
| 163 | C32 | — | 6 | 432 | no | — |
| 164 | C32 | — | 6 | 648 | no | — |
| 165 | C32 | — | 6 | 48 | no | — |
| 166 | C32 | — | 6 | 48 | no | — |
| 167 | C32 | — | 6 | 36 | no | — |
| 168 | C32 | — | 6 | 108 | no | — |
| 169 | C32 | — | 9 | 12 | no | — |
| 170 | C32 | — | 18 | 48 | no | — |
| 171 | C32 | — | 18 | 96 | no | — |
| 172 | C32 | — | 18 | 288 | no | — |
| 173 | C32 | — | 18 | 48 | no | — |
| 174 | C32 | #92301 | 1 | 1 | yes | characteristic, direct product factor |
| 175 | C32 | C32×(C3⋊D3)⋊C3 | 1 | 12 | yes | complemented |
| 176 | C32 | C3×C3⋊((C3⋊D3)⋊C3) | 1 | 16 | yes | — |
| 177 | C32 | C33×C3⋊D3 | 1 | 1 | yes | characteristic |
| 178 | C2×C3 | — | 81 | 9 | no | — |
| 179 | C2×C3 | — | 81 | 4 | no | — |
| 180 | D3 | — | 27 | 4 | no | — |
| 181 | D3 | — | 81 | 12 | no | — |
| 182 | C3 | — | 2 | 16 | no | — |
| 183 | C3 | — | 3 | 12 | no | — |
| 184 | C3 | — | 6 | 48 | no | — |
| 185 | C3 | — | 9 | 9 | no | — |
| 186 | C3 | — | 18 | 36 | no | — |
| 187 | C3 | C3×#92301 | 1 | 4 | yes | direct product factor |
| 188 | C3 | C32×C3⋊((C3⋊D3)⋊C3) | 1 | 4 | yes | — |
| 189 | C2 | — | 81 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #92301 |
| commutator, cosocle, hypocenter | C34 | C2×C33 |
| Frattini | C32 | C33×C3⋊D3 |
| Fitting | C32×C32⋊H(3,𝔽3) | C2 |
| socle | C34 | C3×C3⋊D3 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×C32⋊H(3,𝔽3) | C2 | semidirect |
| 2. C32×C3⋊(C3⋊(C3⋊D3)) | C3 | semidirect |
| 4. C3×#92301 | C3 | direct (N × H) |
| 5. C36 | C2×C3 | semidirect |
| 7. C33×H(3,𝔽3) | D3 | semidirect |
| 9. C3×C32⋊H(3,𝔽3) | C2×C3 | semidirect |
| 10. C3×C3⋊(C3⋊(C3⋊D3)) | C32 | semidirect |
| 16. #92301 | C32 | direct (N × H) |
| 22. C35 | C3⋊D3 | semidirect |
| 23. C35 | C2×C32 | semidirect |
| 31. C32×H(3,𝔽3) | C3×D3 | semidirect |
| 34. C32⋊H(3,𝔽3) | C2×C32 | semidirect |
| 35. C3⋊(C3⋊(C3⋊D3)) | C33 | semidirect |
| 68. C34 | (C3⋊D3)⋊C3 | semidirect |
| 71. C34 | C3×C3⋊D3 | semidirect |
| 72. C34 | C2×C33 | semidirect |
| 84. C3×H(3,𝔽3) | C32×D3 | semidirect |
| 133. C33 | C3×(C3⋊D3)⋊C3 | semidirect |
| 137. C33 | C32×C3⋊D3 | semidirect |
| 174. C32 | #92301 | direct (N × H) |
| 175. C32 | C32×(C3⋊D3)⋊C3 | semidirect |
| 187. C3 | C3×#92301 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C32⋊H(3,𝔽3) | 2 |
| C32×C3⋊(C3⋊(C3⋊D3)) | 3 |
| C32×C3⋊((C3⋊D3)⋊C3) | 3 |
| C3×#92301 | 3 |