Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
1271 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 1271 conjugacy classes of subgroups — too many to draw. Only its 63 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 | C22×#66472 | C2 | 1 | 2 | yes | maximal |
| 2 | C22×#66472 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13049467 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C22×C3⋊(C3⋊(C3⋊D3)) | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×#66472 | — | 2 | 4 | no | — |
| 6 | C2×#66472 | C4 | 1 | 4 | yes | complemented |
| 7 | C2×#66472 | C22 | 1 | 4 | yes | — |
| 8 | C2×#66472 | C22 | 1 | 2 | yes | — |
| 9 | C22×C34 | Q8 | 1 | 1 | yes | characteristic |
| 10 | C2×C3⋊(C3⋊(C3⋊D3)) | C2×C4 | 1 | 2 | yes | — |
| 11 | C2×C3⋊(C3⋊(C3⋊D3)) | D4 | 1 | 2 | yes | — |
| 12 | C2×C3⋊(C3⋊(C3⋊D3)) | Q8 | 1 | 1 | yes | characteristic |
| 13 | C2×C3⋊(C3⋊(C3⋊D3)) | C23 | 1 | 1 | yes | characteristic |
| 14 | #66472 | — | 2 | 4 | no | — |
| 15 | #66472 | C2×C4 | 1 | 4 | yes | complemented |
| 16 | C2×#3777 | — | 9 | 4 | no | maximal |
| 17 | C22×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 18 | C22×C3⋊(C3⋊D3) | — | 12 | 6 | no | — |
| 19 | C2×C34 | C4⋊C4 | 1 | 2 | yes | complemented |
| 20 | C2×C34 | C2×Q8 | 1 | 1 | yes | characteristic |
| 21 | C3⋊(C3⋊(C3⋊D3)) | C4⋊C4 | 1 | 2 | yes | complemented |
| 22 | C3⋊(C3⋊(C3⋊D3)) | C22×C4 | 1 | 1 | yes | characteristic |
| 23 | C3⋊(C3⋊(C3⋊D3)) | C2×D4 | 1 | 1 | yes | characteristic |
| 24 | C22×#157 | — | 9 | 4 | no | — |
| 25 | C22×#157 | — | 9 | 8 | no | — |
| 26 | C22×#157 | — | 18 | 3 | no | — |
| 27 | C22×#157 | — | 18 | 6 | no | — |
| 28 | #3777 | — | 9 | 16 | no | — |
| 29 | C22×C33 | — | 4 | 6 | no | — |
| 30 | C22×C33 | — | 4 | 4 | no | — |
| 31 | C2×C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 32 | C2×C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 33 | C2×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 34 | C2×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 35 | C2×C3⋊(C3⋊D3) | — | 12 | 12 | no | — |
| 36 | C2×C3⋊(C3⋊D3) | — | 12 | 12 | no | — |
| 37 | C2×C3⋊(C3⋊D3) | — | 12 | 6 | no | — |
| 38 | C2×C3⋊(C3⋊D3) | — | 12 | 6 | no | — |
| 39 | C34 | C2×C4⋊C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 40 | C22×C3⋊D3 | — | 9 | 4 | no | — |
| 41 | C22×C3⋊D3 | — | 18 | 3 | no | — |
| 42 | C22×C3⋊D3 | — | 18 | 6 | no | — |
| 43 | C22×C3⋊D3 | — | 36 | 1 | no | — |
| 44 | C22×C3⋊D3 | — | 36 | 18 | no | — |
| 45 | C22×C3⋊D3 | — | 36 | 8 | no | — |
| 46 | C2×#157 | — | 9 | 16 | no | — |
| 47 | C2×#157 | — | 9 | 8 | no | — |
| 48 | C2×#157 | — | 9 | 16 | no | — |
| 49 | C2×#157 | — | 18 | 12 | no | — |
| 50 | C2×#157 | — | 18 | 6 | no | — |
| 51 | C2×#157 | — | 18 | 12 | no | — |
| 52 | C2×#157 | — | 18 | 16 | no | — |
| 53 | C2×#157 | — | 18 | 24 | no | — |
| 54 | C2×C33 | — | 4 | 12 | no | — |
| 55 | C2×C33 | — | 4 | 6 | no | — |
| 56 | C2×C33 | — | 4 | 4 | no | — |
| 57 | C2×C33 | — | 4 | 8 | no | — |
| 58 | C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 59 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 60 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 61 | C3⋊(C3⋊D3) | — | 12 | 12 | no | — |
| 62 | C3⋊(C3⋊D3) | — | 12 | 6 | no | — |
| 63 | C3⋊(C3⋊D3) | — | 12 | 6 | no | — |
| 64 | C22×C32 | — | 2 | 3 | no | — |
| 65 | C22×C32 | — | 2 | 6 | no | — |
| 66 | C22×C32 | — | 4 | 8 | no | — |
| 67 | C22×C32 | — | 4 | 18 | no | — |
| 68 | C22×C32 | — | 4 | 1 | no | — |
| 69 | C22×C32 | PSU(3,𝔽2) | 1 | 4 | yes | — |
| 70 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 71 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 72 | C2×C3⋊D3 | — | 9 | 8 | no | — |
| 73 | C2×C3⋊D3 | — | 9 | 8 | no | — |
| 74 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 75 | C2×C3⋊D3 | — | 18 | 12 | no | — |
| 76 | C2×C3⋊D3 | — | 18 | 3 | no | — |
| 77 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 78 | C2×C3⋊D3 | — | 18 | 12 | no | — |
| 79 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 80 | C2×C3⋊D3 | — | 18 | 3 | no | — |
| 81 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 82 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 83 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 84 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 85 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 86 | C2×C3⋊D3 | — | 36 | 36 | no | — |
| 87 | C2×C3⋊D3 | — | 36 | 36 | no | — |
| 88 | C2×C3⋊D3 | — | 36 | 18 | no | — |
| 89 | C2×C3⋊D3 | — | 36 | 18 | no | — |
| 90 | C2×C3⋊D3 | — | 36 | 16 | no | — |
| 91 | C2×C3⋊D3 | — | 36 | 16 | no | — |
| 92 | C2×C3⋊D3 | — | 36 | 8 | no | — |
| 93 | C2×C3⋊D3 | — | 36 | 8 | no | — |
| 94 | #157 | — | 9 | 16 | no | — |
| 95 | #157 | — | 18 | 12 | no | — |
| 96 | #157 | — | 18 | 16 | no | — |
| 97 | #157 | — | 18 | 24 | no | — |
| 98 | C2×C4⋊C4 | — | 81 | 1 | no | Sylow |
| 99 | C33 | — | 4 | 6 | no | — |
| 100 | C33 | — | 4 | 4 | no | — |
| 101 | C22×D3 | — | 108 | 4 | no | — |
| 102 | C22×D3 | — | 108 | 6 | no | — |
| 103 | C2×C32 | — | 2 | 3 | no | — |
| 104 | C2×C32 | — | 2 | 6 | no | — |
| 105 | C2×C32 | — | 2 | 6 | no | — |
| 106 | C2×C32 | — | 2 | 12 | no | — |
| 107 | C2×C32 | — | 4 | 8 | no | — |
| 108 | C2×C32 | — | 4 | 18 | no | — |
| 109 | C2×C32 | — | 4 | 1 | no | — |
| 110 | C2×C32 | — | 4 | 16 | no | — |
| 111 | C2×C32 | — | 4 | 36 | no | — |
| 112 | C2×C32 | — | 4 | 2 | no | — |
| 113 | C2×C32 | #3777 | 1 | 8 | yes | complemented |
| 114 | C2×C32 | C2×PSU(3,𝔽2) | 1 | 4 | yes | — |
| 115 | C3⋊D3 | — | 9 | 4 | no | — |
| 116 | C3⋊D3 | — | 9 | 4 | no | — |
| 117 | C3⋊D3 | — | 9 | 8 | no | — |
| 118 | C3⋊D3 | — | 18 | 3 | no | — |
| 119 | C3⋊D3 | — | 18 | 6 | no | — |
| 120 | C3⋊D3 | — | 18 | 6 | no | — |
| 121 | C3⋊D3 | — | 18 | 12 | no | — |
| 122 | C3⋊D3 | — | 18 | 6 | no | — |
| 123 | C3⋊D3 | — | 18 | 3 | no | — |
| 124 | C3⋊D3 | — | 36 | 2 | no | — |
| 125 | C3⋊D3 | — | 36 | 1 | no | — |
| 126 | C3⋊D3 | — | 36 | 1 | no | — |
| 127 | C3⋊D3 | — | 36 | 36 | no | — |
| 128 | C3⋊D3 | — | 36 | 18 | no | — |
| 129 | C3⋊D3 | — | 36 | 18 | no | — |
| 130 | C3⋊D3 | — | 36 | 16 | no | — |
| 131 | C3⋊D3 | — | 36 | 8 | no | — |
| 132 | C3⋊D3 | — | 36 | 8 | no | — |
| 133 | C22×C4 | — | 81 | 1 | no | — |
| 134 | C22×C4 | — | 81 | 2 | no | — |
| 135 | C4⋊C4 | — | 81 | 4 | no | — |
| 136 | C22×C3 | — | 4 | 6 | no | — |
| 137 | C22×C3 | — | 4 | 4 | no | — |
| 138 | C2×D3 | — | 108 | 4 | no | — |
| 139 | C2×D3 | — | 108 | 4 | no | — |
| 140 | C2×D3 | — | 108 | 8 | no | — |
| 141 | C2×D3 | — | 108 | 8 | no | — |
| 142 | C2×D3 | — | 108 | 6 | no | — |
| 143 | C2×D3 | — | 108 | 6 | no | — |
| 144 | C2×D3 | — | 108 | 12 | no | — |
| 145 | C2×D3 | — | 108 | 12 | no | — |
| 146 | C32 | — | 2 | 3 | no | — |
| 147 | C32 | — | 2 | 6 | no | — |
| 148 | C32 | — | 4 | 8 | no | — |
| 149 | C32 | — | 4 | 18 | no | — |
| 150 | C32 | — | 4 | 1 | no | — |
| 151 | C32 | C2×#3777 | 1 | 4 | yes | complemented |
| 152 | C23 | — | 81 | 1 | no | — |
| 153 | C2×C4 | — | 81 | 2 | no | — |
| 154 | C2×C4 | — | 81 | 4 | no | — |
| 155 | C2×C4 | — | 81 | 4 | no | — |
| 156 | C2×C4 | — | 162 | 4 | no | — |
| 157 | C2×C3 | — | 4 | 6 | no | — |
| 158 | C2×C3 | — | 4 | 4 | no | — |
| 159 | C2×C3 | — | 4 | 12 | no | — |
| 160 | C2×C3 | — | 4 | 8 | no | — |
| 161 | D3 | — | 108 | 4 | no | — |
| 162 | D3 | — | 108 | 4 | no | — |
| 163 | D3 | — | 108 | 8 | no | — |
| 164 | D3 | — | 108 | 6 | no | — |
| 165 | D3 | — | 108 | 6 | no | — |
| 166 | D3 | — | 108 | 12 | no | — |
| 167 | C22 | — | 81 | 1 | no | — |
| 168 | C22 | — | 81 | 1 | no | — |
| 169 | C22 | — | 81 | 2 | no | — |
| 170 | C22 | — | 81 | 2 | no | — |
| 171 | C22 | #10620211 | 1 | 1 | yes | characteristic |
| 172 | C4 | — | 81 | 4 | no | — |
| 173 | C4 | — | 162 | 4 | no | — |
| 174 | C3 | — | 4 | 6 | no | — |
| 175 | C3 | — | 4 | 4 | no | — |
| 176 | C2 | — | 81 | 1 | no | — |
| 177 | C2 | — | 81 | 1 | no | — |
| 178 | C2 | — | 81 | 2 | no | — |
| 179 | C2 | #13049467 | 1 | 2 | yes | direct product factor |
| 180 | C2 | C2×#10620211 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13049467 | C2 | direct (N × H) |
| 6. C2×#66472 | C4 | semidirect |
| 15. #66472 | C2×C4 | semidirect |
| 19. C2×C34 | C4⋊C4 | semidirect |
| 21. C3⋊(C3⋊(C3⋊D3)) | C4⋊C4 | semidirect |
| 39. C34 | C2×C4⋊C4 | semidirect |
| 113. C2×C32 | #3777 | semidirect |
| 151. C32 | C2×#3777 | semidirect |
| 179. C2 | #13049467 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#66472 | 2 |
| #13049467 | 2 |
| C2×#3777 | 9 |