Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
448 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 102 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 | #10596547 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | Q8×GO+(4,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | Q8×AGL(1,𝔽9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10596409 | C2 | 1 | 3 | yes | maximal |
| 5 | C4×AΓL(1,𝔽9) | C2 | 1 | 3 | yes | maximal |
| 6 | PSU(3,𝔽2)⋊Q8 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #10596404 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×AΓL(1,𝔽9) | C22 | 1 | 3 | yes | — |
| 9 | D32×Q8 | — | 2 | 1 | no | — |
| 10 | Q8×#157 | — | 2 | 1 | no | — |
| 11 | Q8×#157 | C22 | 1 | 1 | yes | characteristic |
| 12 | C4⋊PSU(3,𝔽2) | C22 | 1 | 3 | yes | — |
| 13 | C4×PSU(3,𝔽2) | C22 | 1 | 3 | yes | — |
| 14 | #64243 | — | 2 | 3 | no | — |
| 15 | C4×GO+(4,𝔽2) | C22 | 1 | 3 | yes | — |
| 16 | #64222 | C22 | 1 | 3 | yes | — |
| 17 | D32⋊Q8 | — | 2 | 3 | no | — |
| 18 | C4⋊AGL(1,𝔽9) | C22 | 1 | 3 | yes | — |
| 19 | C4×AGL(1,𝔽9) | C22 | 1 | 3 | yes | — |
| 20 | AGL(1,𝔽9)⋊C4 | C22 | 1 | 3 | yes | — |
| 21 | PSU(3,𝔽2)⋊C4 | C22 | 1 | 3 | yes | — |
| 22 | PSU(3,𝔽2)⋊C4 | C22 | 1 | 6 | yes | — |
| 23 | GO+(4,𝔽2)⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×PSU(3,𝔽2) | C23 | 1 | 3 | yes | — |
| 25 | C2×GO+(4,𝔽2) | C23 | 1 | 1 | yes | characteristic |
| 26 | C2×AGL(1,𝔽9) | C23 | 1 | 3 | yes | — |
| 27 | C2×AGL(1,𝔽9) | C23 | 1 | 1 | yes | characteristic |
| 28 | AΓL(1,𝔽9) | — | 2 | 6 | no | — |
| 29 | Q8×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 30 | C3×D3×Q8 | — | 4 | 1 | no | — |
| 31 | C4×D32 | — | 2 | 3 | no | — |
| 32 | #3797 | — | 2 | 3 | no | — |
| 33 | D3×Dic6 | — | 4 | 3 | no | — |
| 34 | C4⋊#157 | — | 2 | 3 | no | — |
| 35 | C4⋊#157 | D4 | 1 | 3 | yes | — |
| 36 | C4×#157 | — | 2 | 3 | no | — |
| 37 | C4×#157 | C23 | 1 | 3 | yes | — |
| 38 | #3777 | — | 2 | 3 | no | — |
| 39 | #3777 | — | 2 | 3 | no | — |
| 40 | #3777 | C23 | 1 | 1 | yes | characteristic |
| 41 | #3773 | — | 2 | 3 | no | — |
| 42 | #3773 | C23 | 1 | 3 | yes | — |
| 43 | D32⋊C4 | — | 2 | 3 | no | — |
| 44 | #3359 | — | 9 | 1 | no | maximal, Sylow |
| 45 | C2×D3×Q8 | — | 12 | 1 | no | — |
| 46 | C2×D32 | — | 2 | 1 | no | — |
| 47 | C2×#157 | — | 2 | 3 | no | — |
| 48 | C2×#157 | — | 2 | 1 | no | — |
| 49 | C2×#157 | C2×D4 | 1 | 3 | yes | — |
| 50 | C2×#157 | C24 | 1 | 1 | yes | characteristic |
| 51 | PSU(3,𝔽2) | — | 2 | 3 | no | — |
| 52 | PSU(3,𝔽2) | — | 2 | 3 | no | — |
| 53 | GO+(4,𝔽2) | — | 2 | 1 | no | — |
| 54 | GO+(4,𝔽2) | #41 | 1 | 2 | yes | — |
| 55 | AGL(1,𝔽9) | — | 2 | 3 | no | — |
| 56 | AGL(1,𝔽9) | — | 2 | 1 | no | — |
| 57 | C32×Q8 | SD16 | 1 | 1 | yes | characteristic, complemented |
| 58 | C4×C3⋊D3 | C2×D4 | 1 | 3 | yes | — |
| 59 | #634 | SD16 | 1 | 3 | yes | complemented |
| 60 | C3×C4×D3 | — | 4 | 3 | no | — |
| 61 | C3×Dic6 | — | 4 | 3 | no | — |
| 62 | #627 | — | 4 | 3 | no | — |
| 63 | #624 | — | 2 | 3 | no | — |
| 64 | D3×Dic3 | — | 4 | 3 | no | — |
| 65 | #557 | — | 9 | 1 | no | — |
| 66 | D4×Q8 | — | 9 | 1 | no | — |
| 67 | #487 | — | 9 | 3 | no | — |
| 68 | #475 | — | 9 | 3 | no | — |
| 69 | #461 | — | 9 | 3 | no | — |
| 70 | C8×Q8 | — | 9 | 1 | no | — |
| 71 | C4×SD16 | — | 9 | 3 | no | — |
| 72 | C2×C3×Q8 | — | 12 | 1 | no | — |
| 73 | D3×Q8 | — | 4 | 1 | no | — |
| 74 | D3×Q8 | — | 12 | 1 | no | — |
| 75 | D3×Q8 | — | 12 | 3 | no | — |
| 76 | D3×Q8 | — | 12 | 3 | no | — |
| 77 | C2×C4×D3 | — | 12 | 3 | no | — |
| 78 | C2×Dic6 | — | 12 | 3 | no | — |
| 79 | C2×C3⋊D3 | C22×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×C3×D3 | — | 4 | 1 | no | — |
| 81 | D32 | — | 2 | 2 | no | — |
| 82 | D32 | — | 4 | 1 | no | — |
| 83 | #157 | — | 2 | 3 | no | — |
| 84 | #157 | — | 4 | 3 | no | — |
| 85 | #157 | — | 4 | 1 | no | — |
| 86 | #157 | C2×#41 | 1 | 1 | yes | characteristic |
| 87 | #157 | 2−1+4 | 1 | 1 | yes | characteristic |
| 88 | C32×C4 | C2×SD16 | 1 | 3 | yes | — |
| 89 | C3⋊Dic3 | C2×SD16 | 1 | 3 | yes | — |
| 90 | C3×Dic3 | — | 4 | 3 | no | — |
| 91 | C22×Q8 | — | 18 | 1 | no | — |
| 92 | C2×SD16 | — | 9 | 3 | no | — |
| 93 | C4⋊Q8 | — | 9 | 3 | no | — |
| 94 | C4⋊Q8 | — | 9 | 3 | no | — |
| 95 | #125 | — | 18 | 3 | no | — |
| 96 | C22⋊Q8 | — | 18 | 3 | no | — |
| 97 | C4×Q8 | — | 9 | 1 | no | — |
| 98 | C4×Q8 | — | 9 | 3 | no | — |
| 99 | C4×Q8 | — | 18 | 1 | no | — |
| 100 | C4×D4 | — | 9 | 3 | no | — |
| 101 | #106 | — | 9 | 3 | no | — |
| 102 | C4⋊C8 | — | 9 | 3 | no | — |
| 103 | #103 | — | 9 | 3 | no | — |
| 104 | #103 | — | 9 | 6 | no | — |
| 105 | #102 | — | 9 | 1 | no | — |
| 106 | C4×C8 | — | 9 | 3 | no | — |
| 107 | C22×D3 | — | 12 | 1 | no | — |
| 108 | C3×Q8 | — | 4 | 1 | no | — |
| 109 | C3×Q8 | — | 12 | 3 | no | — |
| 110 | C2×C3×C4 | — | 12 | 3 | no | — |
| 111 | C2×Dic3 | — | 12 | 3 | no | — |
| 112 | C4×D3 | — | 4 | 3 | no | — |
| 113 | C4×D3 | — | 12 | 3 | no | — |
| 114 | C4×D3 | — | 12 | 3 | no | — |
| 115 | C4×D3 | — | 12 | 3 | no | — |
| 116 | Dic6 | — | 4 | 3 | no | — |
| 117 | Dic6 | — | 12 | 3 | no | — |
| 118 | Dic6 | — | 12 | 6 | no | — |
| 119 | C2×C32 | C22×SD16 | 1 | 1 | yes | characteristic |
| 120 | C3⋊D3 | #548 | 1 | 1 | yes | characteristic |
| 121 | C3⋊D3 | #578 | 1 | 1 | yes | characteristic |
| 122 | C3×D3 | — | 4 | 2 | no | — |
| 123 | C2×Q8 | — | 9 | 1 | no | — |
| 124 | C2×Q8 | — | 9 | 3 | no | — |
| 125 | C2×Q8 | — | 12 | 1 | no | — |
| 126 | C2×Q8 | — | 18 | 3 | no | — |
| 127 | C2×Q8 | — | 36 | 3 | no | — |
| 128 | C2×D4 | — | 9 | 1 | no | — |
| 129 | C22×C4 | — | 18 | 3 | no | — |
| 130 | SD16 | — | 18 | 6 | no | — |
| 131 | C2×C8 | — | 9 | 1 | no | — |
| 132 | C2×C8 | — | 9 | 3 | no | — |
| 133 | C4⋊C4 | — | 9 | 3 | no | — |
| 134 | C4⋊C4 | — | 9 | 1 | no | — |
| 135 | C4⋊C4 | — | 9 | 3 | no | — |
| 136 | C4⋊C4 | — | 18 | 3 | no | — |
| 137 | C4⋊C4 | — | 18 | 3 | no | — |
| 138 | C4⋊C4 | — | 18 | 3 | no | — |
| 139 | C4⋊C4 | — | 18 | 3 | no | — |
| 140 | C22⋊C4 | — | 18 | 3 | no | — |
| 141 | C42 | — | 9 | 3 | no | — |
| 142 | C42 | — | 18 | 3 | no | — |
| 143 | C22×C3 | — | 12 | 1 | no | — |
| 144 | C2×D3 | — | 4 | 1 | no | — |
| 145 | C2×D3 | — | 12 | 1 | no | — |
| 146 | C2×D3 | — | 12 | 2 | no | — |
| 147 | C2×D3 | — | 12 | 2 | no | — |
| 148 | C3×C4 | — | 4 | 3 | no | — |
| 149 | C3×C4 | — | 12 | 3 | no | — |
| 150 | Dic3 | — | 4 | 3 | no | — |
| 151 | Dic3 | — | 12 | 3 | no | — |
| 152 | C32 | #3359 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 153 | C23 | — | 18 | 1 | no | — |
| 154 | Q8 | — | 9 | 3 | no | — |
| 155 | Q8 | — | 12 | 3 | no | — |
| 156 | Q8 | — | 18 | 3 | no | — |
| 157 | Q8 | — | 18 | 3 | no | — |
| 158 | Q8 | — | 36 | 3 | no | — |
| 159 | Q8 | AΓL(1,𝔽9) | 1 | 1 | yes | characteristic, complemented |
| 160 | D4 | — | 9 | 2 | no | — |
| 161 | D4 | — | 18 | 1 | no | — |
| 162 | C2×C4 | — | 9 | 1 | no | — |
| 163 | C2×C4 | — | 9 | 3 | no | — |
| 164 | C2×C4 | — | 9 | 3 | no | — |
| 165 | C2×C4 | — | 12 | 3 | no | — |
| 166 | C2×C4 | — | 18 | 3 | no | — |
| 167 | C2×C4 | — | 18 | 1 | no | — |
| 168 | C2×C4 | — | 18 | 3 | no | — |
| 169 | C2×C4 | — | 36 | 3 | no | — |
| 170 | C8 | — | 18 | 3 | no | — |
| 171 | C8 | — | 18 | 1 | no | — |
| 172 | C2×C3 | — | 4 | 1 | no | — |
| 173 | C2×C3 | — | 12 | 2 | no | — |
| 174 | D3 | — | 4 | 2 | no | — |
| 175 | D3 | — | 12 | 1 | no | — |
| 176 | D3 | — | 12 | 1 | no | — |
| 177 | C22 | — | 9 | 1 | no | — |
| 178 | C22 | — | 12 | 1 | no | — |
| 179 | C22 | — | 18 | 2 | no | — |
| 180 | C22 | — | 36 | 1 | no | — |
| 181 | C4 | — | 9 | 1 | no | — |
| 182 | C4 | — | 9 | 1 | no | — |
| 183 | C4 | — | 9 | 3 | no | — |
| 184 | C4 | — | 12 | 3 | no | — |
| 185 | C4 | — | 18 | 3 | no | — |
| 186 | C4 | — | 36 | 3 | no | — |
| 187 | C4 | — | 36 | 1 | no | — |
| 188 | C4 | C2×AΓL(1,𝔽9) | 1 | 3 | yes | — |
| 189 | C3 | — | 4 | 1 | no | — |
| 190 | C2 | — | 9 | 1 | no | — |
| 191 | C2 | — | 9 | 1 | no | — |
| 192 | C2 | — | 12 | 2 | no | — |
| 193 | C2 | C22×AΓL(1,𝔽9) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10596547 | C2 | semidirect |
| 3. Q8×AGL(1,𝔽9) | C2 | semidirect |
| 6. PSU(3,𝔽2)⋊Q8 | C2 | semidirect |
| 7. #10596404 | C2 | semidirect |
| 57. C32×Q8 | SD16 | semidirect |
| 59. #634 | SD16 | semidirect |
| 152. C32 | #3359 | semidirect |
| 159. Q8 | AΓL(1,𝔽9) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10596547 | 2 |
| Q8×GO+(4,𝔽2) | 2 |
| Q8×AGL(1,𝔽9) | 2 |
| #10596409 | 2 |
| C4×AΓL(1,𝔽9) | 2 |
| PSU(3,𝔽2)⋊Q8 | 2 |
| #10596404 | 2 |
| #3359 | 9 |