Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
228 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 24 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 | #10593570 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10593272 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10590077 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C22×PSU(3,𝔽2) | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×C22⋊#157 | — | 2 | 1 | no | — |
| 6 | C2×C22⋊#157 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22⋊AGL(1,𝔽9) | — | 2 | 1 | no | — |
| 8 | #63974 | — | 2 | 1 | no | — |
| 9 | #63747 | — | 2 | 1 | no | — |
| 10 | #63747 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C23×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 12 | C22×#157 | — | 2 | 1 | no | — |
| 13 | C22×#157 | — | 2 | 1 | no | — |
| 14 | C22×#157 | C2×C4 | 1 | 1 | yes | characteristic |
| 15 | C22×#157 | D4 | 1 | 1 | yes | characteristic |
| 16 | C2×PSU(3,𝔽2) | — | 4 | 1 | no | — |
| 17 | C2×PSU(3,𝔽2) | — | 4 | 2 | no | — |
| 18 | C2×AGL(1,𝔽9) | — | 4 | 1 | no | — |
| 19 | C2×#624 | — | 4 | 1 | no | — |
| 20 | C22⋊#157 | — | 4 | 1 | no | — |
| 21 | C22⋊#157 | — | 4 | 2 | no | — |
| 22 | C3×#217 | — | 4 | 1 | no | — |
| 23 | #3722 | — | 2 | 2 | no | — |
| 24 | #1351 | — | 9 | 1 | no | maximal, Sylow |
| 25 | D3×C22⋊C4 | — | 12 | 1 | no | — |
| 26 | C23×C32 | SD16 | 1 | 1 | yes | characteristic |
| 27 | C22×C3⋊D3 | — | 2 | 1 | no | — |
| 28 | C22×C3⋊D3 | — | 2 | 1 | no | — |
| 29 | C22×C3⋊D3 | — | 4 | 1 | no | — |
| 30 | C22×C3⋊D3 | — | 4 | 1 | no | — |
| 31 | C22×C3⋊D3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 32 | C22×C3⋊D3 | Q16 | 1 | 1 | yes | characteristic |
| 33 | C2×#157 | — | 2 | 2 | no | — |
| 34 | C2×#157 | — | 2 | 1 | no | — |
| 35 | C2×#157 | — | 2 | 1 | no | — |
| 36 | C2×#157 | — | 4 | 1 | no | — |
| 37 | C2×#157 | — | 4 | 1 | no | — |
| 38 | C2×#157 | — | 4 | 1 | no | — |
| 39 | C2×#157 | — | 4 | 1 | no | — |
| 40 | C2×#157 | — | 4 | 1 | no | — |
| 41 | C2×#157 | — | 4 | 1 | no | — |
| 42 | C2×#157 | — | 4 | 1 | no | — |
| 43 | PSU(3,𝔽2) | — | 4 | 2 | no | — |
| 44 | PSU(3,𝔽2) | — | 4 | 2 | no | — |
| 45 | AGL(1,𝔽9) | — | 8 | 1 | no | — |
| 46 | C2×C3×Dic3 | — | 4 | 2 | no | — |
| 47 | #624 | — | 8 | 1 | no | — |
| 48 | #624 | — | 8 | 1 | no | — |
| 49 | #393 | — | 9 | 1 | no | — |
| 50 | #342 | — | 9 | 1 | no | — |
| 51 | #323 | — | 9 | 1 | no | — |
| 52 | C23×D3 | — | 12 | 1 | no | — |
| 53 | C2×C4×D3 | — | 12 | 2 | no | — |
| 54 | C3×C22⋊C4 | — | 12 | 1 | no | — |
| 55 | #217 | — | 4 | 1 | no | — |
| 56 | (C2×D3)⋊C4 | — | 12 | 2 | no | — |
| 57 | C22×C32 | — | 2 | 1 | no | — |
| 58 | C22×C32 | — | 4 | 1 | no | — |
| 59 | C22×C32 | #103 | 1 | 1 | yes | characteristic |
| 60 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 61 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 62 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 63 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 64 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 65 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 66 | C2×C3⋊D3 | — | 8 | 1 | no | — |
| 67 | C2×C3⋊D3 | #99 | 1 | 1 | yes | characteristic |
| 68 | C2×C3⋊D3 | #104 | 1 | 1 | yes | characteristic |
| 69 | #157 | — | 4 | 1 | no | — |
| 70 | #157 | — | 4 | 2 | no | — |
| 71 | #157 | — | 4 | 1 | no | — |
| 72 | #157 | — | 8 | 1 | no | — |
| 73 | C3×Dic3 | — | 8 | 2 | no | — |
| 74 | C22×Q8 | — | 9 | 1 | no | — |
| 75 | C2×C22⋊C4 | — | 9 | 1 | no | — |
| 76 | C2×C22⋊C4 | — | 18 | 1 | no | — |
| 77 | C2×C22⋊C4 | — | 18 | 1 | no | — |
| 78 | C22⋊C8 | — | 18 | 1 | no | — |
| 79 | #95 | — | 9 | 1 | no | — |
| 80 | #95 | — | 18 | 1 | no | — |
| 81 | C23×C3 | — | 4 | 1 | no | — |
| 82 | C22×D3 | — | 12 | 1 | no | — |
| 83 | C22×D3 | — | 12 | 1 | no | — |
| 84 | C22×D3 | — | 12 | 1 | no | — |
| 85 | C22×D3 | — | 12 | 1 | no | — |
| 86 | C22×D3 | — | 12 | 1 | no | — |
| 87 | C22×D3 | — | 12 | 1 | no | — |
| 88 | C22×D3 | — | 24 | 2 | no | — |
| 89 | C22×D3 | — | 24 | 2 | no | — |
| 90 | C2×C3×C4 | — | 12 | 2 | no | — |
| 91 | C2×Dic3 | — | 4 | 2 | no | — |
| 92 | C4×D3 | — | 24 | 2 | no | — |
| 93 | C4×D3 | — | 24 | 2 | no | — |
| 94 | C2×C32 | — | 2 | 1 | no | — |
| 95 | C2×C32 | — | 4 | 1 | no | — |
| 96 | C2×C32 | #328 | 1 | 1 | yes | characteristic |
| 97 | C3⋊D3 | — | 2 | 1 | no | — |
| 98 | C3⋊D3 | — | 4 | 1 | no | — |
| 99 | C3⋊D3 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 100 | C3⋊D3 | #354 | 1 | 1 | yes | characteristic |
| 101 | C24 | — | 9 | 1 | no | — |
| 102 | C2×Q8 | — | 36 | 2 | no | — |
| 103 | C2×Q8 | — | 36 | 1 | no | — |
| 104 | C22×C4 | — | 9 | 1 | no | — |
| 105 | C22×C4 | — | 9 | 1 | no | — |
| 106 | C22×C4 | — | 18 | 1 | no | — |
| 107 | C22×C4 | — | 18 | 1 | no | — |
| 108 | C22×C4 | — | 36 | 1 | no | — |
| 109 | C2×C8 | — | 36 | 1 | no | — |
| 110 | C22⋊C4 | — | 12 | 1 | no | — |
| 111 | C22⋊C4 | — | 18 | 2 | no | — |
| 112 | C22⋊C4 | — | 36 | 1 | no | — |
| 113 | C22⋊C4 | — | 36 | 2 | no | — |
| 114 | C22×C3 | — | 4 | 1 | no | — |
| 115 | C22×C3 | — | 4 | 1 | no | — |
| 116 | C22×C3 | — | 4 | 1 | no | — |
| 117 | C22×C3 | — | 8 | 2 | no | — |
| 118 | C2×D3 | — | 12 | 1 | no | — |
| 119 | C2×D3 | — | 12 | 1 | no | — |
| 120 | C2×D3 | — | 12 | 1 | no | — |
| 121 | C2×D3 | — | 12 | 1 | no | — |
| 122 | C2×D3 | — | 12 | 2 | no | — |
| 123 | C2×D3 | — | 12 | 2 | no | — |
| 124 | C2×D3 | — | 24 | 1 | no | — |
| 125 | C2×D3 | — | 24 | 1 | no | — |
| 126 | C2×D3 | — | 24 | 2 | no | — |
| 127 | C2×D3 | — | 24 | 1 | no | — |
| 128 | C2×D3 | — | 24 | 1 | no | — |
| 129 | C2×D3 | — | 24 | 2 | no | — |
| 130 | C2×D3 | — | 24 | 2 | no | — |
| 131 | C3×C4 | — | 24 | 2 | no | — |
| 132 | Dic3 | — | 8 | 2 | no | — |
| 133 | C32 | #1351 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 134 | C23 | — | 9 | 1 | no | — |
| 135 | C23 | — | 9 | 1 | no | — |
| 136 | C23 | — | 18 | 1 | no | — |
| 137 | C23 | — | 18 | 1 | no | — |
| 138 | C23 | — | 36 | 1 | no | — |
| 139 | C23 | — | 36 | 1 | no | — |
| 140 | C23 | AΓL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 141 | Q8 | — | 36 | 2 | no | — |
| 142 | Q8 | — | 36 | 2 | no | — |
| 143 | C2×C4 | — | 12 | 2 | no | — |
| 144 | C2×C4 | — | 18 | 1 | no | — |
| 145 | C2×C4 | — | 18 | 1 | no | — |
| 146 | C2×C4 | — | 18 | 2 | no | — |
| 147 | C2×C4 | — | 36 | 1 | no | — |
| 148 | C2×C4 | — | 36 | 1 | no | — |
| 149 | C2×C4 | — | 36 | 1 | no | — |
| 150 | C2×C4 | — | 36 | 1 | no | — |
| 151 | C2×C4 | — | 36 | 1 | no | — |
| 152 | C2×C4 | — | 36 | 1 | no | — |
| 153 | C2×C4 | — | 36 | 1 | no | — |
| 154 | C2×C4 | — | 72 | 1 | no | — |
| 155 | C2×C4 | — | 72 | 1 | no | — |
| 156 | C8 | — | 72 | 1 | no | — |
| 157 | C2×C3 | — | 4 | 2 | no | — |
| 158 | C2×C3 | — | 4 | 1 | no | — |
| 159 | C2×C3 | — | 8 | 1 | no | — |
| 160 | C2×C3 | — | 8 | 1 | no | — |
| 161 | D3 | — | 12 | 1 | no | — |
| 162 | D3 | — | 12 | 1 | no | — |
| 163 | D3 | — | 12 | 2 | no | — |
| 164 | D3 | — | 24 | 1 | no | — |
| 165 | D3 | — | 24 | 1 | no | — |
| 166 | C22 | — | 2 | 1 | no | — |
| 167 | C22 | — | 4 | 1 | no | — |
| 168 | C22 | — | 9 | 1 | no | — |
| 169 | C22 | — | 9 | 1 | no | — |
| 170 | C22 | — | 18 | 1 | no | — |
| 171 | C22 | — | 18 | 1 | no | — |
| 172 | C22 | — | 18 | 1 | no | — |
| 173 | C22 | — | 36 | 1 | no | — |
| 174 | C22 | — | 36 | 1 | no | — |
| 175 | C22 | — | 36 | 1 | no | — |
| 176 | C22 | — | 72 | 1 | no | — |
| 177 | C22 | PSU(3,𝔽2)⋊C4 | 1 | 1 | yes | characteristic |
| 178 | C4 | — | 24 | 2 | no | — |
| 179 | C4 | — | 36 | 1 | no | — |
| 180 | C4 | — | 36 | 2 | no | — |
| 181 | C4 | — | 36 | 1 | no | — |
| 182 | C4 | — | 72 | 1 | no | — |
| 183 | C3 | — | 4 | 1 | no | — |
| 184 | C2 | — | 2 | 1 | no | — |
| 185 | C2 | — | 4 | 1 | no | — |
| 186 | C2 | — | 9 | 1 | no | — |
| 187 | C2 | — | 9 | 1 | no | — |
| 188 | C2 | — | 18 | 1 | no | — |
| 189 | C2 | — | 36 | 1 | no | — |
| 190 | C2 | #10593165 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C22×PSU(3,𝔽2) | C4 | semidirect |
| 10. #63747 | C4 | semidirect |
| 133. C32 | #1351 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10593570 | 2 |
| #10593272 | 2 |
| #10590077 | 2 |
| #1351 | 9 |