Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
2384 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 2384 conjugacy classes of subgroups — too many to draw. Only its 94 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 | C3×#5948 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C22×C3×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #87543 | C3 | 1 | 3 | yes | maximal, direct product factor |
| 4 | C3×C22⋊SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | C3×#994 | — | 4 | 4 | no | maximal |
| 6 | C3×Q8×A4 | — | 4 | 1 | no | maximal |
| 7 | C2×C3×H(5,𝔽2) | — | 3 | 12 | no | — |
| 8 | C2×C3×H(5,𝔽2) | — | 3 | 4 | no | — |
| 9 | C22×C3×#41 | — | 3 | 1 | no | — |
| 10 | C22×C3×#41 | C2×C3 | 1 | 3 | yes | — |
| 11 | C23×C3×D4 | — | 3 | 3 | no | — |
| 12 | #5948 | C2×C3 | 1 | 9 | yes | complemented |
| 13 | C3×#41⋊C3 | — | 4 | 12 | no | — |
| 14 | C3×C4×A4 | — | 4 | 3 | no | — |
| 15 | C22×H(5,𝔽2) | C32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C25×C3 | — | 3 | 2 | no | — |
| 17 | C3×H(5,𝔽2) | — | 3 | 24 | no | — |
| 18 | C3×H(5,𝔽2) | — | 3 | 8 | no | — |
| 19 | C3×H(5,𝔽2) | — | 3 | 4 | no | — |
| 20 | C3×H(5,𝔽2) | — | 3 | 36 | no | — |
| 21 | C3×H(5,𝔽2) | — | 3 | 12 | no | — |
| 22 | C3×H(5,𝔽2) | A4 | 1 | 4 | yes | complemented |
| 23 | C2×C3×#41 | — | 3 | 12 | no | — |
| 24 | C2×C3×#41 | — | 3 | 18 | no | — |
| 25 | C2×C3×#41 | — | 3 | 6 | no | — |
| 26 | C2×C3×#41 | — | 3 | 12 | no | — |
| 27 | C22×C3×Q8 | A4 | 1 | 1 | yes | characteristic |
| 28 | C22×C3×Q8 | C22×C3 | 1 | 1 | yes | characteristic |
| 29 | C22×C3×D4 | — | 3 | 3 | no | — |
| 30 | C22×C3×D4 | — | 3 | 36 | no | — |
| 31 | C22×C3×D4 | — | 3 | 36 | no | — |
| 32 | C22×C3×D4 | — | 3 | 3 | no | — |
| 33 | C23×C3×C4 | — | 3 | 3 | no | — |
| 34 | C22⋊SL(2,𝔽3) | C22×C3 | 1 | 3 | yes | complemented |
| 35 | #994 | — | 4 | 12 | no | — |
| 36 | Q8×A4 | — | 4 | 3 | no | — |
| 37 | C2×C3×A4 | — | 4 | 1 | no | — |
| 38 | C32×Q8 | — | 16 | 1 | no | — |
| 39 | C3×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 40 | C2×H(5,𝔽2) | — | 3 | 12 | no | — |
| 41 | C2×H(5,𝔽2) | — | 3 | 4 | no | — |
| 42 | C22×#41 | — | 3 | 1 | no | — |
| 43 | C22×#41 | C2×C32 | 1 | 3 | yes | — |
| 44 | C23×D4 | — | 3 | 3 | no | — |
| 45 | C24×C3 | — | 3 | 24 | no | — |
| 46 | C24×C3 | — | 3 | 3 | no | — |
| 47 | C24×C3 | — | 12 | 6 | no | — |
| 48 | C24×C3 | — | 12 | 2 | no | — |
| 49 | C3×#41 | — | 3 | 36 | no | — |
| 50 | C3×#41 | — | 3 | 24 | no | — |
| 51 | C3×#41 | — | 3 | 36 | no | — |
| 52 | C3×#41 | — | 3 | 12 | no | — |
| 53 | C3×#41 | — | 3 | 12 | no | — |
| 54 | C3×#41 | — | 3 | 4 | no | — |
| 55 | C3×#41 | C2×A4 | 1 | 12 | yes | — |
| 56 | C2×C3×Q8 | — | 3 | 3 | no | — |
| 57 | C2×C3×Q8 | — | 3 | 4 | no | — |
| 58 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 59 | C2×C3×D4 | — | 3 | 36 | no | — |
| 60 | C2×C3×D4 | — | 3 | 18 | no | — |
| 61 | C2×C3×D4 | — | 3 | 18 | no | — |
| 62 | C2×C3×D4 | — | 3 | 72 | no | — |
| 63 | C2×C3×D4 | — | 3 | 72 | no | — |
| 64 | C2×C3×D4 | — | 3 | 36 | no | — |
| 65 | C2×C3×D4 | — | 3 | 12 | no | — |
| 66 | C22×C3×C4 | — | 3 | 18 | no | — |
| 67 | C22×C3×C4 | — | 3 | 18 | no | — |
| 68 | C22×C3×C4 | — | 3 | 1 | no | — |
| 69 | C22×C3×C4 | C2×A4 | 1 | 3 | yes | — |
| 70 | #41⋊C3 | — | 4 | 36 | no | — |
| 71 | C4×A4 | — | 4 | 9 | no | — |
| 72 | C3×A4 | — | 4 | 1 | no | — |
| 73 | C32×C4 | — | 16 | 3 | no | — |
| 74 | C25 | — | 3 | 2 | no | — |
| 75 | H(5,𝔽2) | — | 3 | 24 | no | — |
| 76 | H(5,𝔽2) | — | 3 | 8 | no | — |
| 77 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 78 | H(5,𝔽2) | — | 3 | 36 | no | — |
| 79 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 80 | H(5,𝔽2) | C3×A4 | 1 | 4 | yes | complemented |
| 81 | C2×#41 | — | 3 | 12 | no | — |
| 82 | C2×#41 | — | 3 | 18 | no | — |
| 83 | C2×#41 | — | 3 | 6 | no | — |
| 84 | C2×#41 | — | 3 | 12 | no | — |
| 85 | C22×Q8 | C3×A4 | 1 | 1 | yes | characteristic |
| 86 | C22×Q8 | C22×C32 | 1 | 1 | yes | characteristic |
| 87 | C22×D4 | — | 3 | 3 | no | — |
| 88 | C22×D4 | — | 3 | 36 | no | — |
| 89 | C22×D4 | — | 3 | 36 | no | — |
| 90 | C22×D4 | — | 3 | 3 | no | — |
| 91 | C23×C4 | — | 3 | 3 | no | — |
| 92 | C23×C3 | — | 3 | 18 | no | — |
| 93 | C23×C3 | — | 3 | 24 | no | — |
| 94 | C23×C3 | — | 3 | 8 | no | — |
| 95 | C23×C3 | — | 6 | 9 | no | — |
| 96 | C23×C3 | — | 6 | 3 | no | — |
| 97 | C23×C3 | — | 12 | 24 | no | — |
| 98 | C23×C3 | — | 12 | 24 | no | — |
| 99 | C23×C3 | C22×A4 | 1 | 1 | yes | characteristic |
| 100 | C2×A4 | — | 4 | 3 | no | — |
| 101 | C3×Q8 | — | 3 | 4 | no | — |
| 102 | C3×Q8 | — | 3 | 2 | no | — |
| 103 | C3×Q8 | — | 3 | 3 | no | — |
| 104 | C3×Q8 | — | 16 | 3 | no | — |
| 105 | C3×Q8 | C22×A4 | 1 | 4 | yes | — |
| 106 | C3×Q8 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 107 | C3×D4 | — | 3 | 12 | no | — |
| 108 | C3×D4 | — | 3 | 36 | no | — |
| 109 | C3×D4 | — | 3 | 36 | no | — |
| 110 | C3×D4 | — | 3 | 12 | no | — |
| 111 | C2×C3×C4 | — | 3 | 6 | no | — |
| 112 | C2×C3×C4 | — | 3 | 36 | no | — |
| 113 | C2×C3×C4 | — | 3 | 3 | no | — |
| 114 | C2×C3×C4 | — | 3 | 12 | no | — |
| 115 | C2×C3×C4 | — | 3 | 3 | no | — |
| 116 | SL(2,𝔽3) | — | 4 | 12 | no | — |
| 117 | C2×C32 | — | 16 | 1 | no | — |
| 118 | C24 | — | 3 | 24 | no | — |
| 119 | C24 | — | 3 | 3 | no | — |
| 120 | C24 | — | 12 | 6 | no | — |
| 121 | C24 | — | 12 | 2 | no | — |
| 122 | #41 | — | 3 | 36 | no | — |
| 123 | #41 | — | 3 | 24 | no | — |
| 124 | #41 | — | 3 | 36 | no | — |
| 125 | #41 | — | 3 | 12 | no | — |
| 126 | #41 | — | 3 | 12 | no | — |
| 127 | #41 | — | 3 | 4 | no | — |
| 128 | #41 | C2×C3×A4 | 1 | 12 | yes | — |
| 129 | C2×Q8 | — | 3 | 3 | no | — |
| 130 | C2×Q8 | — | 3 | 4 | no | — |
| 131 | C2×Q8 | — | 3 | 1 | no | — |
| 132 | C2×D4 | — | 3 | 36 | no | — |
| 133 | C2×D4 | — | 3 | 18 | no | — |
| 134 | C2×D4 | — | 3 | 18 | no | — |
| 135 | C2×D4 | — | 3 | 72 | no | — |
| 136 | C2×D4 | — | 3 | 72 | no | — |
| 137 | C2×D4 | — | 3 | 36 | no | — |
| 138 | C2×D4 | — | 3 | 12 | no | — |
| 139 | C22×C4 | — | 3 | 18 | no | — |
| 140 | C22×C4 | — | 3 | 18 | no | — |
| 141 | C22×C4 | — | 3 | 1 | no | — |
| 142 | C22×C4 | C2×C3×A4 | 1 | 3 | yes | — |
| 143 | C22×C3 | — | 3 | 1 | no | — |
| 144 | C22×C3 | — | 3 | 1 | no | — |
| 145 | C22×C3 | — | 3 | 12 | no | — |
| 146 | C22×C3 | — | 6 | 18 | no | — |
| 147 | C22×C3 | — | 6 | 18 | no | — |
| 148 | C22×C3 | — | 12 | 24 | no | — |
| 149 | C22×C3 | — | 12 | 8 | no | — |
| 150 | C22×C3 | #994 | 1 | 1 | yes | characteristic, complemented |
| 151 | A4 | — | 4 | 3 | no | — |
| 152 | C3×C4 | — | 3 | 4 | no | — |
| 153 | C3×C4 | — | 3 | 3 | no | — |
| 154 | C3×C4 | — | 16 | 9 | no | — |
| 155 | C3×C4 | C2×C22⋊A4 | 1 | 3 | yes | — |
| 156 | C32 | — | 16 | 1 | no | Sylow |
| 157 | C23 | — | 3 | 18 | no | — |
| 158 | C23 | — | 3 | 24 | no | — |
| 159 | C23 | — | 3 | 8 | no | — |
| 160 | C23 | — | 6 | 9 | no | — |
| 161 | C23 | — | 6 | 3 | no | — |
| 162 | C23 | — | 12 | 24 | no | — |
| 163 | C23 | — | 12 | 24 | no | — |
| 164 | C23 | C22×C3×A4 | 1 | 1 | yes | characteristic |
| 165 | Q8 | — | 3 | 4 | no | — |
| 166 | Q8 | — | 3 | 2 | no | — |
| 167 | Q8 | — | 3 | 3 | no | — |
| 168 | Q8 | C22×C3×A4 | 1 | 4 | yes | — |
| 169 | Q8 | C3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 170 | D4 | — | 3 | 12 | no | — |
| 171 | D4 | — | 3 | 36 | no | — |
| 172 | D4 | — | 3 | 36 | no | — |
| 173 | D4 | — | 3 | 12 | no | — |
| 174 | C2×C4 | — | 3 | 6 | no | — |
| 175 | C2×C4 | — | 3 | 36 | no | — |
| 176 | C2×C4 | — | 3 | 3 | no | — |
| 177 | C2×C4 | — | 3 | 12 | no | — |
| 178 | C2×C4 | — | 3 | 3 | no | — |
| 179 | C2×C3 | — | 3 | 1 | no | — |
| 180 | C2×C3 | — | 3 | 1 | no | — |
| 181 | C2×C3 | — | 6 | 12 | no | — |
| 182 | C2×C3 | — | 16 | 3 | no | — |
| 183 | C2×C3 | C22×C22⋊A4 | 1 | 1 | yes | characteristic |
| 184 | C22 | — | 3 | 1 | no | — |
| 185 | C22 | — | 3 | 1 | no | — |
| 186 | C22 | — | 3 | 12 | no | — |
| 187 | C22 | — | 6 | 18 | no | — |
| 188 | C22 | — | 6 | 18 | no | — |
| 189 | C22 | — | 12 | 24 | no | — |
| 190 | C22 | — | 12 | 8 | no | — |
| 191 | C22 | C3×#994 | 1 | 1 | yes | characteristic, complemented |
| 192 | C4 | — | 3 | 4 | no | — |
| 193 | C4 | — | 3 | 3 | no | — |
| 194 | C4 | C2×C3×C22⋊A4 | 1 | 3 | yes | — |
| 195 | C3 | — | 16 | 3 | no | — |
| 196 | C3 | #87543 | 1 | 1 | yes | characteristic, direct product factor |
| 197 | C2 | — | 3 | 1 | no | — |
| 198 | C2 | — | 3 | 1 | no | — |
| 199 | C2 | — | 6 | 12 | no | — |
| 200 | C2 | C22×C3×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C32 |
| Frattini | C2 | C22×C3×C22⋊A4 |
| Fitting | C22×C3×H(5,𝔽2) | C3 |
| socle | C23×C3 | C22×A4 |
| hypercenter | C3×Q8 | C22⋊A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#5948 | C2 | semidirect |
| 2. C22×C3×H(5,𝔽2) | C3 | semidirect |
| 3. #87543 | C3 | direct (N × H) |
| 4. C3×C22⋊SL(2,𝔽3) | C22 | semidirect |
| 12. #5948 | C2×C3 | semidirect |
| 15. C22×H(5,𝔽2) | C32 | semidirect |
| 22. C3×H(5,𝔽2) | A4 | semidirect |
| 34. C22⋊SL(2,𝔽3) | C22×C3 | semidirect |
| 80. H(5,𝔽2) | C3×A4 | semidirect |
| 150. C22×C3 | #994 | semidirect |
| 191. C22 | C3×#994 | semidirect |
| 196. C3 | #87543 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#5948 | 2 |
| C22×C3×H(5,𝔽2) | 3 |
| #87543 | 3 |
| C3×#994 | 4 |
| C3×Q8×A4 | 4 |