Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
188 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 34 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 | #5945 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #568⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C4×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #2901 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 5 | 2−1+4⋊C3 | — | 2 | 1 | no | — |
| 6 | 2−1+4⋊C3 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | #994 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×#41⋊C3 | — | 2 | 1 | no | — |
| 9 | C2×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 10 | #131⋊C3 | — | 2 | 1 | no | — |
| 11 | C4×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 12 | C3×#104 | — | 4 | 1 | no | maximal |
| 13 | #585 | C2×C3 | 1 | 1 | yes | characteristic |
| 14 | #568 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C4×#41 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 16 | #421 | — | 3 | 1 | no | — |
| 17 | #421 | — | 3 | 1 | no | — |
| 18 | C2×#104 | — | 3 | 1 | no | — |
| 19 | C2×#104 | — | 3 | 1 | no | — |
| 20 | C3×#41 | — | 4 | 1 | no | — |
| 21 | #41⋊C3 | — | 2 | 1 | no | — |
| 22 | #41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | #41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 25 | C2×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 26 | C3×M16 | — | 4 | 1 | no | — |
| 27 | C3×C42 | — | 4 | 1 | no | — |
| 28 | 2−1+4 | — | 2 | 1 | no | — |
| 29 | 2−1+4 | — | 3 | 1 | no | — |
| 30 | 2−1+4 | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 31 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 32 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 33 | H(5,𝔽2) | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 34 | C2×#41 | — | 2 | 1 | no | — |
| 35 | C2×#41 | — | 3 | 1 | no | — |
| 36 | C2×#41 | — | 3 | 1 | no | — |
| 37 | C2×#41 | — | 6 | 1 | no | — |
| 38 | C2×#41 | C22×C3 | 1 | 1 | yes | characteristic |
| 39 | #131 | — | 2 | 1 | no | — |
| 40 | #131 | — | 6 | 1 | no | — |
| 41 | C2×M16 | — | 3 | 1 | no | — |
| 42 | C2×M16 | — | 3 | 1 | no | — |
| 43 | C4×Q8 | — | 2 | 1 | no | — |
| 44 | C4×D4 | — | 6 | 1 | no | — |
| 45 | #117 | — | 3 | 1 | no | — |
| 46 | C2×C42 | — | 3 | 1 | no | — |
| 47 | #104 | — | 3 | 2 | no | — |
| 48 | #104 | — | 3 | 1 | no | — |
| 49 | #104 | — | 3 | 1 | no | — |
| 50 | #104 | — | 3 | 1 | no | — |
| 51 | #104 | A4 | 1 | 1 | yes | characteristic |
| 52 | C3×Q8 | — | 4 | 1 | no | — |
| 53 | C3×D4 | — | 4 | 1 | no | — |
| 54 | C3×D4 | — | 8 | 1 | no | — |
| 55 | C2×C3×C4 | — | 4 | 1 | no | — |
| 56 | C2×C3×C4 | — | 8 | 1 | no | — |
| 57 | C2×C3×C4 | — | 8 | 1 | no | — |
| 58 | SL(2,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C3×C8 | — | 8 | 1 | no | — |
| 60 | #41 | — | 2 | 1 | no | — |
| 61 | #41 | — | 3 | 1 | no | — |
| 62 | #41 | — | 3 | 1 | no | — |
| 63 | #41 | — | 3 | 2 | no | — |
| 64 | #41 | — | 3 | 1 | no | — |
| 65 | #41 | — | 3 | 1 | no | — |
| 66 | #41 | — | 3 | 1 | no | — |
| 67 | #41 | — | 6 | 1 | no | — |
| 68 | #41 | — | 6 | 1 | no | — |
| 69 | #41 | — | 6 | 1 | no | — |
| 70 | #41 | — | 6 | 1 | no | — |
| 71 | #41 | — | 6 | 2 | no | — |
| 72 | #41 | — | 6 | 1 | no | — |
| 73 | #41 | — | 6 | 1 | no | — |
| 74 | #41 | — | 6 | 1 | no | — |
| 75 | #41 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 76 | #41 | C3×D4 | 1 | 1 | yes | characteristic |
| 77 | #41 | C2×A4 | 1 | 1 | yes | characteristic |
| 78 | C2×Q8 | — | 2 | 1 | no | — |
| 79 | C2×Q8 | — | 3 | 1 | no | — |
| 80 | C2×Q8 | — | 3 | 1 | no | — |
| 81 | C2×Q8 | — | 6 | 1 | no | — |
| 82 | C2×Q8 | C3×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×D4 | — | 3 | 1 | no | — |
| 84 | C2×D4 | — | 3 | 1 | no | — |
| 85 | C2×D4 | — | 3 | 1 | no | — |
| 86 | C2×D4 | — | 6 | 1 | no | — |
| 87 | C2×D4 | — | 6 | 1 | no | — |
| 88 | C2×D4 | — | 6 | 2 | no | — |
| 89 | C2×D4 | — | 6 | 1 | no | — |
| 90 | C2×D4 | — | 6 | 1 | no | — |
| 91 | C22×C4 | — | 3 | 1 | no | — |
| 92 | C22×C4 | — | 6 | 1 | no | — |
| 93 | C22×C4 | — | 6 | 1 | no | — |
| 94 | C22×C4 | — | 6 | 1 | no | — |
| 95 | M16 | — | 3 | 1 | no | — |
| 96 | M16 | — | 6 | 1 | no | — |
| 97 | M16 | — | 6 | 1 | no | — |
| 98 | M16 | C2×A4 | 1 | 1 | yes | characteristic |
| 99 | C2×C8 | — | 6 | 1 | no | — |
| 100 | C2×C8 | — | 6 | 1 | no | — |
| 101 | C4⋊C4 | — | 6 | 1 | no | — |
| 102 | C22⋊C4 | — | 6 | 1 | no | — |
| 103 | C42 | — | 3 | 1 | no | — |
| 104 | C42 | — | 6 | 1 | no | — |
| 105 | C42 | C2×A4 | 1 | 1 | yes | characteristic |
| 106 | C22×C3 | — | 4 | 1 | no | — |
| 107 | C22×C3 | — | 8 | 1 | no | — |
| 108 | C3×C4 | — | 4 | 1 | no | — |
| 109 | C3×C4 | — | 4 | 1 | no | — |
| 110 | C3×C4 | — | 8 | 1 | no | — |
| 111 | C3×C4 | — | 8 | 2 | no | — |
| 112 | C23 | — | 3 | 1 | no | — |
| 113 | C23 | — | 6 | 1 | no | — |
| 114 | C23 | — | 6 | 1 | no | — |
| 115 | Q8 | — | 3 | 1 | no | — |
| 116 | Q8 | — | 3 | 1 | no | — |
| 117 | Q8 | — | 6 | 1 | no | — |
| 118 | Q8 | — | 6 | 1 | no | — |
| 119 | Q8 | C3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | Q8 | C4×A4 | 1 | 1 | yes | characteristic |
| 121 | D4 | — | 2 | 1 | no | — |
| 122 | D4 | — | 3 | 1 | no | — |
| 123 | D4 | — | 3 | 1 | no | — |
| 124 | D4 | — | 3 | 2 | no | — |
| 125 | D4 | — | 3 | 1 | no | — |
| 126 | D4 | — | 6 | 1 | no | — |
| 127 | D4 | — | 6 | 1 | no | — |
| 128 | D4 | — | 6 | 1 | no | — |
| 129 | D4 | — | 6 | 1 | no | — |
| 130 | D4 | — | 6 | 2 | no | — |
| 131 | D4 | — | 6 | 1 | no | — |
| 132 | D4 | C4×A4 | 1 | 1 | yes | characteristic |
| 133 | C2×C4 | — | 2 | 1 | no | — |
| 134 | C2×C4 | — | 2 | 1 | no | — |
| 135 | C2×C4 | — | 3 | 1 | no | — |
| 136 | C2×C4 | — | 3 | 1 | no | — |
| 137 | C2×C4 | — | 3 | 1 | no | — |
| 138 | C2×C4 | — | 3 | 1 | no | — |
| 139 | C2×C4 | — | 3 | 1 | no | — |
| 140 | C2×C4 | — | 6 | 1 | no | — |
| 141 | C2×C4 | — | 6 | 1 | no | — |
| 142 | C2×C4 | — | 6 | 1 | no | — |
| 143 | C2×C4 | — | 6 | 2 | no | — |
| 144 | C2×C4 | — | 6 | 1 | no | — |
| 145 | C2×C4 | — | 6 | 1 | no | — |
| 146 | C2×C4 | — | 6 | 1 | no | — |
| 147 | C2×C4 | — | 12 | 2 | no | — |
| 148 | C2×C4 | C22×A4 | 1 | 1 | yes | characteristic |
| 149 | C8 | — | 2 | 1 | no | — |
| 150 | C8 | — | 6 | 1 | no | — |
| 151 | C2×C3 | — | 4 | 1 | no | — |
| 152 | C2×C3 | — | 8 | 1 | no | — |
| 153 | C2×C3 | — | 16 | 1 | no | — |
| 154 | C22 | — | 2 | 1 | no | — |
| 155 | C22 | — | 3 | 1 | no | — |
| 156 | C22 | — | 3 | 1 | no | — |
| 157 | C22 | — | 6 | 1 | no | — |
| 158 | C22 | — | 12 | 1 | no | — |
| 159 | C22 | — | 24 | 1 | no | — |
| 160 | C22 | — | 24 | 1 | no | — |
| 161 | C22 | D4×A4 | 1 | 1 | yes | characteristic |
| 162 | C4 | — | 2 | 2 | no | — |
| 163 | C4 | — | 2 | 1 | no | — |
| 164 | C4 | — | 3 | 1 | no | — |
| 165 | C4 | — | 3 | 1 | no | — |
| 166 | C4 | — | 6 | 1 | no | — |
| 167 | C4 | — | 12 | 1 | no | — |
| 168 | C4 | C2×C4×A4 | 1 | 1 | yes | characteristic |
| 169 | C4 | D4×A4 | 1 | 1 | yes | characteristic |
| 170 | C3 | — | 4 | 1 | no | Sylow |
| 171 | C2 | — | 2 | 1 | no | — |
| 172 | C2 | — | 4 | 1 | no | — |
| 173 | C2 | — | 6 | 1 | no | — |
| 174 | C2 | — | 6 | 1 | no | — |
| 175 | C2 | — | 12 | 1 | no | — |
| 176 | C2 | A4×C22⋊C4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #568⋊C3 | C2 | semidirect |
| 3. C4×#41⋊C3 | C2 | semidirect |
| 4. #2901 | C3 | semidirect |
| 6. 2−1+4⋊C3 | C4 | semidirect |
| 7. #994 | C4 | semidirect |
| 14. #568 | C2×C3 | semidirect |
| 15. C4×#41 | C2×C3 | semidirect |
| 30. 2−1+4 | C3×C4 | semidirect |
| 33. H(5,𝔽2) | C3×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5945 | 2 |
| #568⋊C3 | 2 |
| C4×#41⋊C3 | 2 |
| #2901 | 3 |
| C3×#104 | 4 |