Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
259 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 36 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 | Q8×S4 | C2 | 1 | 2 | yes | maximal |
| 2 | A4×Q16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5416 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C8×S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #5398 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | D4×Q16 | — | 3 | 1 | no | maximal, Sylow |
| 7 | Q8×A4 | C22 | 1 | 2 | yes | — |
| 8 | C4×S4 | — | 2 | 2 | no | — |
| 9 | C4×S4 | C22 | 1 | 1 | yes | characteristic |
| 10 | A4⋊Q8 | — | 2 | 2 | no | — |
| 11 | A4⋊Q8 | C22 | 1 | 2 | yes | — |
| 12 | D3×Q16 | — | 4 | 1 | no | maximal |
| 13 | C8×A4 | C22 | 1 | 1 | yes | characteristic |
| 14 | A4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×Q16 | — | 3 | 1 | no | — |
| 16 | C22×Q16 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | D4×Q8 | — | 3 | 2 | no | — |
| 18 | #494 | — | 3 | 1 | no | — |
| 19 | #467 | — | 3 | 1 | no | — |
| 20 | #467 | — | 3 | 1 | no | — |
| 21 | #462 | — | 3 | 2 | no | — |
| 22 | #451 | — | 3 | 2 | no | — |
| 23 | #451 | — | 3 | 2 | no | — |
| 24 | C4×Q16 | — | 3 | 1 | no | — |
| 25 | C8×D4 | — | 3 | 1 | no | — |
| 26 | C2×S4 | D4 | 1 | 1 | yes | characteristic |
| 27 | D3×Q8 | — | 4 | 2 | no | — |
| 28 | C4×A4 | — | 2 | 2 | no | — |
| 29 | C4×A4 | C23 | 1 | 1 | yes | characteristic |
| 30 | SL(2,ℤ/4) | — | 2 | 2 | no | — |
| 31 | SL(2,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 32 | C3×Q16 | — | 4 | 1 | no | — |
| 33 | #216 | — | 4 | 2 | no | — |
| 34 | Dic12 | — | 4 | 1 | no | — |
| 35 | D3×C8 | — | 4 | 1 | no | — |
| 36 | C22×Q8 | — | 3 | 2 | no | — |
| 37 | C22×Q8 | C2×D3 | 1 | 2 | yes | — |
| 38 | C2×Q16 | — | 3 | 1 | no | — |
| 39 | C2×Q16 | — | 3 | 2 | no | — |
| 40 | C2×Q16 | — | 3 | 1 | no | — |
| 41 | C2×Q16 | — | 3 | 2 | no | — |
| 42 | C2×Q16 | — | 3 | 1 | no | — |
| 43 | C2×Q16 | — | 3 | 2 | no | — |
| 44 | C2×Q16 | — | 6 | 2 | no | — |
| 45 | C2×Q16 | — | 6 | 1 | no | — |
| 46 | C2×Q16 | — | 6 | 1 | no | — |
| 47 | C22×C8 | — | 3 | 1 | no | — |
| 48 | C22×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 49 | C4⋊Q8 | — | 3 | 2 | no | — |
| 50 | C4⋊Q8 | — | 6 | 2 | no | — |
| 51 | C22⋊Q8 | — | 3 | 2 | no | — |
| 52 | C22⋊Q8 | — | 3 | 2 | no | — |
| 53 | C22⋊Q8 | — | 6 | 2 | no | — |
| 54 | C22⋊Q8 | — | 6 | 2 | no | — |
| 55 | C4×Q8 | — | 3 | 2 | no | — |
| 56 | C4×D4 | — | 3 | 1 | no | — |
| 57 | C4×D4 | — | 6 | 2 | no | — |
| 58 | #107 | — | 3 | 1 | no | — |
| 59 | C4⋊C8 | — | 3 | 1 | no | — |
| 60 | #103 | — | 3 | 2 | no | — |
| 61 | #103 | — | 3 | 2 | no | — |
| 62 | #103 | — | 3 | 2 | no | — |
| 63 | C22⋊C8 | — | 3 | 1 | no | — |
| 64 | C22⋊C8 | — | 3 | 1 | no | — |
| 65 | C4×C8 | — | 3 | 1 | no | — |
| 66 | C2×A4 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | S4 | Q16 | 1 | 2 | yes | direct product factor |
| 68 | C3×Q8 | — | 4 | 2 | no | — |
| 69 | C4×D3 | — | 4 | 1 | no | — |
| 70 | C4×D3 | — | 8 | 2 | no | — |
| 71 | Dic6 | — | 4 | 2 | no | — |
| 72 | Dic6 | — | 8 | 2 | no | — |
| 73 | C3×C8 | — | 4 | 1 | no | — |
| 74 | C3⋊C8 | — | 4 | 1 | no | — |
| 75 | C2×Q8 | — | 3 | 2 | no | — |
| 76 | C2×Q8 | — | 3 | 2 | no | — |
| 77 | C2×Q8 | — | 3 | 2 | no | — |
| 78 | C2×Q8 | — | 6 | 2 | no | — |
| 79 | C2×Q8 | — | 6 | 2 | no | — |
| 80 | C2×Q8 | — | 6 | 2 | no | — |
| 81 | C2×Q8 | — | 6 | 2 | no | — |
| 82 | C2×Q8 | — | 12 | 2 | no | — |
| 83 | C2×D4 | — | 3 | 1 | no | — |
| 84 | C22×C4 | — | 2 | 2 | no | — |
| 85 | C22×C4 | — | 3 | 1 | no | — |
| 86 | C22×C4 | — | 6 | 2 | no | — |
| 87 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 88 | Q16 | — | 3 | 2 | no | — |
| 89 | Q16 | — | 3 | 1 | no | — |
| 90 | Q16 | — | 6 | 1 | no | — |
| 91 | Q16 | — | 6 | 1 | no | — |
| 92 | Q16 | — | 6 | 2 | no | — |
| 93 | Q16 | — | 6 | 2 | no | — |
| 94 | Q16 | — | 6 | 1 | no | — |
| 95 | Q16 | — | 12 | 2 | no | — |
| 96 | Q16 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 97 | C2×C8 | — | 3 | 1 | no | — |
| 98 | C2×C8 | — | 3 | 1 | no | — |
| 99 | C2×C8 | — | 3 | 1 | no | — |
| 100 | C2×C8 | — | 3 | 1 | no | — |
| 101 | C2×C8 | — | 6 | 1 | no | — |
| 102 | C2×C8 | — | 6 | 1 | no | — |
| 103 | C4⋊C4 | — | 3 | 1 | no | — |
| 104 | C4⋊C4 | — | 3 | 2 | no | — |
| 105 | C4⋊C4 | — | 6 | 2 | no | — |
| 106 | C4⋊C4 | — | 6 | 2 | no | — |
| 107 | C4⋊C4 | — | 6 | 2 | no | — |
| 108 | C4⋊C4 | — | 6 | 2 | no | — |
| 109 | C4⋊C4 | — | 6 | 2 | no | — |
| 110 | C22⋊C4 | — | 3 | 1 | no | — |
| 111 | C22⋊C4 | — | 3 | 1 | no | — |
| 112 | C22⋊C4 | — | 6 | 2 | no | — |
| 113 | C22⋊C4 | — | 6 | 2 | no | — |
| 114 | C42 | — | 3 | 1 | no | — |
| 115 | C42 | — | 6 | 2 | no | — |
| 116 | C2×D3 | — | 4 | 1 | no | — |
| 117 | A4 | C2×Q16 | 1 | 1 | yes | characteristic, complemented |
| 118 | C3×C4 | — | 4 | 1 | no | — |
| 119 | C3×C4 | — | 8 | 2 | no | — |
| 120 | Dic3 | — | 4 | 1 | no | — |
| 121 | Dic3 | — | 8 | 2 | no | — |
| 122 | C23 | — | 3 | 1 | no | — |
| 123 | C23 | D3×D4 | 1 | 1 | yes | characteristic |
| 124 | Q8 | — | 3 | 2 | no | — |
| 125 | Q8 | — | 6 | 2 | no | — |
| 126 | Q8 | — | 6 | 2 | no | — |
| 127 | Q8 | — | 6 | 2 | no | — |
| 128 | Q8 | — | 6 | 2 | no | — |
| 129 | Q8 | — | 12 | 2 | no | — |
| 130 | Q8 | — | 12 | 2 | no | — |
| 131 | Q8 | C2×S4 | 1 | 2 | yes | — |
| 132 | D4 | — | 3 | 2 | no | — |
| 133 | D4 | — | 3 | 2 | no | — |
| 134 | C2×C4 | — | 3 | 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 | — | 6 | 2 | no | — |
| 140 | C2×C4 | — | 6 | 1 | no | — |
| 141 | C2×C4 | — | 6 | 1 | no | — |
| 142 | C2×C4 | — | 6 | 2 | no | — |
| 143 | C2×C4 | — | 6 | 2 | no | — |
| 144 | C2×C4 | — | 6 | 2 | no | — |
| 145 | C2×C4 | — | 12 | 2 | no | — |
| 146 | C2×C4 | — | 12 | 2 | no | — |
| 147 | C8 | — | 3 | 1 | no | — |
| 148 | C8 | — | 6 | 1 | no | — |
| 149 | C8 | — | 6 | 1 | no | — |
| 150 | C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 151 | C2×C3 | — | 4 | 1 | no | — |
| 152 | D3 | — | 4 | 2 | no | — |
| 153 | C22 | — | 3 | 1 | no | — |
| 154 | C22 | — | 3 | 1 | no | — |
| 155 | C22 | — | 3 | 2 | no | — |
| 156 | C22 | — | 6 | 1 | no | — |
| 157 | C22 | — | 6 | 1 | no | — |
| 158 | C22 | D3×Q16 | 1 | 1 | yes | characteristic, complemented |
| 159 | C4 | — | 2 | 2 | no | — |
| 160 | C4 | — | 3 | 2 | no | — |
| 161 | C4 | — | 3 | 1 | no | — |
| 162 | C4 | — | 6 | 1 | no | — |
| 163 | C4 | — | 6 | 1 | no | — |
| 164 | C4 | — | 6 | 2 | no | — |
| 165 | C4 | — | 12 | 2 | no | — |
| 166 | C4 | — | 12 | 2 | no | — |
| 167 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 168 | C3 | — | 4 | 1 | no | Sylow |
| 169 | C2 | — | 3 | 1 | no | — |
| 170 | C2 | — | 3 | 1 | no | — |
| 171 | C2 | — | 6 | 2 | no | — |
| 172 | C2 | D4×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×S4 |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×Q16 | D3 |
| socle | C23 | D3×D4 |
| prime radical | C2×S4 | D4 |
| hypercenter | Q16 | S4 |
| hypocenter | A4 | C2×Q16 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×Q16 | C2 | semidirect |
| 3. #5416 | C2 | semidirect |
| 5. #5398 | C2 | semidirect |
| 16. C22×Q16 | D3 | semidirect |
| 67. S4 | Q16 | direct (N × H) |
| 96. Q16 | S4 | direct (N × H) |
| 117. A4 | C2×Q16 | semidirect |
| 158. C22 | D3×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×S4 | 2 |
| A4×Q16 | 2 |
| #5416 | 2 |
| C8×S4 | 2 |
| #5398 | 2 |
| D4×Q16 | 3 |
| D3×Q16 | 4 |