Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
122 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 23 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 | #5077 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #4544 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #1402 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #1004 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C3×#127 | C22 | 1 | 1 | yes | characteristic |
| 7 | #932 | — | 2 | 1 | no | — |
| 8 | C3×#99 | — | 2 | 1 | no | — |
| 9 | #833 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #805 | — | 2 | 1 | no | — |
| 11 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 12 | #453 | — | 3 | 1 | no | — |
| 13 | #353 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #331 | — | 3 | 1 | no | — |
| 15 | C2×C3×D4 | — | 4 | 1 | no | — |
| 16 | C2×C3×D4 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | C2×C3×D4 | D4 | 1 | 1 | yes | characteristic |
| 18 | C2×#67 | — | 4 | 1 | no | — |
| 19 | #237 | — | 4 | 1 | no | — |
| 20 | D3×D4 | — | 4 | 1 | no | — |
| 21 | #235 | — | 2 | 1 | no | — |
| 22 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 23 | C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 24 | #214 | — | 4 | 1 | no | — |
| 25 | #213 | — | 4 | 1 | no | — |
| 26 | #208 | — | 2 | 1 | no | — |
| 27 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 28 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 29 | AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 30 | #127 | C2×D3 | 1 | 1 | yes | characteristic |
| 31 | C4⋊C8 | — | 6 | 1 | no | — |
| 32 | #104 | — | 6 | 1 | no | — |
| 33 | #100 | — | 3 | 1 | no | — |
| 34 | #99 | — | 2 | 1 | no | — |
| 35 | C23×C3 | — | 2 | 1 | no | — |
| 36 | C23×C3 | — | 2 | 1 | no | — |
| 37 | C22×D3 | — | 4 | 1 | no | — |
| 38 | C3×D4 | — | 4 | 1 | no | — |
| 39 | C3×D4 | — | 4 | 1 | no | — |
| 40 | C3×D4 | — | 8 | 1 | no | — |
| 41 | C3×D4 | — | 8 | 1 | no | — |
| 42 | C2×C3×C4 | — | 2 | 1 | no | — |
| 43 | C2×C3×C4 | — | 8 | 1 | no | — |
| 44 | C2×C3×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | #67 | — | 4 | 2 | no | — |
| 46 | #67 | — | 4 | 1 | no | — |
| 47 | C2×Dic3 | — | 4 | 1 | no | — |
| 48 | D12 | — | 2 | 1 | no | — |
| 49 | C4×D3 | — | 4 | 1 | no | — |
| 50 | Dic6 | — | 2 | 1 | no | — |
| 51 | C3⋊C8 | — | 4 | 1 | no | — |
| 52 | C3⋊C8 | — | 8 | 1 | no | — |
| 53 | #41 | — | 6 | 1 | no | — |
| 54 | #41 | — | 12 | 1 | no | — |
| 55 | C2×D4 | — | 4 | 1 | no | — |
| 56 | C2×D4 | — | 12 | 1 | no | — |
| 57 | C2×D4 | — | 12 | 1 | no | — |
| 58 | C2×D4 | C4×D3 | 1 | 1 | yes | characteristic |
| 59 | C2×D4 | #67 | 1 | 1 | yes | characteristic |
| 60 | SD16 | — | 12 | 1 | no | — |
| 61 | D8 | — | 12 | 1 | no | — |
| 62 | M16 | — | 6 | 1 | no | — |
| 63 | C2×C8 | — | 12 | 1 | no | — |
| 64 | C22⋊C4 | — | 4 | 1 | no | — |
| 65 | C42 | D12 | 1 | 1 | yes | characteristic |
| 66 | C22×C3 | — | 4 | 1 | no | — |
| 67 | C22×C3 | — | 4 | 1 | no | — |
| 68 | C22×C3 | — | 8 | 1 | no | — |
| 69 | C22×C3 | — | 8 | 1 | no | — |
| 70 | C22×C3 | #99 | 1 | 1 | yes | characteristic |
| 71 | C2×D3 | — | 4 | 1 | no | — |
| 72 | C2×D3 | — | 16 | 1 | no | — |
| 73 | C3×C4 | — | 2 | 1 | no | — |
| 74 | C3×C4 | — | 2 | 1 | no | — |
| 75 | C3×C4 | — | 2 | 1 | no | — |
| 76 | C3×C4 | — | 16 | 1 | no | — |
| 77 | Dic3 | — | 4 | 1 | no | — |
| 78 | C23 | — | 2 | 1 | no | — |
| 79 | C23 | — | 2 | 1 | no | — |
| 80 | C23 | — | 12 | 1 | no | — |
| 81 | Q8 | — | 6 | 1 | no | — |
| 82 | D4 | — | 4 | 1 | no | — |
| 83 | D4 | — | 4 | 1 | no | — |
| 84 | D4 | — | 6 | 1 | no | — |
| 85 | D4 | — | 8 | 1 | no | — |
| 86 | D4 | — | 8 | 1 | no | — |
| 87 | D4 | — | 12 | 2 | no | — |
| 88 | D4 | — | 12 | 1 | no | — |
| 89 | C2×C4 | — | 2 | 1 | no | — |
| 90 | C2×C4 | — | 8 | 1 | no | — |
| 91 | C2×C4 | — | 12 | 1 | no | — |
| 92 | C2×C4 | — | 12 | 1 | no | — |
| 93 | C2×C4 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 94 | C8 | — | 12 | 1 | no | — |
| 95 | C8 | — | 24 | 1 | no | — |
| 96 | C2×C3 | — | 2 | 1 | no | — |
| 97 | C2×C3 | — | 8 | 1 | no | — |
| 98 | C2×C3 | — | 8 | 1 | no | — |
| 99 | C2×C3 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 100 | D3 | — | 8 | 1 | no | — |
| 101 | C22 | — | 4 | 1 | no | — |
| 102 | C22 | — | 4 | 1 | no | — |
| 103 | C22 | — | 8 | 1 | no | — |
| 104 | C22 | — | 8 | 1 | no | — |
| 105 | C22 | — | 12 | 1 | no | — |
| 106 | C22 | — | 48 | 1 | no | — |
| 107 | C22 | #806 | 1 | 1 | yes | characteristic |
| 108 | C4 | — | 2 | 1 | no | — |
| 109 | C4 | — | 2 | 1 | no | — |
| 110 | C4 | — | 2 | 1 | no | — |
| 111 | C4 | — | 12 | 1 | no | — |
| 112 | C4 | — | 16 | 1 | no | — |
| 113 | C3 | #1402 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 114 | C2 | — | 2 | 1 | no | — |
| 115 | C2 | — | 8 | 1 | no | — |
| 116 | C2 | — | 8 | 1 | no | — |
| 117 | C2 | — | 24 | 1 | no | — |
| 118 | C2 | #4471 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#353 | C2 | semidirect |
| 3. #4544 | C2 | semidirect |
| 5. #1004 | C4 | semidirect |
| 9. #833 | C4 | semidirect |
| 13. #353 | D3 | semidirect |
| 113. C3 | #1402 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5077 | 2 |
| C3×#353 | 2 |
| #4544 | 2 |
| #1402 | 3 |