Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
398 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 31 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 | C22×C42⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#857 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 3 | C22×#104 | — | 3 | 1 | no | maximal, Sylow |
| 4 | C22×S4 | — | 4 | 1 | no | maximal |
| 5 | C2×C42⋊C3 | C22 | 1 | 3 | yes | complemented |
| 6 | #857 | C22 | 1 | 4 | yes | direct product factor |
| 7 | C22×#41 | — | 3 | 1 | no | — |
| 8 | C22×M16 | — | 3 | 1 | no | — |
| 9 | C22×C42 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×#104 | — | 3 | 6 | no | — |
| 11 | C2×#104 | — | 3 | 6 | no | — |
| 12 | C22×A4 | — | 4 | 1 | no | — |
| 13 | C2×S4 | — | 4 | 6 | no | — |
| 14 | C42⋊C3 | C23 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×#41 | — | 3 | 6 | no | — |
| 16 | C2×#41 | — | 3 | 6 | no | — |
| 17 | C2×#41 | — | 6 | 3 | no | — |
| 18 | C2×#41 | — | 6 | 3 | no | — |
| 19 | C22×Q8 | — | 3 | 1 | no | — |
| 20 | C22×D4 | — | 3 | 1 | no | — |
| 21 | C22×D4 | — | 6 | 1 | no | — |
| 22 | C23×C4 | — | 3 | 1 | no | — |
| 23 | C23×C4 | — | 6 | 1 | no | — |
| 24 | C2×M16 | — | 3 | 6 | no | — |
| 25 | C2×M16 | — | 6 | 3 | no | — |
| 26 | C22×C8 | — | 6 | 1 | no | — |
| 27 | C2×C42 | — | 3 | 3 | no | — |
| 28 | C2×C42 | C2×D3 | 1 | 3 | yes | complemented |
| 29 | #104 | — | 3 | 12 | no | — |
| 30 | #104 | — | 3 | 4 | no | — |
| 31 | C22×D3 | — | 16 | 1 | no | — |
| 32 | C2×A4 | — | 4 | 3 | no | — |
| 33 | S4 | — | 4 | 4 | no | — |
| 34 | C24 | — | 6 | 1 | no | — |
| 35 | C24 | S4 | 1 | 1 | yes | characteristic |
| 36 | #41 | — | 3 | 4 | no | — |
| 37 | #41 | — | 3 | 12 | no | — |
| 38 | #41 | — | 6 | 6 | no | — |
| 39 | #41 | — | 6 | 12 | no | — |
| 40 | #41 | — | 6 | 6 | no | — |
| 41 | C2×Q8 | — | 3 | 6 | no | — |
| 42 | C2×Q8 | — | 6 | 3 | no | — |
| 43 | C2×D4 | — | 3 | 6 | no | — |
| 44 | C2×D4 | — | 6 | 3 | no | — |
| 45 | C2×D4 | — | 6 | 6 | no | — |
| 46 | C2×D4 | — | 6 | 6 | no | — |
| 47 | C22×C4 | — | 3 | 1 | no | — |
| 48 | C22×C4 | — | 3 | 3 | no | — |
| 49 | C22×C4 | — | 3 | 1 | no | — |
| 50 | C22×C4 | — | 3 | 3 | no | — |
| 51 | C22×C4 | — | 3 | 3 | no | — |
| 52 | C22×C4 | — | 3 | 3 | no | — |
| 53 | C22×C4 | — | 6 | 6 | no | — |
| 54 | C22×C4 | — | 6 | 6 | no | — |
| 55 | C22×C4 | — | 6 | 1 | no | — |
| 56 | M16 | — | 3 | 4 | no | — |
| 57 | M16 | — | 6 | 6 | no | — |
| 58 | C2×C8 | — | 6 | 6 | no | — |
| 59 | C42 | — | 3 | 3 | no | — |
| 60 | C42 | — | 6 | 1 | no | — |
| 61 | C42 | C22×D3 | 1 | 1 | yes | characteristic, complemented |
| 62 | C22×C3 | — | 16 | 1 | no | — |
| 63 | C2×D3 | — | 16 | 6 | no | — |
| 64 | A4 | — | 4 | 1 | no | — |
| 65 | C23 | — | 3 | 1 | no | — |
| 66 | C23 | — | 3 | 3 | no | — |
| 67 | C23 | — | 6 | 6 | no | — |
| 68 | C23 | — | 12 | 3 | no | — |
| 69 | C23 | — | 12 | 1 | no | — |
| 70 | C23 | C2×S4 | 1 | 3 | yes | — |
| 71 | Q8 | — | 3 | 4 | no | — |
| 72 | Q8 | — | 6 | 6 | no | — |
| 73 | D4 | — | 3 | 4 | no | — |
| 74 | D4 | — | 6 | 6 | no | — |
| 75 | D4 | — | 6 | 12 | no | — |
| 76 | D4 | — | 6 | 4 | no | — |
| 77 | C2×C4 | — | 3 | 3 | no | — |
| 78 | C2×C4 | — | 3 | 3 | no | — |
| 79 | C2×C4 | — | 3 | 3 | no | — |
| 80 | C2×C4 | — | 3 | 3 | no | — |
| 81 | C2×C4 | — | 3 | 3 | no | — |
| 82 | C2×C4 | — | 3 | 3 | no | — |
| 83 | C2×C4 | — | 3 | 6 | no | — |
| 84 | C2×C4 | — | 3 | 1 | no | — |
| 85 | C2×C4 | — | 3 | 3 | no | — |
| 86 | C2×C4 | — | 6 | 4 | no | — |
| 87 | C2×C4 | — | 6 | 12 | no | — |
| 88 | C2×C4 | — | 6 | 6 | no | — |
| 89 | C8 | — | 6 | 4 | no | — |
| 90 | C2×C3 | — | 16 | 3 | no | — |
| 91 | D3 | — | 16 | 4 | no | — |
| 92 | C22 | — | 3 | 3 | no | — |
| 93 | C22 | — | 3 | 3 | no | — |
| 94 | C22 | — | 3 | 3 | no | — |
| 95 | C22 | — | 6 | 1 | no | — |
| 96 | C22 | — | 6 | 4 | no | — |
| 97 | C22 | — | 12 | 6 | no | — |
| 98 | C22 | — | 12 | 6 | no | — |
| 99 | C22 | #857 | 1 | 1 | yes | characteristic, direct product factor |
| 100 | C22 | C22×S4 | 1 | 1 | yes | characteristic |
| 101 | C4 | — | 3 | 1 | no | — |
| 102 | C4 | — | 3 | 3 | no | — |
| 103 | C4 | — | 3 | 1 | no | — |
| 104 | C4 | — | 3 | 3 | no | — |
| 105 | C4 | — | 6 | 4 | no | — |
| 106 | C3 | — | 16 | 1 | no | Sylow |
| 107 | C2 | — | 3 | 1 | no | — |
| 108 | C2 | — | 3 | 3 | no | — |
| 109 | C2 | — | 12 | 4 | no | — |
| 110 | C2 | C2×#857 | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857 |
| commutator, cosocle, hypocenter | C42⋊C3 | C23 |
| Frattini | C22 | C22×S4 |
| Fitting | C22×C42 | D3 |
| socle | C24 | S4 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C42⋊C3 | C2 | semidirect |
| 2. C2×#857 | C2 | direct (N × H) |
| 5. C2×C42⋊C3 | C22 | semidirect |
| 6. #857 | C22 | direct (N × H) |
| 9. C22×C42 | D3 | semidirect |
| 14. C42⋊C3 | C23 | semidirect |
| 28. C2×C42 | C2×D3 | semidirect |
| 61. C42 | C22×D3 | semidirect |
| 99. C22 | #857 | direct (N × H) |
| 110. C2 | C2×#857 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42⋊C3 | 2 |
| C2×#857 | 2 |
| C22×#104 | 3 |
| C22×S4 | 4 |