Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
342 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C4×Q8 | C2 | 1 | 3 | yes | maximal |
| 2 | C4×C4⋊C4 | C2 | 1 | 9 | yes | maximal |
| 3 | C43 | C2 | 1 | 3 | yes | maximal |
| 4 | C22×Q8 | C22 | 1 | 1 | yes | characteristic |
| 5 | C4×Q8 | C4 | 1 | 24 | yes | direct product factor |
| 6 | C2×C4⋊C4 | C22 | 1 | 9 | yes | — |
| 7 | C2×C42 | C22 | 1 | 9 | yes | — |
| 8 | C2×C42 | C22 | 1 | 9 | yes | — |
| 9 | C2×C42 | C22 | 1 | 1 | yes | characteristic |
| 10 | #95 | C22 | 1 | 6 | yes | — |
| 11 | C2×Q8 | C2×C4 | 1 | 12 | yes | — |
| 12 | C22×C4 | C23 | 1 | 9 | yes | — |
| 13 | C22×C4 | C23 | 1 | 3 | yes | — |
| 14 | C22×C4 | C23 | 1 | 3 | yes | — |
| 15 | C4⋊C4 | C2×C4 | 1 | 36 | yes | — |
| 16 | C42 | — | 2 | 18 | no | — |
| 17 | C42 | C2×C4 | 1 | 36 | yes | — |
| 18 | C42 | Q8 | 1 | 4 | yes | direct product factor |
| 19 | C23 | C24 | 1 | 1 | yes | characteristic |
| 20 | Q8 | C42 | 1 | 16 | yes | direct product factor |
| 21 | C2×C4 | — | 2 | 18 | no | — |
| 22 | C2×C4 | C22×C4 | 1 | 18 | yes | — |
| 23 | C2×C4 | C22×C4 | 1 | 18 | yes | — |
| 24 | C2×C4 | C22×C4 | 1 | 6 | yes | — |
| 25 | C2×C4 | C2×Q8 | 1 | 6 | yes | — |
| 26 | C2×C4 | #41 | 1 | 6 | yes | — |
| 27 | C22 | C23×C4 | 1 | 3 | yes | — |
| 28 | C22 | C22×Q8 | 1 | 1 | yes | characteristic |
| 29 | C22 | C2×#41 | 1 | 3 | yes | — |
| 30 | C4 | — | 2 | 18 | no | — |
| 31 | C4 | C2×C42 | 1 | 12 | yes | — |
| 32 | C4 | C4×Q8 | 1 | 12 | yes | direct product factor |
| 33 | C2 | C22×C42 | 1 | 1 | yes | characteristic |
| 34 | C2 | C2×C4×Q8 | 1 | 3 | yes | — |
| 35 | C2 | C4×#41 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22 |
| commutator | C2 | C22×C42 |
| Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C4×Q8 | C4 | direct (N × H) |
| 18. C42 | Q8 | direct (N × H) |
| 20. Q8 | C42 | direct (N × H) |
| 32. C4 | C4×Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×Q8 | 2 |
| C4×C4⋊C4 | 2 |
| C43 | 2 |