Subgroups — C₂²×C₄×Q₈ #3423

Order 128.

832 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₄	C₂	1	3	yes	maximal
4	C₂²×C₄²	C₂	1	3	yes	maximal
5	C₂²×Q₈	C₄	1	16	yes	direct product factor
6	C₂²×Q₈	C₂²	1	12	yes	—
7	C₂³×C₄	C₂²	1	3	yes	—
8	C₂³×C₄	C₂²	1	3	yes	—
9	C₂³×C₄	C₂²	1	1	yes	characteristic
10	C₄×Q₈	C₂²	1	64	yes	direct product factor
11	C₂×C₄⋊C₄	C₂²	1	36	yes	—
12	C₂×C₄²	C₂²	1	36	yes	—
13	C₂⁴	C₂³	1	1	yes	characteristic
14	C₂×Q₈	C₂×C₄	1	96	yes	direct product factor
15	C₂×Q₈	C₂³	1	16	yes	—
16	C₂²×C₄	—	2	12	no	—
17	C₂²×C₄	C₂×C₄	1	24	yes	—
18	C₂²×C₄	Q₈	1	8	yes	direct product factor
19	C₂²×C₄	C₂³	1	18	yes	—
20	C₂²×C₄	C₂³	1	18	yes	—
21	C₂²×C₄	C₂³	1	6	yes	—
22	C₄⋊C₄	C₂³	1	48	yes	—
23	C₄²	C₂³	1	48	yes	—
24	C₂³	C₂²×C₄	1	4	yes	—
25	C₂³	C₂×Q₈	1	4	yes	—
26	C₂³	#41	1	4	yes	—
27	C₂³	C₂⁴	1	3	yes	—
28	Q₈	C₂²×C₄	1	64	yes	direct product factor
29	C₂×C₄	—	2	36	no	—
30	C₂×C₄	C₂²×C₄	1	72	yes	—
31	C₂×C₄	C₂×Q₈	1	24	yes	direct product factor
32	C₂×C₄	C₂⁴	1	12	yes	—
33	C₂×C₄	C₂⁴	1	12	yes	—
34	C₂×C₄	C₂⁴	1	4	yes	—
35	C₂²	C₄×Q₈	1	16	yes	direct product factor
36	C₂²	C₂³×C₄	1	6	yes	—
37	C₂²	C₂²×Q₈	1	6	yes	—
38	C₂²	C₂×#41	1	6	yes	—
39	C₂²	C₂⁵	1	1	yes	characteristic
40	C₄	—	2	12	no	—
41	C₄	C₂³×C₄	1	24	yes	—
42	C₄	C₂²×Q₈	1	8	yes	direct product factor
43	C₂	C₂×C₄×Q₈	1	12	yes	direct product factor
44	C₂	C₂⁴×C₄	1	1	yes	characteristic
45	C₂	C₂³×Q₈	1	1	yes	characteristic
46	C₂	C₂²×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²
commutator	C₂	C₂⁴×C₄
Frattini, cosocle	C₂²	C₂⁵
socle, prime radical	C₂⁴	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×Q₈	C₂	direct (N × H)
5. C₂²×Q₈	C₄	direct (N × H)
10. C₄×Q₈	C₂²	direct (N × H)
14. C₂×Q₈	C₂×C₄	direct (N × H)
18. C₂²×C₄	Q₈	direct (N × H)
28. Q₈	C₂²×C₄	direct (N × H)
31. C₂×C₄	C₂×Q₈	direct (N × H)
35. C₂²	C₄×Q₈	direct (N × H)
42. C₄	C₂²×Q₈	direct (N × H)
43. C₂	C₂×C₄×Q₈	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×Q₈	2
C₂×C₄×Q₈	2
C₂²×C₄⋊C₄	2
C₂²×C₄²	2

Subgroups — C22×C4×Q8 #3423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Q₈ #3423