Subgroups — C₄²×D₄ #2271

Order 128.

462 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄	C₂	1	3	yes	maximal
2	C₂²×C₄²	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₄×C₂²⋊C₄	C₂	1	6	yes	maximal, complemented
5	C₄³	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄	C₂²	1	1	yes	characteristic
7	C₂³×C₄	C₂²	1	6	yes	—
8	C₄×D₄	C₄	1	24	yes	direct product factor
9	C₂×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
11	C₂×C₄²	—	2	8	no	—
12	C₂×C₄²	C₂²	1	3	yes	—
13	C₂×C₄²	C₂²	1	6	yes	—
14	C₂×C₄²	C₂²	1	3	yes	—
15	C₂×C₄²	C₂²	1	1	yes	characteristic
16	#95	C₂²	1	6	yes	—
17	C₂⁴	C₂³	1	2	yes	—
18	C₂×D₄	C₂×C₄	1	12	yes	—
19	C₂²×C₄	—	2	12	no	—
20	C₂²×C₄	—	2	12	no	—
21	C₂²×C₄	C₂×C₄	1	24	yes	complemented
22	C₂²×C₄	C₂³	1	3	yes	—
23	C₂²×C₄	C₂³	1	6	yes	—
24	C₂²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₄	C₂³	1	3	yes	—
26	C₄⋊C₄	C₂×C₄	1	12	yes	complemented
27	C₂²⋊C₄	C₂×C₄	1	24	yes	complemented
28	C₄²	—	2	12	no	—
29	C₄²	—	2	6	no	—
30	C₄²	C₂×C₄	1	12	yes	complemented
31	C₄²	D₄	1	4	yes	direct product factor
32	C₂³	—	2	2	no	—
33	C₂³	—	2	6	no	—
34	C₂³	C₂²×C₄	1	12	yes	—
35	C₂³	C₂⁴	1	1	yes	characteristic
36	D₄	C₄²	1	16	yes	direct product factor
37	C₂×C₄	—	2	6	no	—
38	C₂×C₄	—	2	6	no	—
39	C₂×C₄	—	2	6	no	—
40	C₂×C₄	—	2	48	no	—
41	C₂×C₄	C₂²×C₄	1	6	yes	—
42	C₂×C₄	C₂²×C₄	1	12	yes	—
43	C₂×C₄	C₂²×C₄	1	6	yes	—
44	C₂×C₄	C₂²×C₄	1	6	yes	—
45	C₂×C₄	C₂×D₄	1	6	yes	—
46	C₂×C₄	#41	1	6	yes	—
47	C₂²	—	2	12	no	—
48	C₂²	—	2	12	no	—
49	C₂²	C₂×C₄²	1	8	yes	complemented
50	C₂²	C₂³×C₄	1	3	yes	—
51	C₂²	C₂²×D₄	1	1	yes	characteristic
52	C₂²	C₂×#41	1	3	yes	—
53	C₄	—	2	12	no	—
54	C₄	—	2	6	no	—
55	C₄	C₂×C₄²	1	4	yes	complemented
56	C₄	C₄×D₄	1	12	yes	direct product factor
57	C₂	—	2	8	no	—
58	C₂	C₂²×C₄²	1	1	yes	characteristic
59	C₂	C₂×C₄×D₄	1	3	yes	—
60	C₂	C₄×#41	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²
commutator	C₂	C₂²×C₄²
Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²	C₂	semidirect
3. C₄×C₄⋊C₄	C₂	semidirect
4. C₄×C₂²⋊C₄	C₂	semidirect
5. C₄³	C₂	semidirect
8. C₄×D₄	C₄	direct (N × H)
21. C₂²×C₄	C₂×C₄	semidirect
26. C₄⋊C₄	C₂×C₄	semidirect
27. C₂²⋊C₄	C₂×C₄	semidirect
30. C₄²	C₂×C₄	semidirect
31. C₄²	D₄	direct (N × H)
36. D₄	C₄²	direct (N × H)
49. C₂²	C₂×C₄²	semidirect
55. C₄	C₂×C₄²	semidirect
56. C₄	C₄×D₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄	2
C₂²×C₄²	2
C₄×C₄⋊C₄	2
C₄×C₂²⋊C₄	2
C₄³	2

Subgroups — C42×D4 #2271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×D₄ #2271