Subgroups — D₄×Q₁₆ #3286

Order 128.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈	C₂	1	2	yes	maximal
3	#494	C₂	1	1	yes	maximal, characteristic, complemented
4	#467	C₂	1	2	yes	maximal, complemented
5	#462	C₂	1	2	yes	maximal, complemented
6	#451	C₂	1	4	yes	maximal, complemented
7	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈	C₂²	1	4	yes	—
10	C₂×Q₁₆	—	2	8	no	—
11	C₂×Q₁₆	C₂²	1	4	yes	—
12	C₂×Q₁₆	C₂²	1	2	yes	—
13	C₂×Q₁₆	C₂²	1	2	yes	—
14	C₂×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈	—	2	2	no	—
17	C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂²⋊Q₈	—	2	4	no	—
19	C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₄×Q₈	C₂²	1	2	yes	—
21	C₄×D₄	—	2	2	no	—
22	C₄×D₄	C₂²	1	1	yes	characteristic
23	#107	C₂²	1	1	yes	characteristic
24	C₄⋊C₈	C₂²	1	1	yes	characteristic
25	#103	C₂²	1	2	yes	—
26	#103	C₂²	1	4	yes	—
27	C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈	C₂²	1	1	yes	characteristic
29	C₂×Q₈	—	2	4	no	—
30	C₂×Q₈	—	2	8	no	—
31	C₂×Q₈	—	4	4	no	—
32	C₂×Q₈	C₂³	1	4	yes	—
33	C₂×Q₈	C₂³	1	2	yes	—
34	C₂×D₄	D₄	1	1	yes	characteristic
35	C₂²×C₄	—	2	4	no	—
36	C₂²×C₄	C₂³	1	2	yes	—
37	Q₁₆	—	2	8	no	—
38	Q₁₆	—	2	4	no	—
39	Q₁₆	—	4	2	no	—
40	Q₁₆	D₄	1	4	yes	direct product factor
41	C₂×C₈	—	2	4	no	—
42	C₂×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₈	C₂³	1	2	yes	—
44	C₂×C₈	C₂³	1	1	yes	characteristic
45	C₄⋊C₄	—	2	2	no	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	2	4	no	—
49	C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₄⋊C₄	C₂³	1	2	yes	—
51	C₂²⋊C₄	—	2	4	no	—
52	C₂²⋊C₄	D₄	1	2	yes	—
53	C₄²	—	2	2	no	—
54	C₄²	C₂³	1	1	yes	characteristic
55	C₂³	C₂×D₄	1	2	yes	—
56	Q₈	—	2	2	no	—
57	Q₈	—	2	4	no	—
58	Q₈	—	2	4	no	—
59	Q₈	—	4	8	no	—
60	Q₈	C₂×D₄	1	4	yes	—
61	D₄	Q₁₆	1	4	yes	direct product factor
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	4	8	no	—
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄	1	2	yes	—
70	C₂×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄	C₂⁴	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	2	1	no	—
74	C₈	C₂×D₄	1	2	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	C₂×Q₁₆	1	4	yes	complemented
78	C₂²	C₂²×D₄	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	4	no	—
81	C₄	—	2	1	no	—
82	C₄	—	4	4	no	—
83	C₄	—	4	2	no	—
84	C₄	C₂×Q₁₆	1	2	yes	complemented
85	C₄	C₂²×D₄	1	1	yes	characteristic
86	C₄	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂	—	2	4	no	—
88	C₂	D₄²	1	1	yes	characteristic
89	C₂	C₂²×Q₁₆	1	1	yes	characteristic
90	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×D₄	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₁₆	C₂	semidirect
3. #494	C₂	semidirect
4. #467	C₂	semidirect
5. #462	C₂	semidirect
6. #451	C₂	semidirect
7. C₄×Q₁₆	C₂	semidirect
40. Q₁₆	D₄	direct (N × H)
61. D₄	Q₁₆	direct (N × H)
77. C₂²	C₂×Q₁₆	semidirect
84. C₄	C₂×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆	2
D₄×Q₈	2
#494	2
#467	2
#462	2
#451	2
C₄×Q₁₆	2
C₈×D₄	2

Subgroups — D4×Q16 #3286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₁₆ #3286