Subgroups — Q₈×#41 #3478

Order 128.

534 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	#557	C₂	1	3	yes	maximal, complemented
3	#554	C₂	1	9	yes	maximal, complemented
4	D₄×Q₈	C₂	1	3	yes	maximal, complemented
5	#533	C₂	1	9	yes	maximal, complemented
6	C₄×#41	C₂	1	3	yes	maximal
7	C₂×C₄×Q₈	C₂	1	3	yes	maximal, complemented
8	C₂×#41	C₂²	1	1	yes	characteristic
9	C₂²×Q₈	C₂²	1	3	yes	—
10	C₄⋊Q₈	C₂²	1	9	yes	—
11	C₄⋊Q₈	C₂²	1	9	yes	—
12	#125	C₂²	1	18	yes	—
13	C₂²⋊Q₈	C₂²	1	18	yes	—
14	C₂²⋊Q₈	C₂²	1	18	yes	—
15	C₄×Q₈	C₂²	1	1	yes	characteristic
16	C₄×Q₈	C₂²	1	9	yes	—
17	C₄×Q₈	C₂²	1	3	yes	—
18	C₄×Q₈	C₂²	1	9	yes	—
19	C₄×Q₈	C₂²	1	3	yes	—
20	C₄×Q₈	C₂²	1	3	yes	—
21	C₄×Q₈	C₂²	1	6	yes	—
22	C₄×D₄	C₂²	1	9	yes	—
23	C₄×D₄	C₂²	1	9	yes	—
24	#117	C₂²	1	9	yes	—
25	C₂×C₄⋊C₄	C₂²	1	9	yes	—
26	C₂×C₄²	C₂²	1	9	yes	—
27	#41	Q₈	1	8	yes	direct product factor
28	C₂×Q₈	—	2	12	no	—
29	C₂×Q₈	C₂³	1	9	yes	—
30	C₂×Q₈	C₂³	1	6	yes	—
31	C₂×Q₈	C₂³	1	1	yes	characteristic
32	C₂×Q₈	C₂³	1	1	yes	characteristic
33	C₂×D₄	C₂³	1	3	yes	—
34	C₂²×C₄	C₂³	1	9	yes	—
35	C₂²×C₄	C₂³	1	9	yes	—
36	C₂²×C₄	C₂³	1	3	yes	—
37	C₄⋊C₄	C₂³	1	3	yes	—
38	C₄⋊C₄	C₂³	1	18	yes	—
39	C₄⋊C₄	C₂³	1	9	yes	—
40	C₄⋊C₄	C₂³	1	9	yes	—
41	C₄⋊C₄	C₂³	1	9	yes	—
42	C₄⋊C₄	C₂³	1	18	yes	—
43	C₂²⋊C₄	C₂³	1	18	yes	—
44	C₄²	C₂³	1	9	yes	—
45	C₄²	C₂³	1	9	yes	—
46	C₄²	C₂³	1	3	yes	—
47	C₄²	C₂³	1	9	yes	—
48	C₂³	C₂⁴	1	3	yes	—
49	Q₈	—	2	18	no	—
50	Q₈	—	4	6	no	—
51	Q₈	C₂×Q₈	1	4	yes	complemented
52	Q₈	#41	1	4	yes	direct product factor
53	D₄	C₂×Q₈	1	12	yes	complemented
54	C₂×C₄	—	2	18	no	—
55	C₂×C₄	—	4	9	no	—
56	C₂×C₄	C₂×Q₈	1	12	yes	complemented
57	C₂×C₄	C₂⁴	1	3	yes	—
58	C₂×C₄	C₂⁴	1	9	yes	—
59	C₂×C₄	C₂⁴	1	9	yes	—
60	C₂×C₄	C₂⁴	1	3	yes	—
61	C₂×C₄	C₂⁴	1	3	yes	—
62	C₂×C₄	C₂⁴	1	1	yes	characteristic
63	C₂²	—	2	3	no	—
64	C₂²	—	2	3	no	—
65	C₂²	C₂²×Q₈	1	6	yes	—
66	C₂²	C₂⁵	1	1	yes	characteristic
67	C₄	—	2	9	no	—
68	C₄	—	2	3	no	—
69	C₄	—	2	9	no	—
70	C₄	C₂²×Q₈	1	2	yes	—
71	C₄	C₂²×Q₈	1	6	yes	—
72	C₄	C₂×#41	1	6	yes	—
73	C₂	—	2	6	no	—
74	C₂	C₂³×Q₈	1	1	yes	characteristic
75	C₂	C₂²×#41	1	1	yes	characteristic
76	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. Q₈²	C₂	semidirect
2. #557	C₂	semidirect
3. #554	C₂	semidirect
4. D₄×Q₈	C₂	semidirect
5. #533	C₂	semidirect
7. C₂×C₄×Q₈	C₂	semidirect
27. #41	Q₈	direct (N × H)
51. Q₈	C₂×Q₈	semidirect
52. Q₈	#41	direct (N × H)
53. D₄	C₂×Q₈	semidirect
56. C₂×C₄	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈²	2
#557	2
#554	2
D₄×Q₈	2
#533	2
C₄×#41	2
C₂×C₄×Q₈	2

Subgroups — Q8×#41 #3478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#41 #3478