Subgroups — C₂×#489 #3132

Order 128.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#124	C₂	1	1	yes	maximal, characteristic
3	#489	C₂	1	8	yes	maximal, direct product factor
4	C₂×#103	C₂	1	2	yes	maximal, complemented
5	C₂×#102	C₂	1	2	yes	maximal
6	C₂×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈	—	2	1	no	—
8	C₂²×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	2	yes	—
11	C₄⋊Q₈	—	2	2	no	—
12	C₄⋊Q₈	C₂²	1	4	yes	complemented
13	#124	—	2	2	no	—
14	#124	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄	—	2	1	no	—
16	C₂×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄	—	2	1	no	—
18	C₂×C₂²⋊C₄	—	2	1	no	—
19	C₂×C₄²	C₂²	1	1	yes	characteristic
20	#103	C₂²	1	8	yes	complemented
21	#102	C₂²	1	8	yes	—
22	#97	C₂²	1	4	yes	complemented
23	C₂⁴	—	2	1	no	—
24	C₂×Q₈	—	2	1	no	—
25	C₂×Q₈	—	2	4	no	—
26	C₂×Q₈	—	2	4	no	—
27	C₂×Q₈	—	4	4	no	—
28	C₂×Q₈	C₂³	1	2	yes	—
29	C₂×D₄	—	2	1	no	—
30	C₂×D₄	—	2	4	no	—
31	C₂×D₄	C₂³	1	2	yes	—
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	—	2	2	no	—
34	C₂²×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₄	D₄	1	1	yes	characteristic
36	C₂²×C₄	C₂³	1	1	yes	characteristic
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	C₂³	1	4	yes	—
39	C₄⋊C₄	—	2	4	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄⋊C₄	C₂³	1	4	yes	—
42	C₂²⋊C₄	—	2	4	no	—
43	C₂²⋊C₄	—	2	4	no	—
44	C₄²	D₄	1	2	yes	—
45	C₄²	C₂³	1	2	yes	—
46	C₂³	—	2	2	no	—
47	C₂³	—	4	2	no	—
48	C₂³	—	4	2	no	—
49	C₂³	—	4	2	no	—
50	C₂³	C₂×D₄	1	1	yes	characteristic
51	Q₈	—	2	4	no	—
52	Q₈	—	4	8	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	2	4	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	4	4	no	—
62	C₂×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₄	C₂×D₄	1	2	yes	—
64	C₂×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄	#41	1	4	yes	—
66	C₂×C₄	C₂⁴	1	1	yes	characteristic
67	C₈	—	2	4	no	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	8	4	no	—
72	C₂²	#124	1	2	yes	—
73	C₂²	AGL(1,ℤ/8)	1	2	yes	—
74	C₂²	#137	1	2	yes	—
75	C₂²	C₂²×D₄	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	4	2	no	—
79	C₄	—	4	4	no	—
80	C₄	#124	1	2	yes	—
81	C₄	C₂×#41	1	2	yes	—
82	C₂	—	8	2	no	—
83	C₂	#489	1	4	yes	direct product factor
84	C₂	C₂×#124	1	1	yes	characteristic
85	C₂	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂	C₂×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈	C₂	semidirect
3. #489	C₂	direct (N × H)
4. C₂×#103	C₂	semidirect
6. C₂×#97	C₂	semidirect
12. C₄⋊Q₈	C₂²	semidirect
20. #103	C₂²	semidirect
22. #97	C₂²	semidirect
83. C₂	#489	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⋊Q₈	2
C₂×#124	2
#489	2
C₂×#103	2
C₂×#102	2
C₂×#97	2

Subgroups — C2×#489 #3132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#489 #3132