Subgroups — C₂×#492 #3143

Order 128.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#127	C₂	1	1	yes	maximal, characteristic
4	#492	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈	C₂²	1	2	yes	—
7	C₂²×D₄	—	2	2	no	—
8	C₂²×D₄	C₂²	1	2	yes	—
9	C₂×SD₁₆	—	2	16	no	—
10	C₂×SD₁₆	C₂²	1	16	yes	complemented
11	C₂²×C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈	—	2	2	no	—
13	C₄⋊Q₈	C₂²	1	4	yes	complemented
14	#127	—	2	2	no	—
15	#127	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄²	C₂²	1	1	yes	characteristic
18	C₄×C₈	C₂²	1	4	yes	complemented
19	C₂⁴	—	2	2	no	—
20	C₂×Q₈	—	2	2	no	—
21	C₂×Q₈	—	2	8	no	—
22	C₂×Q₈	C₂³	1	4	yes	—
23	C₂×D₄	—	2	8	no	—
24	C₂×D₄	—	2	2	no	—
25	C₂×D₄	—	2	8	no	—
26	C₂×D₄	—	4	8	no	—
27	C₂×D₄	C₂³	1	4	yes	—
28	C₂²×C₄	—	2	2	no	—
29	C₂²×C₄	D₄	1	2	yes	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	SD₁₆	—	2	32	no	—
32	C₂×C₈	D₄	1	8	yes	complemented
33	C₂×C₈	C₂³	1	4	yes	—
34	C₄⋊C₄	—	2	8	no	—
35	C₄²	D₄	1	2	yes	—
36	C₄²	C₂³	1	2	yes	—
37	C₂³	—	2	4	no	—
38	C₂³	—	4	4	no	—
39	C₂³	—	4	8	no	—
40	C₂³	C₂×D₄	1	1	yes	characteristic
41	Q₈	—	2	8	no	—
42	Q₈	—	4	4	no	—
43	D₄	—	2	8	no	—
44	D₄	—	4	4	no	—
45	D₄	—	4	16	no	—
46	C₂×C₄	—	2	4	no	—
47	C₂×C₄	—	4	4	no	—
48	C₂×C₄	SD₁₆	1	8	yes	complemented
49	C₂×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄	C₂×D₄	1	4	yes	—
51	C₂×C₄	C₂×D₄	1	4	yes	—
52	C₂×C₄	C₂⁴	1	1	yes	characteristic
53	C₈	C₂×D₄	1	8	yes	complemented
54	C₂²	—	4	4	no	—
55	C₂²	—	4	8	no	—
56	C₂²	—	8	8	no	—
57	C₂²	#127	1	2	yes	—
58	C₂²	C₂×SD₁₆	1	4	yes	—
59	C₂²	C₂²×D₄	1	1	yes	characteristic
60	C₄	—	4	4	no	—
61	C₄	#127	1	2	yes	—
62	C₄	C₂×SD₁₆	1	8	yes	complemented
63	C₄	C₂²×D₄	1	2	yes	—
64	C₂	—	8	4	no	—
65	C₂	#492	1	4	yes	direct product factor
66	C₂	C₂×#127	1	1	yes	characteristic
67	C₂	C₂²×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×#127	C₂
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. C₂²×SD₁₆	C₂	semidirect
2. C₂×C₄⋊Q₈	C₂	semidirect
4. #492	C₂	direct (N × H)
5. C₂×C₄×C₈	C₂	semidirect
10. C₂×SD₁₆	C₂²	semidirect
13. C₄⋊Q₈	C₂²	semidirect
18. C₄×C₈	C₂²	semidirect
32. C₂×C₈	D₄	semidirect
48. C₂×C₄	SD₁₆	semidirect
53. C₈	C₂×D₄	semidirect
62. C₄	C₂×SD₁₆	semidirect
65. C₂	#492	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆	2
C₂×C₄⋊Q₈	2
C₂×#127	2
#492	2
C₂×C₄×C₈	2

Subgroups — C2×#492 #3143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#492 #3143