Subgroups — C₂×#466 #3048

Order 128.

308 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#121	C₂	1	2	yes	maximal, complemented
4	#466	C₂	1	8	yes	maximal, direct product factor
5	C₂×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#102	C₂	1	2	yes	maximal, complemented
7	C₂²×D₄	—	2	2	no	—
8	C₂²×D₄	C₂²	1	2	yes	—
9	C₂³×C₄	C₂²	1	1	yes	characteristic
10	C₂×D₈	—	2	4	no	—
11	C₂×D₈	C₂²	1	4	yes	complemented
12	C₂²×C₈	—	2	4	no	—
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂²	1	4	yes	complemented
16	#121	—	2	4	no	—
17	#121	C₂²	1	8	yes	complemented
18	C₂×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄	—	2	2	no	—
20	#107	C₂²	1	4	yes	complemented
21	#102	C₂²	1	8	yes	complemented
22	C₂⁴	—	2	2	no	—
23	C₂⁴	D₄	1	1	yes	characteristic
24	C₂×D₄	—	2	8	no	—
25	C₂×D₄	—	2	4	no	—
26	C₂×D₄	—	2	2	no	—
27	C₂×D₄	—	2	4	no	—
28	C₂×D₄	—	4	8	no	—
29	C₂×D₄	C₂³	1	4	yes	—
30	C₂²×C₄	—	2	2	no	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	—	2	2	no	—
33	C₂²×C₄	D₄	1	2	yes	—
34	C₂²×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₄	C₂³	1	2	yes	—
36	C₂²×C₄	C₂³	1	1	yes	characteristic
37	D₈	—	2	8	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	—	2	8	no	—
40	C₂×C₈	D₄	1	4	yes	complemented
41	C₂×C₈	C₂³	1	2	yes	—
42	C₂×C₈	C₂³	1	2	yes	—
43	C₄⋊C₄	—	2	2	no	—
44	C₄⋊C₄	C₂³	1	4	yes	—
45	C₂²⋊C₄	—	2	8	no	—
46	C₂³	—	2	2	no	—
47	C₂³	—	2	2	no	—
48	C₂³	—	2	4	no	—
49	C₂³	—	4	4	no	—
50	C₂³	—	4	4	no	—
51	C₂³	—	4	4	no	—
52	C₂³	D₈	1	4	yes	complemented
53	C₂³	C₂×D₄	1	2	yes	—
54	C₂³	C₂×D₄	1	1	yes	characteristic
55	D₄	—	2	4	no	—
56	D₄	—	2	4	no	—
57	D₄	—	4	16	no	—
58	D₄	—	4	4	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	4	4	no	—
65	C₂×C₄	C₂×D₄	1	2	yes	—
66	C₂×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄	C₂×D₄	1	2	yes	—
68	C₂×C₄	#41	1	2	yes	—
69	C₂×C₄	C₂⁴	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	C₂×D₄	1	4	yes	complemented
72	C₂²	—	2	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	2	8	no	—
75	C₂²	—	4	4	no	—
76	C₂²	—	4	4	no	—
77	C₂²	—	4	4	no	—
78	C₂²	—	8	8	no	—
79	C₂²	#121	1	2	yes	—
80	C₂²	C₂×D₈	1	2	yes	—
81	C₂²	C₂×D₈	1	4	yes	complemented
82	C₂²	#135	1	2	yes	—
83	C₂²	C₂²×D₄	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	4	4	no	—
86	C₄	#121	1	2	yes	—
87	C₄	C₂²×D₄	1	1	yes	characteristic
88	C₄	C₂×#41	1	1	yes	characteristic
89	C₂	—	2	4	no	—
90	C₂	—	8	4	no	—
91	C₂	#466	1	4	yes	direct product factor
92	C₂	C₂×#121	1	1	yes	characteristic
93	C₂	C₂²×D₈	1	1	yes	characteristic
94	C₂	C₂×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ 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×D₈	C₂	semidirect
2. C₂³×C₈	C₂	semidirect
3. C₂×#121	C₂	semidirect
4. #466	C₂	direct (N × H)
5. C₂×#107	C₂	semidirect
6. C₂×#102	C₂	semidirect
11. C₂×D₈	C₂²	semidirect
15. C₂²×C₈	C₂²	semidirect
17. #121	C₂²	semidirect
20. #107	C₂²	semidirect
21. #102	C₂²	semidirect
40. C₂×C₈	D₄	semidirect
52. C₂³	D₈	semidirect
71. C₈	C₂×D₄	semidirect
81. C₂²	C₂×D₈	semidirect
91. C₂	#466	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×D₈	2
C₂³×C₈	2
C₂×#121	2
#466	2
C₂×#107	2
C₂×#102	2

Subgroups — C2×#466 #3048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#466 #3048