Subgroups — C₂×#509 #3412

Order 128.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#509	C₂	1	8	yes	maximal, direct product factor
4	C₂×SD₃₂	C₂	1	2	yes	maximal, complemented
5	C₂×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41	—	2	1	no	—
8	C₂²×D₄	—	2	1	no	—
9	#135	—	2	2	no	—
10	#135	C₂²	1	4	yes	complemented
11	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
12	C₂×SD₁₆	—	2	1	no	—
13	C₂×D₈	—	2	1	no	—
14	C₂×D₈	—	2	2	no	—
15	C₂×D₈	C₂²	1	2	yes	—
16	C₂×D₈	C₂²	1	4	yes	complemented
17	C₂×D₈	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₈	C₂²	1	1	yes	characteristic
19	SD₃₂	C₂²	1	8	yes	complemented
20	D₁₆	C₂²	1	8	yes	complemented
21	M₃₂	C₂²	1	4	yes	complemented
22	C₂×C₁₆	C₂²	1	2	yes	complemented
23	C₂⁴	—	4	1	no	—
24	#41	—	2	4	no	—
25	#41	—	4	2	no	—
26	C₂×Q₈	—	2	1	no	—
27	C₂×D₄	—	2	2	no	—
28	C₂×D₄	—	2	4	no	—
29	C₂×D₄	—	2	1	no	—
30	C₂×D₄	—	4	1	no	—
31	C₂×D₄	—	4	2	no	—
32	C₂×D₄	—	4	1	no	—
33	C₂²×C₄	—	4	1	no	—
34	C₂²×C₄	D₄	1	1	yes	characteristic
35	Q₁₆	—	2	1	no	—
36	Q₁₆	C₂³	1	2	yes	complemented
37	SD₁₆	—	2	4	no	—
38	D₈	—	2	2	no	—
39	D₈	—	2	4	no	—
40	D₈	—	2	1	no	—
41	D₈	C₂³	1	4	yes	—
42	D₈	C₂³	1	2	yes	complemented
43	C₂×C₈	D₄	1	1	yes	characteristic
44	C₂×C₈	D₄	1	2	yes	—
45	C₂×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₈	C₂³	1	2	yes	—
47	C₁₆	C₂³	1	4	yes	complemented
48	C₂³	—	4	4	no	—
49	C₂³	—	4	1	no	—
50	C₂³	—	4	2	no	—
51	C₂³	—	8	4	no	—
52	C₂³	D₈	1	1	yes	characteristic
53	Q₈	—	2	2	no	—
54	Q₈	—	4	1	no	—
55	D₄	—	2	4	no	—
56	D₄	—	2	2	no	—
57	D₄	—	4	4	no	—
58	D₄	—	4	4	no	—
59	D₄	—	4	2	no	—
60	D₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	4	no	—
63	C₂×C₄	D₈	1	2	yes	—
64	C₂×C₄	D₈	1	1	yes	characteristic
65	C₂×C₄	C₂×D₄	1	2	yes	—
66	C₂×C₄	C₂×D₄	1	1	yes	characteristic
67	C₈	C₂×D₄	1	1	yes	characteristic
68	C₈	C₂×D₄	1	2	yes	—
69	C₈	C₂⁴	1	1	yes	characteristic
70	C₂²	—	2	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	4	4	no	—
73	C₂²	—	8	8	no	—
74	C₂²	—	8	2	no	—
75	C₂²	—	8	4	no	—
76	C₂²	C₂×D₈	1	2	yes	—
77	C₂²	C₂×D₈	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	C₂×D₈	1	2	yes	—
80	C₄	C₂×D₈	1	1	yes	characteristic
81	C₄	C₂²×D₄	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	—	8	2	no	—
84	C₂	—	8	4	no	—
85	C₂	#509	1	2	yes	direct product factor
86	C₂	C₂²×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#135	C₂	semidirect
2. C₂²×D₈	C₂	semidirect
3. #509	C₂	direct (N × H)
4. C₂×SD₃₂	C₂	semidirect
5. C₂×D₁₆	C₂	semidirect
6. C₂×M₃₂	C₂	semidirect
10. #135	C₂²	semidirect
11. C₂×Q₁₆	C₂²	semidirect
16. C₂×D₈	C₂²	semidirect
17. C₂×D₈	C₂²	semidirect
19. SD₃₂	C₂²	semidirect
20. D₁₆	C₂²	semidirect
21. M₃₂	C₂²	semidirect
22. C₂×C₁₆	C₂²	semidirect
36. Q₁₆	C₂³	semidirect
42. D₈	C₂³	semidirect
47. C₁₆	C₂³	semidirect
85. C₂	#509	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×#135	2
C₂²×D₈	2
#509	2
C₂×SD₃₂	2
C₂×D₁₆	2
C₂×M₃₂	2

Subgroups — C2×#509 #3412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#509 #3412