Subgroups — #3414

Order 128.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	2	yes	maximal, complemented
2	#510	C₂	1	4	yes	maximal, complemented
3	#509	C₂	1	4	yes	maximal, complemented
4	#508	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41	—	2	2	no	—
7	#135	—	2	2	no	—
8	#135	—	2	2	no	—
9	#135	C₂²	1	4	yes	—
10	#135	C₂²	1	4	yes	complemented
11	C₂×Q₁₆	C₂²	1	2	yes	complemented
12	C₂×SD₁₆	—	2	2	no	—
13	C₂×D₈	C₂²	1	2	yes	complemented
14	C₂²×C₈	C₂²	1	1	yes	characteristic
15	Q₃₂	C₂²	1	4	yes	complemented
16	SD₃₂	C₂²	1	8	yes	complemented
17	D₁₆	C₂²	1	4	yes	complemented
18	M₃₂	C₂²	1	4	yes	complemented
19	C₂×C₁₆	C₂²	1	2	yes	complemented
20	#41	—	2	4	no	—
21	#41	—	2	4	no	—
22	#41	—	4	2	no	—
23	#41	—	4	2	no	—
24	C₂×Q₈	—	2	2	no	—
25	C₂×D₄	—	2	2	no	—
26	C₂×D₄	—	4	2	no	—
27	C₂²×C₄	—	4	2	no	—
28	C₂²×C₄	D₄	1	1	yes	characteristic
29	Q₁₆	—	2	2	no	—
30	Q₁₆	C₂³	1	4	yes	—
31	SD₁₆	—	2	4	no	—
32	SD₁₆	—	2	4	no	—
33	D₈	—	2	2	no	—
34	D₈	C₂³	1	4	yes	—
35	C₂×C₈	D₄	1	2	yes	—
36	C₂×C₈	D₄	1	1	yes	characteristic
37	C₂×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₈	C₂³	1	2	yes	—
39	C₁₆	C₂³	1	4	yes	—
40	C₂³	—	4	2	no	—
41	C₂³	D₈	1	1	yes	characteristic
42	Q₈	—	2	4	no	—
43	Q₈	—	4	2	no	—
44	D₄	—	2	4	no	—
45	D₄	—	4	4	no	—
46	D₄	—	4	4	no	—
47	D₄	—	4	2	no	—
48	C₂×C₄	—	4	2	no	—
49	C₂×C₄	—	4	4	no	—
50	C₂×C₄	—	4	4	no	—
51	C₂×C₄	D₈	1	1	yes	characteristic
52	C₂×C₄	D₈	1	2	yes	—
53	C₂×C₄	C₂×D₄	1	2	yes	—
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₈	C₂×D₄	1	2	yes	—
56	C₈	C₂×D₄	1	1	yes	characteristic
57	C₈	C₂⁴	1	1	yes	characteristic
58	C₂²	—	4	1	no	—
59	C₂²	—	4	4	no	—
60	C₂²	—	8	4	no	—
61	C₂²	C₂×D₈	1	2	yes	—
62	C₂²	C₂×D₈	1	1	yes	characteristic
63	C₄	—	4	4	no	—
64	C₄	C₂×D₈	1	2	yes	—
65	C₄	C₂×D₈	1	1	yes	characteristic
66	C₄	C₂²×D₄	1	1	yes	characteristic
67	C₂	—	2	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	8	4	no	—
70	C₂	C₂²×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴
socle	C₂	C₂²×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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. #510	C₂	semidirect
3. #509	C₂	semidirect
4. #508	C₂	semidirect
5. C₂×M₃₂	C₂	semidirect
10. #135	C₂²	semidirect
11. C₂×Q₁₆	C₂²	semidirect
13. C₂×D₈	C₂²	semidirect
15. Q₃₂	C₂²	semidirect
16. SD₃₂	C₂²	semidirect
17. D₁₆	C₂²	semidirect
18. M₃₂	C₂²	semidirect
19. C₂×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135	2
#510	2
#509	2
#508	2
C₂×M₃₂	2