Subgroups — #3169

Order 128.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532	C₂	1	2	yes	maximal, complemented
2	#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#493	C₂	1	1	yes	maximal, characteristic, complemented
4	#486	C₂	1	2	yes	maximal, complemented
5	#466	C₂	1	4	yes	maximal, complemented
6	C₄×D₈	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	—	2	2	no	—
9	C₂×D₈	C₂²	1	4	yes	—
10	C₂²×C₈	C₂²	1	2	yes	—
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#127	C₂²	1	2	yes	—
13	#124	—	2	2	no	—
14	#121	—	2	2	no	—
15	#121	C₂²	1	4	yes	—
16	C₄×D₄	—	2	2	no	—
17	C₄×D₄	C₂²	1	4	yes	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#107	C₂²	1	4	yes	—
20	#102	C₂²	1	8	yes	—
21	C₄×C₈	C₂²	1	2	yes	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₄×C₈	C₂²	1	1	yes	characteristic
24	#41	—	4	8	no	—
25	C₂×Q₈	—	2	2	no	—
26	C₂×D₄	—	2	4	no	—
27	C₂×D₄	—	2	2	no	—
28	C₂×D₄	—	2	2	no	—
29	C₂×D₄	C₂³	1	4	yes	—
30	C₂²×C₄	—	2	4	no	—
31	C₂²×C₄	D₄	1	1	yes	characteristic
32	C₂²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	D₈	—	2	8	no	—
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	C₂³	1	2	yes	—
37	C₂×C₈	C₂³	1	2	yes	—
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	C₂³	1	4	yes	—
40	C₂²⋊C₄	—	2	4	no	—
41	C₂²⋊C₄	—	2	4	no	—
42	C₄²	D₄	1	1	yes	characteristic
43	C₄²	D₄	1	1	yes	characteristic
44	C₄²	C₂³	1	1	yes	characteristic
45	C₄²	C₂³	1	1	yes	characteristic
46	C₂³	—	2	4	no	—
47	C₂³	C₂×D₄	1	1	yes	characteristic
48	Q₈	—	4	4	no	—
49	D₄	—	2	8	no	—
50	D₄	—	4	8	no	—
51	D₄	—	4	4	no	—
52	D₄	—	4	4	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	4	8	no	—
57	C₂×C₄	D₈	1	4	yes	complemented
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄	C₂⁴	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	#41	1	4	yes	complemented
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	4	4	no	—
71	C₂²	C₂×D₈	1	2	yes	—
72	C₂²	C₂²×D₄	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	4	4	no	—
76	C₄	C₂×D₈	1	2	yes	—
77	C₄	C₂×D₈	1	2	yes	—
78	C₄	#135	1	2	yes	—
79	C₄	C₂×#41	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	8	4	no	—
82	C₂	#532	1	1	yes	characteristic
83	C₂	C₂²×D₈	1	1	yes	characteristic
84	C₂	C₂×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂²	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. #532	C₂	semidirect
2. #500	C₂	semidirect
3. #493	C₂	semidirect
4. #486	C₂	semidirect
5. #466	C₂	semidirect
6. C₄×D₈	C₂	semidirect
7. C₂×C₄×C₈	C₂	semidirect
57. C₂×C₄	D₈	semidirect
65. C₈	#41	semidirect

Maximal subgroups

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

subgroup	index
#532	2
#500	2
#493	2
#486	2
#466	2
C₄×D₈	2
C₂×C₄×C₈	2