Subgroups — #5157

Order 192.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#938	C₂	1	2	yes	maximal, complemented
3	C₂×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#832	C₂	1	2	yes	maximal, complemented
5	#818	C₂	1	1	yes	maximal, characteristic, complemented
6	#466	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	2	yes	—
8	C₂×#67	—	2	2	no	—
9	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₃×D₈	—	2	2	no	—
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	#217	—	2	2	no	—
13	C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	D₃×C₈	—	2	2	no	—
16	C₂×D₈	D₃	1	1	yes	characteristic, complemented
17	C₂²×C₈	—	3	1	no	—
18	#121	—	3	2	no	—
19	#107	—	3	1	no	—
20	#102	—	3	2	no	—
21	C₂³×C₃	—	2	2	no	—
22	C₂²×D₃	D₄	1	1	yes	characteristic
23	C₃×D₄	—	2	2	no	—
24	C₃×D₄	—	2	2	no	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	#67	—	4	4	no	—
27	C₂×Dic₃	—	2	2	no	—
28	C₂×Dic₃	D₄	1	1	yes	characteristic
29	C₄×D₃	—	2	1	no	—
30	C₄×D₃	—	2	1	no	—
31	C₃×C₈	D₄	1	2	yes	complemented
32	C₃⋊C₈	—	2	1	no	—
33	C₂×D₄	—	6	2	no	—
34	C₂×D₄	C₂×D₃	1	2	yes	—
35	C₂²×C₄	—	3	1	no	—
36	D₈	—	2	2	no	—
37	C₂×C₈	—	3	1	no	—
38	C₂×C₈	—	6	2	no	—
39	C₂×C₈	C₂×D₃	1	1	yes	characteristic
40	C₄⋊C₄	—	3	2	no	—
41	C₂²⋊C₄	—	6	2	no	—
42	C₂²×C₃	—	4	2	no	—
43	C₂²×C₃	—	4	2	no	—
44	C₂²×C₃	—	4	2	no	—
45	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
46	C₂×D₃	—	2	1	no	—
47	C₂×D₃	—	2	1	no	—
48	C₂×D₃	D₈	1	2	yes	complemented
49	C₃×C₄	C₂×D₄	1	1	yes	characteristic
50	C₃×C₄	#41	1	1	yes	characteristic
51	Dic₃	—	2	1	no	—
52	Dic₃	—	4	2	no	—
53	C₂³	—	2	2	no	—
54	C₂³	—	3	1	no	—
55	D₄	—	2	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	12	4	no	—
58	C₂×C₄	—	3	1	no	—
59	C₂×C₄	—	6	1	no	—
60	C₂×C₄	—	6	1	no	—
61	C₂×C₄	—	6	2	no	—
62	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₈	—	6	1	no	—
64	C₈	#67	1	2	yes	complemented
65	C₂×C₃	—	8	2	no	—
66	C₂×C₃	#121	1	1	yes	characteristic
67	C₂×C₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₃	#135	1	1	yes	characteristic
69	D₃	—	2	2	no	—
70	C₂²	—	3	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	6	1	no	—
75	C₂²	—	6	1	no	—
76	C₂²	D₃×D₄	1	1	yes	characteristic
77	C₄	—	6	1	no	—
78	C₄	—	12	2	no	—
79	C₄	#237	1	1	yes	characteristic
80	C₄	C₂×#67	1	1	yes	characteristic
81	C₃	#466	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	6	2	no	—
83	C₂	—	8	2	no	—
84	C₂	D₃×D₈	1	1	yes	characteristic
85	C₂	#912	1	1	yes	characteristic
86	C₂	#938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×D₈	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	C₂×D₈	D₃
hypocenter	C₃	#466
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₈	C₂	semidirect
2. #938	C₂	semidirect
3. C₂×D₃×C₈	C₂	semidirect
4. #832	C₂	semidirect
5. #818	C₂	semidirect
16. C₂×D₈	D₃	semidirect
31. C₃×C₈	D₄	semidirect
48. C₂×D₃	D₈	semidirect
64. C₈	#67	semidirect
81. C₃	#466	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₈	2
#938	2
C₂×D₃×C₈	2
#832	2
#818	2
#466	3