Subgroups — #5160

Order 192.

108 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	#932	C₂	1	2	yes	maximal, complemented
3	#901	C₂	1	1	yes	maximal, characteristic, complemented
4	#833	C₂	1	2	yes	maximal, complemented
5	#819	C₂	1	1	yes	maximal, characteristic, complemented
6	#472	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	2	yes	—
8	#235	C₂²	1	1	yes	characteristic
9	C₃×D₈	—	2	1	no	—
10	C₃×D₈	—	2	1	no	—
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	#214	—	2	2	no	—
13	#213	—	2	2	no	—
14	#208	C₂²	1	2	yes	—
15	#208	C₂²	1	1	yes	characteristic
16	ΓL(1,𝔽₂₅)	—	2	1	no	—
17	D₃×C₈	—	2	1	no	—
18	AGL(1,ℤ/8)	—	3	2	no	—
19	C₂×D₈	D₃	1	1	yes	characteristic, complemented
20	#131	—	3	1	no	—
21	#108	—	3	1	no	—
22	#100	—	3	2	no	—
23	C₂³×C₃	—	2	2	no	—
24	C₃×D₄	—	2	2	no	—
25	C₃×D₄	—	2	2	no	—
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	#67	—	2	1	no	—
28	D₁₂	D₄	1	1	yes	characteristic
29	C₄×D₃	—	2	1	no	—
30	Dic₆	D₄	1	1	yes	characteristic
31	C₃×C₈	D₄	1	1	yes	characteristic
32	C₃×C₈	D₄	1	1	yes	characteristic
33	C₃⋊C₈	—	2	1	no	—
34	C₃⋊C₈	—	2	2	no	—
35	#41	—	3	1	no	—
36	C₂×D₄	C₂×D₃	1	2	yes	—
37	SD₁₆	—	6	2	no	—
38	D₈	—	2	1	no	—
39	D₈	—	2	1	no	—
40	D₈	—	6	2	no	—
41	M₁₆	—	3	1	no	—
42	M₁₆	—	3	2	no	—
43	M₁₆	—	6	1	no	—
44	C₂×C₈	—	6	1	no	—
45	C₂×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃	—	4	2	no	—
47	C₂²×C₃	—	8	2	no	—
48	C₂²×C₃	#41	1	1	yes	characteristic
49	C₂×D₃	—	2	1	no	—
50	C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	Dic₃	—	2	1	no	—
53	C₂³	—	2	2	no	—
54	Q₈	—	3	1	no	—
55	D₄	—	2	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	3	1	no	—
58	D₄	—	6	1	no	—
59	C₂×C₄	—	6	1	no	—
60	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₈	—	6	2	no	—
62	C₈	—	6	1	no	—
63	C₈	#67	1	1	yes	characteristic
64	C₈	#67	1	1	yes	characteristic
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	—	8	2	no	—
67	C₂×C₃	#121	1	1	yes	characteristic
68	D₃	—	4	1	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	6	1	no	—
71	C₂²	—	8	2	no	—
72	C₂²	#237	1	1	yes	characteristic
73	C₄	—	6	1	no	—
74	C₄	D₃×D₄	1	1	yes	characteristic
75	C₄	C₂×#67	1	1	yes	characteristic
76	C₃	#472	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	12	1	no	—
80	C₂	#938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#938
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×D₈	C₂
socle	C₂×C₃	#121
hypercenter	C₂×D₈	D₃
hypocenter	C₃	#472
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ 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. #932	C₂	semidirect
3. #901	C₂	semidirect
4. #833	C₂	semidirect
5. #819	C₂	semidirect
19. C₂×D₈	D₃	semidirect
76. C₃	#472	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
#932	2
#901	2
#833	2
#819	2
#472	3