Subgroups — #5072

Order 192.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#213	C₂	1	2	yes	maximal, complemented
3	C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#832	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#459	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	—	2	1	no	—
8	C₂×C₃×D₄	—	2	1	no	—
9	C₂×C₃×D₄	C₂²	1	2	yes	—
10	C₂×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃	—	2	1	no	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	#213	—	2	4	no	—
14	(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×D₈	—	3	2	no	—
18	#127	D₃	1	1	yes	characteristic, complemented
19	C₄×D₄	—	3	1	no	—
20	C₄⋊C₈	—	3	1	no	—
21	#102	—	3	2	no	—
22	C₂³×C₃	—	2	2	no	—
23	C₂²×D₃	—	2	1	no	—
24	C₃×D₄	—	2	2	no	—
25	C₃×D₄	—	2	2	no	—
26	C₃×D₄	—	4	2	no	—
27	C₃×D₄	—	4	2	no	—
28	C₂×C₃×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂×Dic₃	—	2	1	no	—
32	D₁₂	—	2	1	no	—
33	D₁₂	D₄	1	2	yes	complemented
34	C₄×D₃	—	4	2	no	—
35	C₃⋊C₈	—	2	2	no	—
36	C₂×D₄	—	2	1	no	—
37	C₂×D₄	—	2	1	no	—
38	C₂×D₄	—	3	1	no	—
39	C₂×D₄	C₂×D₃	1	2	yes	—
40	C₂²×C₄	—	6	1	no	—
41	D₈	—	6	4	no	—
42	C₂×C₈	—	3	2	no	—
43	C₄⋊C₄	—	3	1	no	—
44	C₂²⋊C₄	—	6	1	no	—
45	C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃	—	4	2	no	—
47	C₂²×C₃	—	4	2	no	—
48	C₂²×C₃	—	4	2	no	—
49	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
50	C₂×D₃	—	2	2	no	—
51	C₂×D₃	—	4	1	no	—
52	C₂×D₃	—	4	1	no	—
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	D₈	1	2	yes	complemented
55	C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₃×C₄	#41	1	1	yes	characteristic
57	Dic₃	—	4	1	no	—
58	C₂³	—	2	2	no	—
59	C₂³	—	6	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	3	2	no	—
63	D₄	—	4	2	no	—
64	D₄	—	4	2	no	—
65	D₄	—	6	1	no	—
66	C₂×C₄	—	6	1	no	—
67	C₂×C₄	—	12	2	no	—
68	C₂×C₄	#67	1	1	yes	characteristic
69	C₂×C₄	#67	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₈	—	6	2	no	—
72	C₂×C₃	—	8	2	no	—
73	C₂×C₃	#121	1	1	yes	characteristic
74	C₂×C₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	D₃	—	4	2	no	—
77	C₂²	—	4	2	no	—
78	C₂²	—	4	2	no	—
79	C₂²	—	4	2	no	—
80	C₂²	—	6	2	no	—
81	C₂²	—	12	1	no	—
82	C₂²	—	12	1	no	—
83	C₂²	C₂×#67	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	12	1	no	—
86	C₄	#213	1	2	yes	complemented
87	C₄	D₃×D₄	1	1	yes	characteristic
88	C₄	#237	1	1	yes	characteristic
89	C₃	#459	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	8	2	no	—
91	C₂	—	12	2	no	—
92	C₂	C₂×#213	1	1	yes	characteristic
93	C₂	#932	1	1	yes	characteristic
94	C₂	#938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#127	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	#127	D₃
hypocenter	C₃	#459
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₃×#127	C₂	semidirect
2. C₂×#213	C₂	semidirect
3. C₄×D₁₂	C₂	semidirect
4. #832	C₂	semidirect
5. C₄⋊(C₃⋊C₈)	C₂	semidirect
18. #127	D₃	semidirect
33. D₁₂	D₄	semidirect
54. C₃×C₄	D₈	semidirect
86. C₄	#213	semidirect
89. C₃	#459	semidirect

Maximal subgroups

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

subgroup	index
C₃×#127	2
C₂×#213	2
C₄×D₁₂	2
#832	2
C₄⋊(C₃⋊C₈)	2
#459	3