Subgroups — #5066

Order 192.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	#809	C₂	1	1	yes	maximal, characteristic
4	#809	C₂	1	1	yes	maximal, characteristic
5	#808	C₂	1	1	yes	maximal, characteristic, complemented
6	#807	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#478	—	3	1	no	maximal, Sylow
9	C₂×D₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃	—	2	1	no	—
11	C₃×C₄⋊C₄	—	2	1	no	—
12	C₃×C₄⋊C₄	—	2	1	no	—
13	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₄²	C₂²	1	1	yes	characteristic
16	(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#125	D₃	1	1	yes	characteristic, complemented
21	C₄×D₄	—	3	1	no	—
22	#107	—	3	1	no	—
23	#106	—	3	1	no	—
24	C₄⋊C₈	—	3	1	no	—
25	#102	—	3	1	no	—
26	#102	—	3	1	no	—
27	C₂²×D₃	—	2	1	no	—
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×Dic₃	—	2	1	no	—
34	D₁₂	—	2	1	no	—
35	D₁₂	Q₈	1	2	yes	—
36	C₄×D₃	—	4	2	no	—
37	C₃⋊C₈	—	2	1	no	—
38	C₃⋊C₈	—	2	1	no	—
39	C₂×D₄	—	3	1	no	—
40	C₂²×C₄	—	6	1	no	—
41	C₂×C₈	—	3	1	no	—
42	C₂×C₈	—	3	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	3	1	no	—
46	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
48	C₂²⋊C₄	—	6	1	no	—
49	C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
51	C₂×D₃	—	2	2	no	—
52	C₂×D₃	—	4	1	no	—
53	C₂×D₃	—	4	1	no	—
54	C₃×C₄	—	2	1	no	—
55	C₃×C₄	—	2	1	no	—
56	C₃×C₄	—	4	1	no	—
57	C₃×C₄	—	4	1	no	—
58	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
59	C₃×C₄	#41	1	1	yes	characteristic
60	Dic₃	—	4	1	no	—
61	C₂³	—	6	1	no	—
62	D₄	—	3	2	no	—
63	D₄	—	6	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	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	1	no	—
72	C₈	—	6	1	no	—
73	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₃	#135	1	1	yes	characteristic
75	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	D₃	—	4	2	no	—
77	C₂²	—	6	2	no	—
78	C₂²	—	12	1	no	—
79	C₂²	—	12	1	no	—
80	C₂²	C₂×#67	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	12	1	no	—
86	C₄	D₃×Q₈	1	1	yes	characteristic
87	C₄	#239	1	1	yes	characteristic
88	C₃	#478	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	12	2	no	—
90	C₂	#946	1	1	yes	characteristic
91	C₂	#949	1	1	yes	characteristic
92	C₂	#950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#125	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×D₁₂	C₂²
hypercenter	#125	D₃
hypocenter	C₃	#478
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃×#125	C₂	semidirect
5. #808	C₂	semidirect
6. #807	C₂	semidirect
7. C₄⋊(C₃⋊C₈)	C₂	semidirect
20. #125	D₃	semidirect
88. C₃	#478	semidirect

Maximal subgroups

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

subgroup	index
C₃×#125	2
C₄×D₁₂	2
#809	2
#808	2
#807	2
C₄⋊(C₃⋊C₈)	2
#478	3