Subgroups — #5124

Order 192.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×#208	C₂	1	1	yes	maximal, characteristic
4	#822	C₂	1	4	yes	maximal, complemented
5	#430	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₃×M₁₆	—	2	1	no	—
8	C₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₈	—	2	1	no	—
10	#208	—	2	1	no	—
11	#208	C₄	1	2	yes	—
12	#208	C₄	1	2	yes	—
13	#208	C₂²	1	2	yes	—
14	C₂×C₃⋊C₈	—	2	1	no	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×M₁₆	—	3	1	no	—
17	C₂×M₁₆	—	3	1	no	—
18	C₂×M₁₆	D₃	1	1	yes	characteristic
19	#108	—	3	4	no	—
20	C₂³×C₃	Q₈	1	1	yes	characteristic
21	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₃×C₈	—	2	2	no	—
28	C₃⋊C₈	—	2	2	no	—
29	C₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
31	M₁₆	—	2	1	no	—
32	M₁₆	—	3	2	no	—
33	M₁₆	—	3	2	no	—
34	M₁₆	—	3	2	no	—
35	M₁₆	—	6	1	no	—
36	M₁₆	C₂×D₃	1	2	yes	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	3	2	no	—
39	C₂×C₈	—	6	1	no	—
40	C₂²×C₃	—	4	1	no	—
41	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
44	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
45	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
46	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂³	Dic₆	1	1	yes	characteristic
49	C₂×C₄	Dic₆	1	1	yes	characteristic
50	C₂×C₄	C₄×D₃	1	1	yes	characteristic
51	C₂×C₄	C₄×D₃	1	1	yes	characteristic
52	C₂×C₄	#67	1	1	yes	characteristic
53	C₂×C₄	#67	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
55	C₈	—	2	2	no	—
56	C₈	—	3	4	no	—
57	C₈	—	6	2	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
64	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
65	C₂²	C₂×Dic₆	1	1	yes	characteristic
66	C₄	Dic₃⋊C₄	1	1	yes	characteristic
67	C₄	Dic₃⋊C₄	1	1	yes	characteristic
68	C₄	C₂×C₄×D₃	1	1	yes	characteristic
69	C₄	C₂×#67	1	1	yes	characteristic
70	C₃	#430	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×Dic₃⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#67
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	Q₈
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #822	C₂	semidirect
70. C₃	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×M₁₆	2
C₂×#208	2
#822	2
#430	3