Subgroups — #4550

Order 192.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#336	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂²×Dic₃	C₄	1	2	yes	—
7	C₂×C₃×C₈	—	2	1	no	—
8	C₂×C₃×C₈	—	2	1	no	—
9	C₂×C₃×C₈	C₄	1	2	yes	complemented
10	C₄×Dic₃	—	2	1	no	—
11	C₄×Dic₃	—	2	1	no	—
12	C₂×C₃⋊C₈	—	2	1	no	—
13	C₂×C₃⋊C₈	—	2	1	no	—
14	C₂×C₃⋊C₈	C₄	1	2	yes	complemented
15	C₂²×C₈	—	3	1	no	—
16	C₂²×C₈	D₃	1	1	yes	characteristic
17	C₂×C₄²	—	3	1	no	—
18	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
25	C₂×Dic₃	—	2	2	no	—
26	C₂×Dic₃	—	2	2	no	—
27	C₂×Dic₃	C₈	1	4	yes	complemented
28	C₃×C₈	—	2	2	no	—
29	C₃⋊C₈	—	2	2	no	—
30	C₂²×C₄	—	3	2	no	—
31	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	3	2	no	—
35	C₂×C₈	—	6	1	no	—
36	C₂×C₈	—	6	1	no	—
37	C₂×C₈	Dic₃	1	2	yes	complemented
38	C₄²	—	6	1	no	—
39	C₄²	—	6	1	no	—
40	C₂²×C₃	C₄²	1	1	yes	characteristic
41	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₃	C₂×C₈	1	2	yes	—
44	C₂²×C₃	M₁₆	1	2	yes	—
45	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
48	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
49	Dic₃	—	2	4	no	—
50	C₂³	C₄×D₃	1	1	yes	characteristic
51	C₂×C₄	—	3	4	no	—
52	C₂×C₄	—	6	2	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	Dic₆	1	1	yes	characteristic
55	C₂×C₄	C₄×D₃	1	1	yes	characteristic
56	C₂×C₄	D₁₂	1	1	yes	characteristic
57	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
58	C₂×C₄	#67	1	1	yes	characteristic
59	C₂×C₄	#67	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	6	2	no	—
62	C₂×C₃	#95	1	1	yes	characteristic
63	C₂×C₃	C₄×C₈	1	1	yes	characteristic
64	C₂×C₃	#97	1	1	yes	characteristic
65	C₂×C₃	C₂²⋊C₈	1	2	yes	—
66	C₂×C₃	C₄⋊C₈	1	2	yes	—
67	C₂²	D₃×C₈	1	2	yes	—
68	C₂²	ΓL(1,𝔽₂₅)	1	2	yes	—
69	C₂²	C₄×Dic₃	1	1	yes	characteristic
70	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
71	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
72	C₄	—	6	4	no	—
73	C₄	Dic₃⋊C₄	1	1	yes	characteristic
74	C₄	C₄⋊Dic₃	1	1	yes	characteristic
75	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
76	C₄	#217	1	1	yes	characteristic
77	C₃	#336	1	1	yes	characteristic, complemented, Sylow
78	C₂	C₈×Dic₃	1	1	yes	characteristic
79	C₂	Dic₃⋊C₈	1	2	yes	—
80	C₂	#815	1	1	yes	characteristic
81	C₂	(C₂×D₃)⋊C₈	1	2	yes	—
82	C₂	C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃
commutator	C₂×C₃	C₄×C₈
Fitting	C₂²×C₃×C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	#336
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
9. C₂×C₃×C₈	C₄	semidirect
14. C₂×C₃⋊C₈	C₄	semidirect
27. C₂×Dic₃	C₈	semidirect
37. C₂×C₈	Dic₃	semidirect
77. C₃	#336	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈	2
C₂×C₄×Dic₃	2
C₂²×C₃⋊C₈	2
#336	3