Subgroups — #4833

Order 192.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
4	C₈×Dic₃	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#431	—	3	1	no	maximal, Sylow
7	C₂×C₄×D₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂²	1	2	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₄×Dic₃	C₄	1	1	yes	characteristic
13	C₄×Dic₃	C₄	1	1	yes	characteristic
14	C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄²	—	3	1	no	—
17	C₄⋊C₈	—	3	1	no	—
18	C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
19	C₂²⋊C₈	—	3	2	no	—
20	C₄×C₈	—	3	2	no	—
21	C₂²×D₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₄×D₃	—	2	1	no	—
29	C₄×D₃	—	2	1	no	—
30	C₄×D₃	—	2	2	no	—
31	C₄×D₃	C₈	1	4	yes	complemented
32	C₃×C₈	—	2	2	no	—
33	C₃⋊C₈	—	2	2	no	—
34	C₂²×C₄	—	3	1	no	—
35	C₂²×C₄	—	3	1	no	—
36	C₂²×C₄	—	3	1	no	—
37	C₂×C₈	—	3	2	no	—
38	C₂×C₈	C₂×D₃	1	2	yes	—
39	C₄²	—	3	1	no	—
40	C₄²	—	3	1	no	—
41	C₄²	—	3	1	no	—
42	C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×D₃	—	2	1	no	—
45	C₂×D₃	—	2	1	no	—
46	C₂×D₃	C₂×C₈	1	2	yes	—
47	C₃×C₄	—	2	1	no	—
48	C₃×C₄	C₂×C₈	1	2	yes	—
49	C₃×C₄	#41	1	1	yes	characteristic
50	C₃×C₄	#41	1	1	yes	characteristic
51	Dic₃	—	2	1	no	—
52	Dic₃	C₂×C₈	1	2	yes	—
53	Dic₃	M₁₆	1	2	yes	—
54	C₂³	—	3	1	no	—
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	3	1	no	—
57	C₂×C₄	—	3	4	no	—
58	C₂×C₄	—	3	1	no	—
59	C₂×C₄	—	6	2	no	—
60	C₂×C₄	—	6	1	no	—
61	C₂×C₄	—	6	1	no	—
62	C₂×C₄	C₄×D₃	1	1	yes	characteristic
63	C₂×C₄	C₄×D₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₈	—	6	2	no	—
67	C₂×C₃	#117	1	1	yes	characteristic
68	C₂×C₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
70	D₃	—	2	2	no	—
71	C₂²	—	3	2	no	—
72	C₂²	—	6	1	no	—
73	C₂²	—	6	1	no	—
74	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	3	2	no	—
77	C₄	—	3	2	no	—
78	C₄	—	6	1	no	—
79	C₄	D₃×C₈	1	2	yes	—
80	C₄	#237	1	1	yes	characteristic
81	C₄	#239	1	1	yes	characteristic
82	C₃	#431	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	6	2	no	—
84	C₂	#892	1	1	yes	characteristic
85	C₂	C₂×D₃×C₈	1	1	yes	characteristic
86	C₂	D₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₈
Fitting	C₃×C₄⋊C₈	C₂
socle	C₂²×C₃	C₂²×C₄
prime radical	C₂²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#431
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄⋊C₈	C₂	semidirect
4. C₈×Dic₃	C₂	semidirect
5. C₄⋊(C₃⋊C₈)	C₂	semidirect
18. C₄⋊C₈	D₃	semidirect
31. C₄×D₃	C₈	semidirect
82. C₃	#431	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₃	2
C₃×C₄⋊C₈	2
(C₂×D₃)⋊C₈	2
C₈×Dic₃	2
C₄⋊(C₃⋊C₈)	2
#431	3