Subgroups — #4685

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	Dic₃⋊C₈	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	2	no	—
18	C₂²⋊C₈	—	3	2	no	—
19	C₄×C₈	—	3	1	no	—
20	C₄×C₈	D₃	1	1	yes	characteristic, complemented
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	2	no	—
29	C₄×D₃	—	2	2	no	—
30	C₄×D₃	C₈	1	4	yes	complemented
31	C₃×C₈	—	2	2	no	—
32	C₃⋊C₈	—	2	2	no	—
33	C₂²×C₄	—	3	1	no	—
34	C₂²×C₄	—	3	1	no	—
35	C₂²×C₄	—	3	1	no	—
36	C₂×C₈	—	3	2	no	—
37	C₂×C₈	C₂×D₃	1	2	yes	—
38	C₄²	—	3	1	no	—
39	C₄²	—	3	1	no	—
40	C₄²	—	3	1	no	—
41	C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×D₃	—	2	1	no	—
44	C₂×D₃	—	2	1	no	—
45	C₂×D₃	C₂×C₈	1	2	yes	—
46	C₃×C₄	C₂×C₈	1	2	yes	—
47	C₃×C₄	M₁₆	1	2	yes	—
48	C₃×C₄	#41	1	2	yes	—
49	Dic₃	—	2	1	no	—
50	Dic₃	—	2	1	no	—
51	Dic₃	C₂×C₈	1	2	yes	—
52	C₂³	—	3	1	no	—
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	3	4	no	—
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	3	1	no	—
57	C₂×C₄	—	6	2	no	—
58	C₂×C₄	—	6	2	no	—
59	C₂×C₄	C₄×D₃	1	1	yes	characteristic
60	C₂×C₄	C₄×D₃	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	6	2	no	—
64	C₂×C₃	#117	1	1	yes	characteristic
65	C₂×C₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
67	D₃	—	2	2	no	—
68	C₂²	—	3	2	no	—
69	C₂²	—	6	1	no	—
70	C₂²	—	6	1	no	—
71	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
72	C₄	—	3	2	no	—
73	C₄	—	6	1	no	—
74	C₄	—	6	1	no	—
75	C₄	D₃×C₈	1	2	yes	—
76	C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
77	C₄	#235	1	2	yes	—
78	C₃	#431	1	1	yes	characteristic, complemented, Sylow
79	C₂	—	6	2	no	—
80	C₂	#872	1	1	yes	characteristic
81	C₂	C₂×D₃×C₈	1	1	yes	characteristic
82	C₂	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₃
commutator	C₂×C₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₃
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 ◁ C₄² ◁ 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. Dic₃⋊C₈	C₂	semidirect
5. C₄×C₃⋊C₈	C₂	semidirect
20. C₄×C₈	D₃	semidirect
30. C₄×D₃	C₈	semidirect
78. 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
Dic₃⋊C₈	2
C₄×C₃⋊C₈	2
#431	3