Subgroups — #4551

Order 192.

114 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	2	yes	maximal
3	#340	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
5	C₂²×Dic₃	—	2	2	no	—
6	C₂×C₃×C₈	—	2	2	no	—
7	C₂×C₃×C₈	C₄	1	2	yes	complemented
8	C₄⋊Dic₃	—	2	2	no	—
9	C₄⋊Dic₃	C₄	1	4	yes	complemented
10	C₂²×C₈	D₃	1	1	yes	characteristic
11	C₂×C₄⋊C₄	—	3	2	no	—
12	C₂³×C₃	D₄	1	1	yes	characteristic
13	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
14	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₄	D₄	1	2	yes	—
16	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
17	C₂×Dic₃	—	2	4	no	—
18	C₂×Dic₃	—	4	2	no	—
19	C₂×Dic₃	—	4	2	no	—
20	C₃×C₈	—	2	2	no	—
21	C₂²×C₄	—	6	2	no	—
22	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
23	C₂×C₈	—	2	2	no	—
24	C₂×C₈	Dic₃	1	2	yes	complemented
25	C₄⋊C₄	—	3	4	no	—
26	C₄⋊C₄	—	6	2	no	—
27	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
28	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₃	D₈	1	1	yes	characteristic
30	C₂²×C₃	SD₁₆	1	2	yes	—
31	C₂²×C₃	Q₁₆	1	1	yes	characteristic
32	C₃×C₄	C₄²	1	1	yes	characteristic
33	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₃×C₄	C₄⋊C₄	1	2	yes	—
35	Dic₃	—	4	4	no	—
36	C₂³	D₁₂	1	1	yes	characteristic
37	C₂×C₄	—	6	4	no	—
38	C₂×C₄	—	12	2	no	—
39	C₂×C₄	—	12	2	no	—
40	C₂×C₄	Dic₆	1	1	yes	characteristic
41	C₂×C₄	C₄×D₃	1	2	yes	—
42	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
43	C₂×C₄	#67	1	2	yes	—
44	C₈	—	2	2	no	—
45	C₂×C₃	#95	1	1	yes	characteristic
46	C₂×C₃	#102	1	2	yes	—
47	C₂×C₃	#103	1	2	yes	—
48	C₂×C₃	#106	1	1	yes	characteristic
49	C₂×C₃	#107	1	1	yes	characteristic
50	C₂²	#204	1	2	yes	—
51	C₂²	D₂₄	1	1	yes	characteristic
52	C₂²	Dic₁₂	1	1	yes	characteristic
53	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
54	C₂²	(C₂×D₃)⋊C₄	1	2	yes	—
55	C₄	—	12	4	no	—
56	C₄	C₄×Dic₃	1	1	yes	characteristic
57	C₄	Dic₃⋊C₄	1	2	yes	—
58	C₄	#217	1	1	yes	characteristic
59	C₃	#340	1	1	yes	characteristic, complemented, Sylow
60	C₂	#816	1	2	yes	—
61	C₂	#817	1	1	yes	characteristic
62	C₂	#818	1	1	yes	characteristic
63	C₂	#821	1	2	yes	—
64	C₂	C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₂
commutator	C₃×C₄	C₄²
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂²×C₃×C₈	C₂
socle, prime radical	C₂³×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	#340
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
7. C₂×C₃×C₈	C₄	semidirect
9. C₄⋊Dic₃	C₄	semidirect
24. C₂×C₈	Dic₃	semidirect
59. C₃	#340	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
#340	3