Subgroups — #4924

Order 192.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
4	#423	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₄	1	2	yes	—
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₃×C₄²	C₄	1	1	yes	characteristic
8	C₃×C₄²	C₄	1	1	yes	characteristic
9	C₃×C₄²	C₂²	1	2	yes	—
10	#208	—	2	4	no	—
11	C₂×C₃⋊C₈	C₂²	1	4	yes	—
12	C₂×M₁₆	—	3	2	no	—
13	C₂×C₄²	D₃	1	1	yes	characteristic
14	C₄⋊C₈	—	3	4	no	—
15	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₄	—	2	4	no	—
17	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
18	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₄	D₄	1	2	yes	—
21	C₂×C₃×C₄	Q₈	1	2	yes	—
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₃⋊C₈	—	2	4	no	—
24	C₂²×C₄	Dic₃	1	2	yes	—
25	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
26	M₁₆	—	6	4	no	—
27	C₂×C₈	—	3	4	no	—
28	C₄²	Dic₃	1	1	yes	characteristic
29	C₄²	Dic₃	1	1	yes	characteristic
30	C₄²	C₂×D₃	1	2	yes	—
31	C₂²×C₃	—	2	2	no	—
32	C₂²×C₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
34	C₃×C₄	—	2	2	no	—
35	C₃×C₄	C₄⋊C₄	1	2	yes	—
36	C₃×C₄	M₁₆	1	4	yes	complemented
37	C₃×C₄	C₂×D₄	1	1	yes	characteristic
38	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
39	C₂³	C₂×Dic₃	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	Dic₆	1	2	yes	—
42	C₂×C₄	D₁₂	1	2	yes	—
43	C₂×C₄	C₂×Dic₃	1	2	yes	—
44	C₂×C₄	C₂×Dic₃	1	2	yes	—
45	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
46	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
47	C₈	—	6	4	no	—
48	C₂×C₃	—	2	2	no	—
49	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃	C₂×M₁₆	1	2	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄⋊Dic₃	1	2	yes	—
53	C₂²	C₂²×Dic₃	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	#208	1	4	yes	complemented
56	C₄	C₄⋊Dic₃	1	2	yes	—
57	C₄	C₂×Dic₆	1	1	yes	characteristic
58	C₄	C₂×D₁₂	1	1	yes	characteristic
59	C₃	#423	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	2	2	no	—
61	C₂	C₂×#208	1	2	yes	—
62	C₂	C₂×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₂×C₃×C₄²	C₂
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×C₄²	D₃
hypocenter	C₃	#423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊C₈)	C₂	semidirect
36. C₃×C₄	M₁₆	semidirect
55. C₄	#208	semidirect
59. C₃	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²	2
C₂×#208	2
C₄⋊(C₃⋊C₈)	2
#423	3