Subgroups — #4557

Order 192.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#926	C₂	1	2	yes	maximal
3	H(3,ℤ/4)	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
5	C₃×M₁₆	—	2	2	no	—
6	C₂×C₃×C₈	C₄	1	2	yes	complemented
7	#217	—	2	2	no	—
8	C₄⋊Dic₃	—	2	2	no	—
9	C₄×Dic₃	C₄	1	4	yes	complemented
10	C₂×M₁₆	D₃	1	1	yes	characteristic
11	#117	—	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	C₃×C₈	—	2	2	no	—
19	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
20	M₁₆	—	2	2	no	—
21	C₂×C₈	Dic₃	1	2	yes	complemented
22	C₄⋊C₄	—	6	2	no	—
23	C₂²⋊C₄	—	6	2	no	—
24	C₄²	—	3	4	no	—
25	C₂²×C₃	—	4	1	no	—
26	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
27	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
28	C₃×C₄	C₄²	1	1	yes	characteristic
29	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
30	C₃×C₄	C₄⋊C₄	1	2	yes	—
31	Dic₃	—	4	4	no	—
32	C₂³	D₁₂	1	1	yes	characteristic
33	C₂×C₄	—	6	4	no	—
34	C₂×C₄	Dic₆	1	1	yes	characteristic
35	C₂×C₄	C₄×D₃	1	2	yes	—
36	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
37	C₂×C₄	#67	1	2	yes	—
38	C₈	—	2	2	no	—
39	C₂×C₃	—	2	2	no	—
40	C₂×C₃	—	2	1	no	—
41	C₂×C₃	#95	1	1	yes	characteristic
42	C₂²	—	4	1	no	—
43	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
44	C₂²	(C₂×D₃)⋊C₄	1	2	yes	—
45	C₄	—	12	4	no	—
46	C₄	C₄×Dic₃	1	1	yes	characteristic
47	C₄	Dic₃⋊C₄	1	2	yes	—
48	C₄	#217	1	1	yes	characteristic
49	C₃	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₃
commutator	C₃×C₄	C₄²
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂×C₃	#95
prime radical	C₂³×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	H(3,ℤ/4)
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
6. C₂×C₃×C₈	C₄	semidirect
9. C₄×Dic₃	C₄	semidirect
21. C₂×C₈	Dic₃	semidirect
49. C₃	H(3,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×M₁₆	2
#926	2
H(3,ℤ/4)	3