Subgroups — #4508

Order 192.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#812	C₂	1	1	yes	maximal, characteristic, complemented
4	#350	—	3	1	no	maximal, Sylow
5	#235	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
7	#208	C₄	1	1	yes	characteristic
8	ΓL(1,𝔽₂₅)	—	2	1	no	—
9	D₃×C₈	—	2	1	no	—
10	C₃×C₁₆	—	2	1	no	—
11	C₃⋊C₁₆	—	2	1	no	—
12	#131	—	3	1	no	—
13	M₃₂	—	3	1	no	—
14	C₂×C₁₆	D₃	1	1	yes	characteristic, complemented
15	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	#67	—	2	1	no	—
17	D₁₂	C₈	1	1	yes	characteristic
18	C₄×D₃	—	2	1	no	—
19	Dic₆	C₈	1	1	yes	characteristic
20	C₃×C₈	D₄	1	1	yes	characteristic
21	C₃×C₈	D₄	1	1	yes	characteristic
22	C₃⋊C₈	—	2	1	no	—
23	#41	—	3	1	no	—
24	M₁₆	—	3	1	no	—
25	M₁₆	—	6	1	no	—
26	C₂×C₈	—	6	1	no	—
27	C₂×C₈	C₂×D₃	1	1	yes	characteristic
28	C₁₆	—	2	1	no	—
29	C₁₆	—	6	1	no	—
30	C₂²×C₃	M₁₆	1	1	yes	characteristic
31	C₂×D₃	—	2	1	no	—
32	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₄	C₂×C₈	1	1	yes	characteristic
34	Dic₃	—	2	1	no	—
35	Q₈	—	3	1	no	—
36	D₄	—	3	1	no	—
37	D₄	—	6	1	no	—
38	C₂×C₄	—	6	1	no	—
39	C₂×C₄	C₄×D₃	1	1	yes	characteristic
40	C₈	—	6	1	no	—
41	C₈	D₁₂	1	1	yes	characteristic
42	C₈	#67	1	1	yes	characteristic
43	C₂×C₃	—	2	1	no	—
44	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
45	D₃	—	4	1	no	—
46	C₂²	—	6	1	no	—
47	C₂²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
48	C₄	—	6	1	no	—
49	C₄	D₃×C₈	1	1	yes	characteristic
50	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
51	C₃	#350	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	2	1	no	—
53	C₂	—	12	1	no	—
54	C₂	(C₂×D₃)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₂
commutator	C₃×C₄	C₂×C₈
Frattini	C₂×C₈	C₂×D₃
Fitting	C₂×C₃×C₁₆	C₂
socle	C₂×C₃	C₂²⋊C₈
prime radical	#235	C₄
cosocle	C₂×C₃×C₈	C₂²
hypercenter	C₂×C₁₆	D₃
hypocenter	C₃	#350
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₁₆	C₂	semidirect
3. #812	C₂	semidirect
14. C₂×C₁₆	D₃	semidirect
51. C₃	#350	semidirect

Maximal subgroups

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

subgroup	index
#901	2
C₂×C₃×C₁₆	2
#812	2
#350	3