Subgroups — #4509

Order 192.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#818	C₂	1	1	yes	maximal, characteristic, complemented
4	#357	—	3	1	no	maximal, Sylow
5	C₂×D₁₂	—	2	1	no	—
6	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₃	—	2	1	no	—
8	D₂₄	—	2	1	no	—
9	D₂₄	C₄	1	2	yes	complemented
10	C₃×C₁₆	—	2	1	no	—
11	C₂×D₈	—	3	1	no	—
12	C₂×C₁₆	D₃	1	1	yes	characteristic, complemented
13	#107	—	3	1	no	—
14	C₂²×D₃	—	4	1	no	—
15	C₂×C₃×C₄	D₄	1	1	yes	characteristic
16	C₂×Dic₃	—	4	1	no	—
17	D₁₂	—	2	2	no	—
18	D₁₂	—	4	1	no	—
19	C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃×C₈	D₄	1	1	yes	characteristic
21	C₂×D₄	—	6	1	no	—
22	D₈	—	3	2	no	—
23	D₈	—	6	1	no	—
24	C₂×C₈	C₂×D₃	1	1	yes	characteristic
25	C₄⋊C₄	—	6	1	no	—
26	C₁₆	—	2	1	no	—
27	C₂²×C₃	D₈	1	1	yes	characteristic
28	C₂×D₃	—	4	2	no	—
29	C₂×D₃	—	8	1	no	—
30	C₂×D₃	—	8	1	no	—
31	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₃×C₄	SD₁₆	1	1	yes	characteristic
33	Dic₃	—	8	1	no	—
34	C₂³	—	12	1	no	—
35	D₄	—	6	2	no	—
36	D₄	—	12	1	no	—
37	C₂×C₄	—	12	1	no	—
38	C₂×C₄	D₁₂	1	1	yes	characteristic
39	C₈	C₄×D₃	1	1	yes	characteristic
40	C₈	#67	1	1	yes	characteristic
41	C₂×C₃	#102	1	1	yes	characteristic
42	C₂×C₃	D₁₆	1	1	yes	characteristic
43	C₂×C₃	SD₃₂	1	1	yes	characteristic
44	D₃	—	8	2	no	—
45	C₂²	—	12	2	no	—
46	C₂²	—	24	1	no	—
47	C₂²	—	24	1	no	—
48	C₂²	D₂₄	1	1	yes	characteristic
49	C₄	—	24	1	no	—
50	C₄	#204	1	1	yes	characteristic
51	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
52	C₃	#357	1	1	yes	characteristic, complemented, Sylow
53	C₂	—	24	2	no	—
54	C₂	D₄₈	1	1	yes	characteristic
55	C₂	#800	1	1	yes	characteristic
56	C₂	#821	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₄
commutator	C₃×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₃
Fitting	C₂×C₃×C₁₆	C₂
socle	C₂²×C₃	D₈
prime radical	C₂×D₂₄	C₂
cosocle	C₂×C₃×C₈	C₂²
hypercenter	C₂×C₁₆	D₃
hypocenter	C₃	#357
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ 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. #818	C₂	semidirect
9. D₂₄	C₄	semidirect
12. C₂×C₁₆	D₃	semidirect
52. C₃	#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₄	2
C₂×C₃×C₁₆	2
#818	2
#357	3