Subgroups — #4562

Order 192.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₈	C₂	1	1	yes	maximal, characteristic
2	#818	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#357	—	3	1	no	maximal, Sylow
5	C₂×C₃×D₄	—	2	1	no	—
6	C₃×D₈	—	2	1	no	—
7	C₃×D₈	C₄	1	2	yes	complemented
8	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₃	—	2	1	no	—
10	C₃⋊C₁₆	—	2	1	no	—
11	C₂×D₈	D₃	1	1	yes	characteristic
12	C₂×C₁₆	—	3	1	no	—
13	#107	—	3	1	no	—
14	C₂³×C₃	—	4	1	no	—
15	C₃×D₄	—	2	2	no	—
16	C₃×D₄	—	4	1	no	—
17	C₂×C₃×C₄	D₄	1	1	yes	characteristic
18	C₂×Dic₃	—	4	1	no	—
19	C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃×C₈	D₄	1	1	yes	characteristic
21	C₂×D₄	—	2	1	no	—
22	D₈	—	2	1	no	—
23	D₈	Dic₃	1	2	yes	complemented
24	C₂×C₈	C₂×D₃	1	1	yes	characteristic
25	C₄⋊C₄	—	6	1	no	—
26	C₁₆	—	6	1	no	—
27	C₂²×C₃	—	4	2	no	—
28	C₂²×C₃	—	8	1	no	—
29	C₂²×C₃	—	8	1	no	—
30	C₂²×C₃	D₈	1	1	yes	characteristic
31	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₃×C₄	SD₁₆	1	1	yes	characteristic
33	Dic₃	—	8	1	no	—
34	C₂³	—	4	1	no	—
35	D₄	—	2	2	no	—
36	D₄	—	4	1	no	—
37	C₂×C₄	—	12	1	no	—
38	C₂×C₄	#67	1	1	yes	characteristic
39	C₈	C₂×Dic₃	1	1	yes	characteristic
40	C₈	#67	1	1	yes	characteristic
41	C₂×C₃	—	8	2	no	—
42	C₂×C₃	#102	1	1	yes	characteristic
43	C₂×C₃	D₁₆	1	1	yes	characteristic
44	C₂×C₃	SD₃₂	1	1	yes	characteristic
45	C₂²	—	4	2	no	—
46	C₂²	—	8	1	no	—
47	C₂²	—	8	1	no	—
48	C₂²	#213	1	1	yes	characteristic
49	C₄	—	24	1	no	—
50	C₄	#214	1	1	yes	characteristic
51	C₄	#217	1	1	yes	characteristic
52	C₃	#357	1	1	yes	characteristic, complemented, Sylow
53	C₂	—	8	2	no	—
54	C₂	#826	1	1	yes	characteristic
55	C₂	#827	1	1	yes	characteristic
56	C₂	#832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#213
commutator	C₃×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₃
Fitting, prime radical	C₂×C₃×D₈	C₂
socle	C₂²×C₃	D₈
cosocle	C₂×C₃×C₈	C₂²
hypercenter	C₂×D₈	D₃
hypocenter	C₃	#357
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. #818	C₂	semidirect
3. C₂×C₃⋊C₁₆	C₂	semidirect
7. C₃×D₈	C₄	semidirect
23. D₈	Dic₃	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×C₃×D₈	2
#818	2
C₂×C₃⋊C₁₆	2
#357	3