Subgroups — #4549

Order 192.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₁₆	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
5	C₂×C₃×C₈	—	2	2	no	—
6	C₂×C₃×C₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₃⋊C₁₆	—	2	2	no	—
9	C₂²×C₈	D₃	1	1	yes	characteristic
10	C₂×C₁₆	—	3	2	no	—
11	C₂³×C₃	C₈	1	1	yes	characteristic
12	C₂×C₃×C₄	—	2	1	no	—
13	C₂×C₃×C₄	—	2	1	no	—
14	C₂×C₃×C₄	C₈	1	1	yes	characteristic
15	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	C₃×C₈	—	2	1	no	—
17	C₃×C₈	D₄	1	2	yes	—
18	C₂²×C₄	Dic₃	1	1	yes	characteristic
19	C₂×C₈	—	2	2	no	—
20	C₂×C₈	Dic₃	1	1	yes	characteristic
21	C₂×C₈	C₂×D₃	1	1	yes	characteristic
22	C₁₆	—	6	2	no	—
23	C₂²×C₃	—	2	1	no	—
24	C₂²×C₃	—	2	1	no	—
25	C₂²×C₃	C₁₆	1	2	yes	complemented
26	C₂²×C₃	C₂×C₈	1	1	yes	characteristic
27	C₃×C₄	—	2	1	no	—
28	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₃×C₄	M₁₆	1	1	yes	characteristic
30	C₂³	C₃⋊C₈	1	1	yes	characteristic
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	C₃⋊C₈	1	1	yes	characteristic
34	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
35	C₈	—	2	1	no	—
36	C₈	#67	1	2	yes	—
37	C₂×C₃	—	2	2	no	—
38	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₃	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₃	M₃₂	1	1	yes	characteristic
41	C₂²	—	2	1	no	—
42	C₂²	—	2	1	no	—
43	C₂²	C₃⋊C₁₆	1	2	yes	complemented
44	C₂²	C₂×C₃⋊C₈	1	1	yes	characteristic
45	C₄	—	2	1	no	—
46	C₄	#208	1	1	yes	characteristic
47	C₄	#217	1	1	yes	characteristic
48	C₃	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	2	2	no	—
50	C₂	C₂×C₃⋊C₁₆	1	1	yes	characteristic
51	C₂	#812	1	1	yes	characteristic
52	C₂	#830	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₁₆
Fitting	C₂²×C₃×C₈	C₂
socle	C₂²×C₃	C₂×C₈
prime radical	C₂³×C₃	C₈
cosocle	C₂×C₃×C₈	C₂²
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	C₂²⋊C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊C₁₆	C₂	semidirect
25. C₂²×C₃	C₁₆	semidirect
43. C₂²	C₃⋊C₁₆	semidirect
48. C₃	C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈	2
C₂×C₃⋊C₁₆	2
C₂²⋊C₁₆	3