Subgroups — #4552

Order 192.

90 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×#208	C₂	1	2	yes	maximal
3	#341	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
5	C₂×C₃×C₈	—	2	2	no	—
6	C₂×C₃×C₈	C₄	1	2	yes	—
7	#208	—	2	2	no	—
8	#208	C₄	1	4	yes	—
9	C₂×C₃⋊C₈	—	2	2	no	—
10	C₂×M₁₆	—	3	2	no	—
11	C₂²×C₈	D₃	1	1	yes	characteristic
12	C₂³×C₃	Q₈	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	1	yes	characteristic
16	C₂×C₃×C₄	D₄	1	2	yes	—
17	C₃×C₈	—	2	2	no	—
18	C₃⋊C₈	—	2	4	no	—
19	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
20	M₁₆	—	3	4	no	—
21	M₁₆	—	6	2	no	—
22	C₂×C₈	—	2	2	no	—
23	C₂×C₈	—	6	2	no	—
24	C₂×C₈	Dic₃	1	2	yes	—
25	C₂²×C₃	—	2	2	no	—
26	C₂²×C₃	C₄⋊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	2	yes	—
30	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂³	Dic₆	1	1	yes	characteristic
32	C₂×C₄	C₄×D₃	1	2	yes	—
33	C₂×C₄	D₁₂	1	1	yes	characteristic
34	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
35	C₂×C₄	#67	1	2	yes	—
36	C₈	—	2	2	no	—
37	C₈	—	6	4	no	—
38	C₂×C₃	—	2	2	no	—
39	C₂×C₃	#95	1	1	yes	characteristic
40	C₂×C₃	#108	1	2	yes	—
41	C₂²	—	2	2	no	—
42	C₂²	Dic₃⋊C₄	1	2	yes	—
43	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
44	C₄	C₄×Dic₃	1	1	yes	characteristic
45	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
46	C₄	#217	1	1	yes	characteristic
47	C₃	#341	1	1	yes	characteristic, complemented, Sylow
48	C₂	—	2	2	no	—
49	C₂	#819	1	2	yes	—
50	C₂	C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂
commutator	C₃×C₄	C₄²
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂²×C₃	C₄⋊C₄
prime radical	C₂³×C₃	Q₈
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	#341
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
47. C₃	#341	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×#208	2
#341	3