Subgroups — #4480

Order 192.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#334	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₈	—	2	1	no	—
6	C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₈	—	2	1	no	—
8	C₂×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₄⋊C₈	—	3	1	no	—
11	C₄⋊C₈	D₃	1	1	yes	characteristic
12	C₄×C₈	—	3	1	no	—
13	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₄	D₄	1	1	yes	characteristic
15	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
16	C₃×C₈	—	4	1	no	—
17	C₃⋊C₈	—	2	1	no	—
18	C₃⋊C₈	—	4	1	no	—
19	C₃⋊C₈	C₈	1	2	yes	complemented
20	C₂×C₈	—	2	1	no	—
21	C₂×C₈	—	3	1	no	—
22	C₂×C₈	—	3	1	no	—
23	C₂×C₈	—	6	1	no	—
24	C₄²	C₂×D₃	1	1	yes	characteristic
25	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
26	C₃×C₄	—	2	1	no	—
27	C₃×C₄	C₂×C₈	1	1	yes	characteristic
28	C₃×C₄	M₁₆	1	1	yes	characteristic
29	C₃×C₄	SD₁₆	1	1	yes	characteristic
30	C₃×C₄	SD₁₆	1	1	yes	characteristic
31	C₂×C₄	Dic₆	1	1	yes	characteristic
32	C₂×C₄	C₄×D₃	1	1	yes	characteristic
33	C₂×C₄	#67	1	1	yes	characteristic
34	C₈	—	3	2	no	—
35	C₈	—	4	1	no	—
36	C₈	—	6	1	no	—
37	C₈	—	12	1	no	—
38	C₂×C₃	C₄⋊C₈	1	1	yes	characteristic
39	C₂×C₃	#106	1	1	yes	characteristic
40	C₂×C₃	#108	1	1	yes	characteristic
41	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	D₃×C₈	1	1	yes	characteristic
44	C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
45	C₄	#214	1	1	yes	characteristic
46	C₄	#215	1	1	yes	characteristic
47	C₃	#334	1	1	yes	characteristic, complemented, Sylow
48	C₂	#808	1	1	yes	characteristic
49	C₂	Dic₃⋊C₈	1	1	yes	characteristic
50	C₂	#822	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#67
commutator	C₃×C₄	C₂×C₈
Frattini	C₄²	C₂×D₃
Fitting	C₃×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂²
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#334
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
19. C₃⋊C₈	C₈	semidirect
47. C₃	#334	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄⋊C₈	2
C₄⋊(C₃⋊C₈)	2
C₄×C₃⋊C₈	2
#334	3