Subgroups — #88550

Order 432.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#6229	C₂	1	1	yes	maximal, characteristic
3	#6180	C₂	1	1	yes	maximal, characteristic
4	C₃×#216	—	3	1	no	maximal
5	C₃×#216	—	3	3	no	maximal
6	C₄×H(3,𝔽₃)	—	2	1	no	—
7	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	#1084	—	2	1	no	—
9	C₃²×Q₈	D₃	1	1	yes	characteristic
10	C₃²×Q₈	D₃	1	3	yes	—
11	C₃×Dic₆	—	3	1	no	—
12	C₃×Dic₆	—	3	3	no	—
13	C₃×C₃⋊C₈	—	3	1	no	—
14	C₃×C₃⋊C₈	—	3	3	no	—
15	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
16	C₃×Q₁₆	—	9	1	no	—
17	#216	—	3	1	no	—
18	#216	—	3	3	no	—
19	C₃²×C₄	—	2	1	no	—
20	C₃²×C₄	—	2	3	no	—
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃²×C₄	C₂×D₃	1	3	yes	—
23	C₃×Dic₃	—	6	1	no	—
24	C₃×Dic₃	—	6	3	no	—
25	H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₃×Q₈	—	3	1	no	—
27	C₃×Q₈	—	6	3	no	—
28	C₃×Q₈	—	9	1	no	—
29	C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
30	Dic₆	—	3	1	no	—
31	Dic₆	—	3	3	no	—
32	C₃×C₈	—	9	1	no	—
33	C₃⋊C₈	—	3	1	no	—
34	C₃⋊C₈	—	3	3	no	—
35	C₂×C₃²	#67	1	1	yes	characteristic
36	C₂×C₃²	#67	1	3	yes	—
37	Q₁₆	—	9	1	no	Sylow
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	6	1	no	—
41	C₃×C₄	—	6	3	no	—
42	C₃×C₄	—	6	3	no	—
43	C₃×C₄	—	18	1	no	—
44	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
45	Dic₃	—	6	1	no	—
46	Dic₃	—	6	3	no	—
47	C₃²	#216	1	1	yes	characteristic, complemented
48	C₃²	#216	1	3	yes	complemented
49	Q₈	—	9	1	no	—
50	Q₈	#272	1	1	yes	characteristic
51	C₈	—	9	1	no	—
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	6	3	no	—
54	C₂×C₃	#638	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	—	18	1	no	—
57	C₄	C₂×#272	1	1	yes	characteristic
58	C₃	—	3	1	no	—
59	C₃	—	6	3	no	—
60	C₃	#3756	1	1	yes	characteristic
61	C₂	#6235	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#638
commutator, cosocle	C₄×H(3,𝔽₃)	C₂²
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	Q₈×H(3,𝔽₃)	C₂
prime radical	C₂×H(3,𝔽₃)	D₄
hypercenter	C₃×Q₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	Q₁₆
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

Complements

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

N	H ≅ G/N	type
25. H(3,𝔽₃)	Q₁₆	semidirect
47. C₃²	#216	semidirect
48. C₃²	#216	semidirect

Maximal subgroups

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

subgroup	index
Q₈×H(3,𝔽₃)	2
#6229	2
#6180	2
C₃×#216	3