Subgroups — #88748

Order 432.

220 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, complemented
2	C₄⋊#272	C₂	1	3	yes	maximal, complemented
3	C₄×#272	C₂	1	3	yes	maximal, complemented
4	C₃×#239	—	3	2	no	maximal
5	C₃×#239	—	3	2	no	maximal
6	C₂×#272	C₂²	1	3	yes	—
7	C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
8	#1084	C₂²	1	1	yes	characteristic
9	C₃²×Q₈	D₃	1	2	yes	complemented
10	C₃²×Q₈	D₃	1	2	yes	complemented
11	C₃×D₁₂	—	3	6	no	—
12	C₃×D₁₂	—	3	6	no	—
13	C₃×C₄×D₃	—	3	6	no	—
14	C₃×C₄×D₃	—	3	6	no	—
15	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
16	#272	—	2	3	no	—
17	C₃×#41	—	9	1	no	—
18	#239	—	3	2	no	—
19	#239	—	3	2	no	—
20	C₂×C₃×D₃	—	3	6	no	—
21	C₂×C₃×D₃	—	3	6	no	—
22	C₃²×C₄	C₂×D₃	1	6	yes	—
23	C₃²×C₄	C₂×D₃	1	6	yes	—
24	C₃×Dic₃	—	3	2	no	—
25	C₃×Dic₃	—	3	2	no	—
26	H(3,𝔽₃)	#41	1	1	yes	characteristic, complemented, Sylow
27	C₃×Q₈	—	3	2	no	—
28	C₃×Q₈	—	6	2	no	—
29	C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
30	C₃×D₄	—	9	3	no	—
31	C₂×C₃×C₄	—	9	3	no	—
32	D₁₂	—	3	6	no	—
33	D₁₂	—	3	6	no	—
34	C₄×D₃	—	3	6	no	—
35	C₄×D₃	—	3	6	no	—
36	C₂×C₃²	C₂²×D₃	1	2	yes	—
37	C₂×C₃²	C₂²×D₃	1	2	yes	—
38	C₃×D₃	—	6	6	no	—
39	C₃×D₃	—	6	6	no	—
40	#41	—	9	1	no	Sylow
41	C₂²×C₃	—	9	3	no	—
42	C₂×D₃	—	3	6	no	—
43	C₂×D₃	—	3	6	no	—
44	C₃×C₄	—	3	6	no	—
45	C₃×C₄	—	6	6	no	—
46	C₃×C₄	—	9	1	no	—
47	C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
48	Dic₃	—	3	2	no	—
49	Dic₃	—	3	2	no	—
50	C₃²	#239	1	2	yes	complemented
51	C₃²	#239	1	2	yes	complemented
52	Q₈	#272	1	1	yes	characteristic, complemented
53	D₄	—	9	3	no	—
54	C₂×C₄	—	9	3	no	—
55	C₂×C₃	—	3	2	no	—
56	C₂×C₃	—	6	2	no	—
57	C₂×C₃	—	18	3	no	—
58	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
59	D₃	—	6	6	no	—
60	D₃	—	6	6	no	—
61	C₂²	—	9	3	no	—
62	C₄	—	9	1	no	—
63	C₄	C₂×#272	1	3	yes	—
64	C₃	—	3	2	no	—
65	C₃	—	6	2	no	—
66	C₃	#3832	1	1	yes	characteristic
67	C₂	—	18	3	no	—
68	C₂	C₂²×#272	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Fitting	Q₈×H(3,𝔽₃)	C₂
hypercenter	C₃×Q₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#41
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×H(3,𝔽₃)	C₂	semidirect
2. C₄⋊#272	C₂	semidirect
3. C₄×#272	C₂	semidirect
9. C₃²×Q₈	D₃	semidirect
10. C₃²×Q₈	D₃	semidirect
26. H(3,𝔽₃)	#41	semidirect
50. C₃²	#239	semidirect
51. C₃²	#239	semidirect
52. Q₈	#272	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
C₄⋊#272	2
C₄×#272	2
C₃×#239	3