Subgroups — #88622

Order 432.

71 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	#3832	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(#41⋊C₃)	—	3	1	no	maximal
4	(C₃⋊Dic₃)⋊C₃	—	4	1	no	maximal
5	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
6	C₄⋊(C₃⋊D₃)	—	3	1	no	—
7	C₄×C₃⋊D₃	—	3	1	no	—
8	C₃×SL(2,𝔽₃)	—	2	1	no	—
9	C₃×SL(2,𝔽₃)	D₃	1	1	yes	characteristic, complemented
10	C₂×H(3,𝔽₃)	—	4	1	no	—
11	#239	—	3	1	no	—
12	#239	—	9	1	no	—
13	#41⋊C₃	—	9	1	no	—
14	C₂×C₃⋊D₃	—	3	1	no	—
15	C₃²×C₄	—	3	1	no	—
16	C₃⋊Dic₃	A₄	1	1	yes	characteristic
17	C₃×Dic₃	—	12	1	no	—
18	H(3,𝔽₃)	—	4	1	no	Sylow
19	C₃×Q₈	—	3	1	no	—
20	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
21	D₁₂	—	9	1	no	—
22	D₁₂	—	9	3	no	—
23	C₄×D₃	—	9	1	no	—
24	C₄×D₃	—	9	3	no	—
25	SL(2,𝔽₃)	—	3	1	no	—
26	SL(2,𝔽₃)	—	6	1	no	—
27	C₂×C₃²	—	4	1	no	—
28	C₂×C₃²	—	8	1	no	—
29	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
30	C₃⋊D₃	—	6	1	no	—
31	#41	—	9	1	no	Sylow
32	C₂×D₃	—	9	3	no	—
33	C₂×D₃	—	9	1	no	—
34	C₃×C₄	—	3	3	no	—
35	C₃×C₄	—	3	1	no	—
36	C₃×C₄	—	36	1	no	—
37	Dic₃	—	3	1	no	—
38	Dic₃	—	9	1	no	—
39	C₃²	—	4	1	no	—
40	C₃²	—	8	1	no	—
41	C₃²	#41⋊C₃	1	1	yes	characteristic, complemented
42	Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
43	D₄	—	27	1	no	—
44	C₂×C₄	—	27	1	no	—
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	12	1	no	—
47	C₂×C₃	—	24	1	no	—
48	C₂×C₃	D₃×A₄	1	1	yes	characteristic
49	D₃	—	18	3	no	—
50	D₃	—	18	1	no	—
51	C₂²	—	27	1	no	—
52	C₄	—	3	1	no	—
53	C₄	—	9	1	no	—
54	C₃	—	3	1	no	—
55	C₃	—	12	1	no	—
56	C₃	—	24	1	no	—
57	C₃	C₃⋊(#41⋊C₃)	1	1	yes	characteristic
58	C₂	—	54	1	no	—
59	C₂	(C₃⋊D₃)⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	(C₃⋊D₃)⋊A₄
commutator, Fitting, cosocle, hypocenter	C₃²×Q₈	C₂×C₃
Frattini, socle	C₂×C₃	D₃×A₄
LCS	G ▷ C₃²×Q₈
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. Q₈⋊H(3,𝔽₃)	C₂	semidirect
2. #3832	C₃	semidirect
9. C₃×SL(2,𝔽₃)	D₃	semidirect
41. C₃²	#41⋊C₃	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
#3832	3
C₃⋊(#41⋊C₃)	3
(C₃⋊Dic₃)⋊C₃	4