Subgroups — #88623

Order 432.

79 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	Q₈×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
3	D₃×SL(2,𝔽₃)	—	3	1	no	maximal
4	C₂×(C₃⋊D₃)⋊C₃	—	4	1	no	maximal
5	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
6	C₄×C₃⋊D₃	—	3	1	no	—
7	#634	—	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	(C₃⋊D₃)⋊C₃	—	4	2	no	—
12	D₃×Q₈	—	3	1	no	—
13	D₃×Q₈	—	9	1	no	—
14	C₂×SL(2,𝔽₃)	—	9	1	no	—
15	C₂×C₃⋊D₃	A₄	1	1	yes	characteristic
16	C₂×C₃×D₃	—	12	1	no	—
17	C₃²×C₄	—	3	1	no	—
18	C₃⋊Dic₃	—	3	1	no	—
19	H(3,𝔽₃)	—	4	1	no	Sylow
20	C₃×Q₈	—	3	1	no	—
21	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
22	C₄×D₃	—	9	1	no	—
23	C₄×D₃	—	9	3	no	—
24	Dic₆	—	9	3	no	—
25	Dic₆	—	9	1	no	—
26	SL(2,𝔽₃)	—	3	1	no	—
27	SL(2,𝔽₃)	—	6	1	no	—
28	C₂×C₃²	—	4	1	no	—
29	C₂×C₃²	—	8	1	no	—
30	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
31	C₃⋊D₃	SL(2,𝔽₃)	1	2	yes	complemented
32	C₃×D₃	—	12	2	no	—
33	C₂×Q₈	—	9	1	no	Sylow
34	C₂²×C₃	—	36	1	no	—
35	C₂×D₃	—	3	1	no	—
36	C₂×D₃	—	9	1	no	—
37	C₃×C₄	—	3	3	no	—
38	C₃×C₄	—	3	1	no	—
39	Dic₃	—	9	3	no	—
40	Dic₃	—	9	1	no	—
41	C₃²	—	4	1	no	—
42	C₃²	—	8	1	no	—
43	C₃²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
44	Q₈	—	27	1	no	—
45	Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
46	C₂×C₄	—	27	1	no	—
47	C₂×C₃	—	3	1	no	—
48	C₂×C₃	—	12	1	no	—
49	C₂×C₃	—	24	1	no	—
50	C₂×C₃	—	36	2	no	—
51	C₂×C₃	D₃×A₄	1	1	yes	characteristic
52	D₃	—	3	2	no	—
53	D₃	—	9	2	no	—
54	C₂²	—	9	1	no	—
55	C₄	—	3	1	no	—
56	C₄	—	27	1	no	—
57	C₃	—	3	1	no	—
58	C₃	—	12	1	no	—
59	C₃	—	24	1	no	—
60	C₃	D₃×SL(2,𝔽₃)	1	1	yes	characteristic
61	C₂	—	9	2	no	—
62	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. Q₈×C₃⋊D₃	C₃	semidirect
5. C₃²×Q₈	C₂×C₃	semidirect
9. C₃×SL(2,𝔽₃)	D₃	semidirect
31. C₃⋊D₃	SL(2,𝔽₃)	semidirect
43. C₃²	C₂×SL(2,𝔽₃)	semidirect
45. Q₈	(C₃⋊D₃)⋊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
Q₈×C₃⋊D₃	3
D₃×SL(2,𝔽₃)	3
C₂×(C₃⋊D₃)⋊C₃	4