Subgroups — #11722557

Order 864.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88683	C₂	1	3	yes	maximal, complemented
2	C₃×#994	C₃	1	1	yes	maximal, characteristic
3	C₉×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented
4	#63700	C₃	1	2	yes	maximal, complemented
5	Q₈×3₋¹⁺²	—	4	1	no	maximal
6	C₉⋊SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₃×#41⋊C₃	C₂×C₃	1	3	yes	—
8	C₉×#41	—	3	1	no	—
9	C₉×#41	C₂×C₃	1	3	yes	—
10	C₂×D₄×C₉	—	3	3	no	—
11	#41⋊C₉	C₂×C₃	1	6	yes	—
12	C₄×3₋¹⁺²	—	4	3	no	—
13	C₃×H(5,𝔽₂)	C₃²	1	1	yes	characteristic
14	#994	—	3	1	no	—
15	C₃²×Q₈	—	4	1	no	—
16	C₃×SL(2,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
17	C₂³×C₉	—	3	2	no	—
18	Q₈×C₉	—	4	2	no	—
19	Q₈×C₉	A₄	1	1	yes	characteristic
20	Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
21	D₄×C₉	—	3	3	no	—
22	D₄×C₉	—	3	3	no	—
23	C₂×C₄×C₉	—	3	3	no	—
24	Q₈⋊C₉	C₂²×C₃	1	2	yes	—
25	C₂×3₋¹⁺²	—	4	1	no	—
26	C₃×#41	—	3	1	no	—
27	C₃×#41	C₂×C₃²	1	3	yes	—
28	C₂×C₃×D₄	—	3	3	no	—
29	#41⋊C₃	—	3	3	no	—
30	C₃²×C₄	—	4	3	no	—
31	C₂²×C₉	—	3	3	no	—
32	C₂²×C₉	—	12	2	no	—
33	C₄×C₉	—	3	1	no	—
34	C₄×C₉	—	4	6	no	—
35	C₄×C₉	C₂×A₄	1	3	yes	—
36	H(5,𝔽₂)	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
37	3₋¹⁺²	—	4	1	no	Sylow
38	C₂³×C₃	—	3	2	no	—
39	C₃×Q₈	—	12	1	no	—
40	C₃×Q₈	C₃×A₄	1	1	yes	characteristic
41	C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
42	C₃×D₄	—	3	3	no	—
43	C₃×D₄	—	3	3	no	—
44	C₂×C₃×C₄	—	3	3	no	—
45	SL(2,𝔽₃)	—	3	1	no	—
46	C₂×C₃²	—	4	1	no	—
47	C₂×C₉	—	4	2	no	—
48	C₂×C₉	—	6	3	no	—
49	C₂×C₉	C₂²×A₄	1	1	yes	characteristic
50	#41	—	3	1	no	—
51	#41	C₂×3₋¹⁺²	1	3	yes	—
52	C₂×D₄	—	3	3	no	—
53	C₂²×C₃	—	3	3	no	—
54	C₂²×C₃	—	12	2	no	—
55	C₃×C₄	—	3	1	no	—
56	C₃×C₄	—	12	3	no	—
57	C₃×C₄	C₂×C₃×A₄	1	3	yes	—
58	C₃²	—	4	1	no	—
59	C₉	—	4	2	no	—
60	C₉	#994	1	1	yes	characteristic, complemented
61	C₂³	—	3	2	no	—
62	Q₈	C₉⋊A₄	1	1	yes	characteristic
63	Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
64	D₄	—	3	3	no	—
65	D₄	—	3	3	no	—
66	C₂×C₄	—	3	3	no	—
67	C₂×C₃	—	6	3	no	—
68	C₂×C₃	—	12	1	no	—
69	C₂×C₃	C₂²×C₃×A₄	1	1	yes	characteristic
70	C₂²	—	3	3	no	—
71	C₂²	—	12	2	no	—
72	C₄	—	3	1	no	—
73	C₄	C₂×C₉⋊A₄	1	3	yes	—
74	C₃	—	12	1	no	—
75	C₃	C₃×#994	1	1	yes	characteristic
76	C₂	—	6	3	no	—
77	C₂	C₂²×C₉⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×C₃×A₄
commutator, cosocle	C₃×Q₈	C₂²×C₃²
Fitting	C₉×H(5,𝔽₂)	C₃
prime radical	C₃×#994	C₃
hypercenter	Q₈×C₉	A₄
hypocenter	Q₈	C₂²×3₋¹⁺²
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. #88683	C₂	semidirect
3. C₉×H(5,𝔽₂)	C₃	semidirect
4. #63700	C₃	semidirect
6. C₉⋊SL(2,𝔽₃)	C₂²	semidirect
36. H(5,𝔽₂)	3₋¹⁺²	semidirect
60. C₉	#994	semidirect

Maximal subgroups

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

subgroup	index
#88683	2
C₃×#994	3
C₉×H(5,𝔽₂)	3
#63700	3
Q₈×3₋¹⁺²	4