Subgroups — C₃²×#984 #11724447

Order 864.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×#226	C₂	1	2	yes	maximal, complemented
3	C₃×#984	C₃	1	4	yes	maximal, direct product factor
4	C₃²×#137	—	3	1	no	maximal
5	C₃²×Dic₆	—	4	1	no	maximal
6	C₃²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₃²×#41	D₃	1	1	yes	characteristic
8	C₂×C₃²×Q₈	—	3	1	no	—
9	C₃×#41⋊C₃	—	2	4	no	—
10	C₃×#41⋊C₃	C₂×C₃	1	4	yes	—
11	C₃×#226	C₂×C₃	1	8	yes	complemented
12	C₃²×Q₁₆	—	3	2	no	—
13	C₃²×SD₁₆	—	3	2	no	—
14	C₃²×M₁₆	—	3	1	no	—
15	C₃³×C₄	—	4	1	no	—
16	C₃²×Dic₃	—	4	2	no	—
17	#984	C₃²	1	1	yes	characteristic, direct product factor
18	C₃×#137	—	3	4	no	—
19	C₃²×Q₈	—	3	2	no	—
20	C₃²×Q₈	—	6	1	no	—
21	C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
22	C₃²×D₄	—	3	1	no	—
23	C₂×C₃²×C₄	—	3	1	no	—
24	C₂×C₃²×C₄	—	6	1	no	—
25	C₃×Dic₆	—	4	4	no	—
26	C₃×SL(2,𝔽₃)	—	2	4	no	—
27	C₃×SL(2,𝔽₃)	C₂²×C₃	1	4	yes	—
28	C₃²×C₈	—	3	2	no	—
29	C₂×C₃³	—	4	1	no	—
30	C₃×#41	C₃×D₃	1	4	yes	—
31	C₂×C₃×Q₈	—	3	4	no	—
32	#41⋊C₃	—	2	4	no	—
33	#41⋊C₃	C₂×C₃²	1	1	yes	characteristic
34	#226	C₂×C₃²	1	2	yes	complemented
35	C₃×Q₁₆	—	3	8	no	—
36	C₃×SD₁₆	—	3	8	no	—
37	C₃×M₁₆	—	3	4	no	—
38	C₂²×C₃²	—	3	1	no	—
39	C₃²×C₄	—	3	1	no	—
40	C₃²×C₄	—	4	4	no	—
41	C₃²×C₄	—	6	2	no	—
42	C₃²×C₄	—	8	4	no	—
43	C₃²×C₄	S₄	1	1	yes	characteristic
44	C₃×Dic₃	—	4	8	no	—
45	#137	—	3	1	no	Sylow
46	C₃³	—	4	1	no	Sylow
47	C₃×Q₈	—	3	8	no	—
48	C₃×Q₈	—	6	4	no	—
49	C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
50	C₃×D₄	—	3	4	no	—
51	C₂×C₃×C₄	—	3	4	no	—
52	C₂×C₃×C₄	—	6	4	no	—
53	Dic₆	—	4	1	no	—
54	SL(2,𝔽₃)	—	2	4	no	—
55	SL(2,𝔽₃)	C₂²×C₃²	1	1	yes	characteristic
56	C₃×C₈	—	3	8	no	—
57	C₂×C₃²	—	4	4	no	—
58	C₂×C₃²	—	6	1	no	—
59	C₂×C₃²	—	8	4	no	—
60	C₂×C₃²	C₂×S₄	1	1	yes	characteristic
61	#41	C₃²×D₃	1	1	yes	characteristic
62	C₂×Q₈	—	3	1	no	—
63	Q₁₆	—	3	2	no	—
64	SD₁₆	—	3	2	no	—
65	M₁₆	—	3	1	no	—
66	C₂²×C₃	—	3	4	no	—
67	C₃×C₄	—	3	4	no	—
68	C₃×C₄	—	4	1	no	—
69	C₃×C₄	—	6	8	no	—
70	C₃×C₄	—	8	4	no	—
71	C₃×C₄	C₃×S₄	1	4	yes	—
72	Dic₃	—	4	2	no	—
73	C₃²	—	4	4	no	—
74	C₃²	—	8	4	no	—
75	C₃²	#984	1	1	yes	characteristic, direct product factor
76	Q₈	—	3	2	no	—
77	Q₈	—	6	1	no	—
78	Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
79	D₄	—	3	1	no	—
80	C₂×C₄	—	3	1	no	—
81	C₂×C₄	—	6	1	no	—
82	C₈	—	3	2	no	—
83	C₂×C₃	—	4	1	no	—
84	C₂×C₃	—	6	4	no	—
85	C₂×C₃	—	8	4	no	—
86	C₂×C₃	C₂×C₃×S₄	1	4	yes	—
87	C₂²	—	3	1	no	—
88	C₄	—	3	1	no	—
89	C₄	—	6	2	no	—
90	C₄	C₃²×S₄	1	1	yes	characteristic
91	C₃	—	4	1	no	—
92	C₃	—	8	4	no	—
93	C₃	C₃×#984	1	4	yes	direct product factor
94	C₂	—	6	1	no	—
95	C₂	C₂×C₃²×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂²×C₃²
Frattini	C₂	C₂×C₃²×S₄
Fitting	C₃²×#41	D₃
prime radical	C₃²×#41⋊C₃	C₂
hypercenter	C₃²×C₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₃²×#226	C₂	semidirect
3. C₃×#984	C₃	direct (N × H)
11. C₃×#226	C₂×C₃	semidirect
17. #984	C₃²	direct (N × H)
34. #226	C₂×C₃²	semidirect
75. C₃²	#984	direct (N × H)
93. C₃	C₃×#984	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#41⋊C₃	2
C₃²×#226	2
C₃×#984	3
C₃²×#137	3
C₃²×Dic₆	4

Subgroups — C32×#984 #11724447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#984 #11724447