Subgroups — #13049595

Order 1296.

436 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊#622)	C₂	1	2	yes	maximal, complemented
3	#88993	—	3	4	no	maximal
4	C₂²×C₃⁴	C₄	1	1	yes	characteristic
5	C₃²×C₃⋊Dic₃	C₄	1	1	yes	characteristic
6	C₃²×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₃⋊Dic₃	D₃	1	4	yes	—
8	C₂×C₃²×Dic₃	—	6	2	no	—
9	C₃⋊#622	—	3	8	no	—
10	C₂×C₃⁴	—	2	1	no	—
11	C₂×C₃⁴	C₂×C₄	1	1	yes	characteristic
12	#3792	—	9	1	no	—
13	#3748	—	9	1	no	maximal
14	C₂²×C₃³	—	2	2	no	—
15	C₂²×C₃³	—	4	8	no	—
16	C₂²×C₃³	Dic₃	1	4	yes	—
17	C₃×C₃⋊Dic₃	Dic₃	1	4	yes	—
18	C₃×C₃⋊Dic₃	C₂×D₃	1	4	yes	—
19	C₃²×Dic₃	—	6	2	no	—
20	C₃²×Dic₃	—	6	2	no	—
21	C₃⁴	M₁₆	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃²×C₄	—	9	1	no	—
23	C₂×C₃⋊Dic₃	C₃⋊D₃	1	1	yes	characteristic
24	C₂×C₃×Dic₃	—	6	8	no	—
25	#622	—	9	2	no	—
26	C₃⋊(C₃⋊C₈)	—	9	2	no	—
27	C₂×C₃³	—	2	4	no	—
28	C₂×C₃³	—	2	4	no	—
29	C₂×C₃³	—	2	2	no	—
30	C₂×C₃³	—	4	16	no	—
31	C₂×C₃³	—	4	8	no	—
32	C₂×C₃³	C₂×Dic₃	1	4	yes	—
33	#208	—	27	4	no	—
34	C₂²×C₃²	—	2	8	no	—
35	C₂²×C₃²	—	4	8	no	—
36	C₂²×C₃²	—	4	12	no	—
37	C₂²×C₃²	—	4	8	no	—
38	C₂²×C₃²	C₃⋊Dic₃	1	1	yes	characteristic
39	C₂²×C₃²	#157	1	1	yes	characteristic
40	C₃²×C₄	—	9	1	no	—
41	C₃²×C₄	—	9	1	no	—
42	C₃⋊Dic₃	C₃⋊Dic₃	1	1	yes	characteristic
43	C₃⋊Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
44	C₃×Dic₃	—	6	8	no	—
45	C₃×Dic₃	—	6	8	no	—
46	C₃³	—	2	2	no	—
47	C₃³	—	4	8	no	—
48	C₃³	#208	1	4	yes	complemented
49	C₂×C₃×C₄	—	9	4	no	—
50	C₂×Dic₃	—	6	2	no	—
51	C₃⋊C₈	—	27	8	no	—
52	C₂×C₃²	—	2	1	no	—
53	C₂×C₃²	—	2	1	no	—
54	C₂×C₃²	—	2	16	no	—
55	C₂×C₃²	—	2	8	no	—
56	C₂×C₃²	—	4	16	no	—
57	C₂×C₃²	—	4	16	no	—
58	C₂×C₃²	—	4	8	no	—
59	C₂×C₃²	—	4	24	no	—
60	C₂×C₃²	—	4	12	no	—
61	C₂×C₃²	—	4	8	no	—
62	C₂×C₃²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
63	C₂×C₃²	C₂×#157	1	1	yes	characteristic
64	M₁₆	—	81	1	no	Sylow
65	C₂²×C₃	—	2	2	no	—
66	C₂²×C₃	—	4	8	no	—
67	C₂²×C₃	C₃⋊#157	1	4	yes	—
68	C₃×C₄	—	9	4	no	—
69	C₃×C₄	—	9	4	no	—
70	Dic₃	—	6	2	no	—
71	Dic₃	—	6	2	no	—
72	C₃²	—	2	8	no	—
73	C₃²	—	4	8	no	—
74	C₃²	—	4	12	no	—
75	C₃²	—	4	8	no	—
76	C₃²	#3748	1	1	yes	characteristic, complemented
77	C₃²	#3792	1	1	yes	characteristic, complemented
78	C₂×C₄	—	9	1	no	—
79	C₈	—	81	2	no	—
80	C₂×C₃	—	2	4	no	—
81	C₂×C₃	—	2	4	no	—
82	C₂×C₃	—	2	2	no	—
83	C₂×C₃	—	4	16	no	—
84	C₂×C₃	—	4	8	no	—
85	C₂×C₃	C₂×C₃⋊#157	1	4	yes	—
86	C₂²	C₃⋊(C₃⋊#157)	1	1	yes	characteristic
87	C₄	—	9	1	no	—
88	C₄	—	9	1	no	—
89	C₃	—	2	2	no	—
90	C₃	—	4	8	no	—
91	C₃	#88993	1	4	yes	complemented
92	C₂	—	2	1	no	—
93	C₂	C₂×C₃⋊(C₃⋊#157)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊(C₃⋊#157)
commutator, socle	C₂×C₃⁴	C₂×C₄
Fitting, prime radical	C₂²×C₃⁴	C₄
cosocle	C₃²×C₃⋊Dic₃	C₂²
hypercenter	C₂²	C₃⋊(C₃⋊#157)
hypocenter	C₃⁴	M₁₆
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊#622)	C₂	semidirect
21. C₃⁴	M₁₆	semidirect
48. C₃³	#208	semidirect
76. C₃²	#3748	semidirect
77. C₃²	#3792	semidirect
91. C₃	#88993	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₃⋊Dic₃	2
C₃⋊(C₃⋊#622)	2
#88993	3
#3748	9