Subgroups — #13047021

Order 1296.

145 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10619584	C₂	1	1	yes	maximal, characteristic
2	C₃×#3756	—	3	1	no	maximal
3	C₃×#3688	—	3	3	no	maximal
4	#66328	—	4	1	no	maximal
5	C₃³×Q₈	D₃	1	1	yes	characteristic
6	C₃×#634	—	3	1	no	—
7	C₃×C₃⋊(C₃⋊C₈)	—	3	1	no	—
8	C₃×Q₈⋊C₉	D₃	1	3	yes	—
9	C₂×C₃²⋊C₉	—	4	1	no	—
10	#3756	—	3	1	no	—
11	C₃×#216	—	9	3	no	—
12	C₃×#216	—	9	1	no	—
13	#3688	—	3	3	no	—
14	C₃³×C₄	—	3	1	no	—
15	C₃×C₃⋊Dic₃	—	6	1	no	—
16	C₃×Dic₉	—	12	3	no	—
17	C₃²⋊C₉	—	4	1	no	Sylow
18	C₃²×Q₈	—	3	1	no	—
19	C₃²×Q₈	—	6	1	no	—
20	C₃²×Q₈	D₉	1	3	yes	—
21	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
22	#634	—	3	1	no	—
23	C₃×Dic₆	—	9	3	no	—
24	C₃×Dic₆	—	9	1	no	—
25	C₃⋊(C₃⋊C₈)	—	3	1	no	—
26	C₃×C₃⋊C₈	—	9	3	no	—
27	C₃×C₃⋊C₈	—	9	1	no	—
28	Q₈⋊C₉	—	3	3	no	—
29	C₂×C₃³	S₄	1	1	yes	characteristic
30	C₂×C₃×C₉	—	4	3	no	—
31	C₃×Q₁₆	—	27	1	no	—
32	#216	—	9	3	no	—
33	#216	—	9	1	no	—
34	C₃²×C₄	—	3	1	no	—
35	C₃²×C₄	—	3	3	no	—
36	C₃²×C₄	—	3	1	no	—
37	C₃²×C₄	—	6	1	no	—
38	C₃²×C₄	—	6	3	no	—
39	C₃⋊Dic₃	—	6	1	no	—
40	C₃×Dic₃	—	18	3	no	—
41	C₃×Dic₃	—	18	1	no	—
42	Dic₉	—	12	3	no	—
43	C₃³	#226	1	1	yes	characteristic
44	C₃×C₉	—	4	3	no	—
45	C₃×Q₈	—	2	1	no	—
46	C₃×Q₈	—	3	3	no	—
47	C₃×Q₈	—	27	1	no	—
48	C₃×Q₈	C₃⋊D₉	1	1	yes	characteristic
49	C₃×Q₈	#272	1	1	yes	characteristic
50	Dic₆	—	9	3	no	—
51	Dic₆	—	9	1	no	—
52	C₃×C₈	—	27	1	no	—
53	C₃⋊C₈	—	9	3	no	—
54	C₃⋊C₈	—	9	1	no	—
55	C₂×C₃²	—	3	1	no	—
56	C₂×C₃²	—	6	1	no	—
57	C₂×C₃²	(C₂²⋊C₉)⋊C₂	1	3	yes	—
58	C₂×C₃²	C₃⋊S₄	1	1	yes	characteristic
59	C₂×C₉	—	12	3	no	—
60	Q₁₆	—	27	1	no	Sylow
61	C₃×C₄	—	3	3	no	—
62	C₃×C₄	—	3	1	no	—
63	C₃×C₄	—	3	1	no	—
64	C₃×C₄	—	6	1	no	—
65	C₃×C₄	—	6	3	no	—
66	C₃×C₄	—	54	1	no	—
67	Dic₃	—	18	3	no	—
68	Dic₃	—	18	1	no	—
69	C₃²	—	3	1	no	—
70	C₃²	—	6	1	no	—
71	C₃²	#3688	1	3	yes	complemented
72	C₃²	C₃⋊#226	1	1	yes	characteristic
73	C₉	—	12	3	no	—
74	Q₈	—	27	1	no	—
75	Q₈	#4181	1	1	yes	characteristic
76	C₈	—	27	1	no	—
77	C₂×C₃	—	2	1	no	—
78	C₂×C₃	—	3	3	no	—
79	C₂×C₃	C₃⋊((C₂²⋊C₉)⋊C₂)	1	1	yes	characteristic
80	C₂×C₃	#6258	1	1	yes	characteristic
81	C₄	—	3	1	no	—
82	C₄	—	54	1	no	—
83	C₃	—	2	1	no	—
84	C₃	—	3	3	no	—
85	C₃	C₃⋊#3688	1	1	yes	characteristic
86	C₃	#88610	1	1	yes	characteristic
87	C₂	#10619761	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊((C₂²⋊C₉)⋊C₂)
commutator, cosocle, hypocenter	#10619584	C₂
Frattini, socle	C₂×C₃²	C₃⋊S₄
Fitting	C₃³×Q₈	D₃
prime radical	C₂×C₃³	S₄
LCS	G ▷ #10619584
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
71. C₃²	#3688	semidirect

Maximal subgroups

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

subgroup	index
#10619584	2
C₃×#3756	3
C₃×#3688	3
#66328	4