Subgroups — #13166202

Order 1728.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#63764	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#63725	C₂	1	2	yes	maximal, complemented
3	#10593248	—	3	1	no	maximal
4	C₃×C₈×C₃⋊D₃	C₄	1	1	yes	characteristic
5	C₃×#3708	C₄	1	1	yes	characteristic
6	C₃×#3708	C₂²	1	1	yes	characteristic
7	#63764	D₃	1	1	yes	characteristic
8	#63725	—	3	2	no	—
9	C₃×C₄×C₃⋊D₃	C₈	1	1	yes	characteristic
10	C₃³×C₈	C₈	1	1	yes	characteristic
11	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃⋊C₃₂	—	9	1	no	maximal
13	C₈×C₃⋊D₃	Dic₃	1	1	yes	characteristic
14	C₃×D₃×C₈	—	12	1	no	—
15	#3708	Dic₃	1	1	yes	characteristic
16	#3708	C₂×D₃	1	1	yes	characteristic
17	C₂×C₃×C₃⋊D₃	C₁₆	1	1	yes	characteristic
18	C₃³×C₄	C₂×C₈	1	1	yes	characteristic
19	C₃×C₃⋊Dic₃	C₁₆	1	1	yes	characteristic
20	C₂×C₃×C₁₆	—	9	1	no	—
21	C₃⋊C₃₂	—	9	2	no	—
22	C₄×C₃⋊D₃	C₃⋊C₈	1	1	yes	characteristic
23	C₃×C₄×D₃	—	12	1	no	—
24	C₃²×C₈	—	4	1	no	—
25	C₃²×C₈	—	8	1	no	—
26	C₃²×C₈	C₃⋊C₈	1	1	yes	characteristic
27	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	1	yes	characteristic
28	C₃×C₃⋊C₈	—	12	1	no	—
29	C₂×C₃₂	—	27	1	no	Sylow
30	C₂×C₃³	C₂×C₁₆	1	1	yes	characteristic
31	C₃×C₃⋊D₃	C₃₂	1	2	yes	complemented
32	C₂×C₃×C₈	—	9	1	no	—
33	D₃×C₈	—	12	1	no	—
34	C₃×C₁₆	—	9	1	no	—
35	C₃×C₁₆	—	9	1	no	—
36	C₂×C₃⋊D₃	C₃⋊C₁₆	1	1	yes	characteristic
37	C₂×C₃×D₃	—	12	1	no	—
38	C₃²×C₄	—	4	1	no	—
39	C₃²×C₄	—	8	1	no	—
40	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
41	C₃⋊Dic₃	C₃⋊C₁₆	1	1	yes	characteristic
42	C₃×Dic₃	—	12	1	no	—
43	C₂×C₁₆	—	9	1	no	—
44	C₃₂	—	27	2	no	—
45	C₃³	C₂×C₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃×C₄	—	9	1	no	—
47	C₄×D₃	—	12	1	no	—
48	C₃×C₈	—	4	1	no	—
49	C₃×C₈	—	8	1	no	—
50	C₃×C₈	—	9	1	no	—
51	C₃×C₈	AGL(1,𝔽₉)	1	1	yes	characteristic
52	C₃⋊C₈	—	12	1	no	—
53	C₂×C₃²	—	4	1	no	—
54	C₂×C₃²	—	8	1	no	—
55	C₂×C₃²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
56	C₃⋊D₃	C₃⋊C₃₂	1	2	yes	complemented
57	C₃×D₃	—	12	2	no	—
58	C₂×C₈	—	9	1	no	—
59	C₁₆	—	9	1	no	—
60	C₁₆	—	9	1	no	—
61	C₂²×C₃	—	9	1	no	—
62	C₂×D₃	—	12	1	no	—
63	C₃×C₄	—	4	1	no	—
64	C₃×C₄	—	8	1	no	—
65	C₃×C₄	—	9	1	no	—
66	C₃×C₄	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
67	Dic₃	—	12	1	no	—
68	C₃²	—	4	1	no	—
69	C₃²	—	8	1	no	—
70	C₃²	C₂×C₃⋊C₃₂	1	1	yes	characteristic, complemented
71	C₂×C₄	—	9	1	no	—
72	C₈	—	9	1	no	—
73	C₈	C₃⋊AGL(1,𝔽₉)	1	1	yes	characteristic
74	C₂×C₃	—	4	1	no	—
75	C₂×C₃	—	8	1	no	—
76	C₂×C₃	—	9	2	no	—
77	C₂×C₃	#64213	1	1	yes	characteristic
78	D₃	—	12	2	no	—
79	C₂²	—	9	1	no	—
80	C₄	—	9	1	no	—
81	C₄	C₂×C₃⋊AGL(1,𝔽₉)	1	1	yes	characteristic
82	C₃	—	4	1	no	—
83	C₃	—	8	1	no	—
84	C₃	#10593248	1	1	yes	characteristic, complemented
85	C₂	—	9	2	no	—
86	C₂	#11724237	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	C₃⋊AGL(1,𝔽₉)
commutator, hypocenter	C₃³	C₂×C₃₂
Frattini	C₄	C₂×C₃⋊AGL(1,𝔽₉)
Fitting	C₃³×C₈	C₈
socle	C₂×C₃³	C₂×C₁₆
prime radical	C₂×C₃×C₃⋊D₃	C₁₆
cosocle	C₃×#3708	C₂²
LCS	G ▷ C₃³
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₃⋊#63725	C₂	semidirect
31. C₃×C₃⋊D₃	C₃₂	semidirect
45. C₃³	C₂×C₃₂	semidirect
56. C₃⋊D₃	C₃⋊C₃₂	semidirect
70. C₃²	C₂×C₃⋊C₃₂	semidirect
84. C₃	#10593248	semidirect

Maximal subgroups

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

subgroup	index
C₃×#63764	2
C₃⋊#63725	2
#10593248	3
C₂×C₃⋊C₃₂	9