Subgroups — #13135759

Order 1728.

253 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#97	C₂	1	1	yes	maximal, characteristic
2	#11720545	C₂	1	2	yes	maximal
3	C₃×#4466	—	3	4	no	maximal
4	C₂×C₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄²×H(3,𝔽₃)	C₄	1	1	yes	characteristic
7	#88386	C₄	1	4	yes	complemented
8	C₃²×#97	D₃	1	4	yes	—
9	C₃×#812	—	3	8	no	—
10	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
11	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₈×H(3,𝔽₃)	—	2	1	no	—
13	C₈×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
14	C₃×#347	—	9	1	no	—
15	#4466	—	3	4	no	—
16	C₂×C₃²×C₈	Dic₃	1	4	yes	—
17	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
18	C₃²×C₄²	Dic₃	1	4	yes	—
19	C₃×C₃⋊C₁₆	—	3	16	no	—
20	C₂²×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
21	C₄×H(3,𝔽₃)	—	4	1	no	—
22	C₄×H(3,𝔽₃)	C₄²	1	1	yes	characteristic
23	C₄×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
24	C₃×M₃₂	—	9	2	no	—
25	C₃×#97	—	3	4	no	—
26	C₃×#97	C₃⋊D₃	1	1	yes	characteristic
27	#812	—	3	8	no	—
28	C₂×C₃²×C₄	—	2	4	no	—
29	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
30	C₃²×C₈	—	2	4	no	—
31	C₃²×C₈	C₄×D₃	1	8	yes	—
32	#347	—	9	1	no	Sylow
33	C₂×H(3,𝔽₃)	—	2	1	no	—
34	C₂×H(3,𝔽₃)	#97	1	1	yes	characteristic
35	C₂×C₃×C₈	—	3	4	no	—
36	C₂×C₃×C₈	—	3	4	no	—
37	C₂×C₃×C₈	C₃⋊Dic₃	1	1	yes	characteristic
38	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
39	C₃×C₄²	—	3	4	no	—
40	C₃×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
41	C₃×C₁₆	—	9	4	no	—
42	C₃⋊C₁₆	—	3	16	no	—
43	C₂²×C₃²	#208	1	4	yes	—
44	C₃²×C₄	—	4	4	no	—
45	C₃²×C₄	#208	1	4	yes	—
46	C₃²×C₄	C₄×Dic₃	1	4	yes	—
47	M₃₂	—	9	2	no	—
48	#97	#272	1	1	yes	characteristic
49	H(3,𝔽₃)	#347	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃×C₄	—	2	1	no	—
51	C₂×C₃×C₄	—	3	4	no	—
52	C₂×C₃×C₄	—	6	4	no	—
53	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
54	C₃×C₈	—	2	1	no	—
55	C₃×C₈	—	3	8	no	—
56	C₃×C₈	—	6	4	no	—
57	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
58	C₂×C₃²	—	2	4	no	—
59	C₂×C₃²	#803	1	4	yes	—
60	C₂×C₈	#1084	1	1	yes	characteristic
61	C₂×C₈	C₂×#272	1	1	yes	characteristic
62	C₄²	#1084	1	1	yes	characteristic
63	C₁₆	—	9	4	no	—
64	C₂²×C₃	—	3	4	no	—
65	C₂²×C₃	#3748	1	1	yes	characteristic
66	C₃×C₄	—	3	4	no	—
67	C₃×C₄	—	3	4	no	—
68	C₃×C₄	—	4	1	no	—
69	C₃×C₄	—	12	4	no	—
70	C₃×C₄	#3748	1	1	yes	characteristic
71	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
72	C₃²	#4466	1	4	yes	complemented
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	C₂×#1084	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	C₄×#272	1	2	yes	—
77	C₂×C₃	—	2	1	no	—
78	C₂×C₃	—	3	4	no	—
79	C₂×C₃	—	6	4	no	—
80	C₂×C₃	#63630	1	1	yes	characteristic
81	C₂²	#88538	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	#88538	1	1	yes	characteristic
84	C₄	C₄×#1084	1	1	yes	characteristic
85	C₃	—	3	4	no	—
86	C₃	#10589332	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	#11720520	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₄×C₃⋊Dic₃
commutator	C₄×H(3,𝔽₃)	C₄²
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×#97	C₂
socle	C₂×C₃	#63630
prime radical	C₂²×H(3,𝔽₃)	M₁₆
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₃×#97	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#347
LCS	G ▷ C₄×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
7. #88386	C₄	semidirect
49. H(3,𝔽₃)	#347	semidirect
72. C₃²	#4466	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)×#97	2
#11720545	2
C₃×#4466	3