Subgroups — #13135915

Order 1728.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#125	C₂	1	1	yes	maximal, characteristic
2	#11720520	C₂	1	2	yes	maximal
3	C₃×#4543	—	3	4	no	maximal
4	C₄⋊C₄×H(3,𝔽₃)	—	2	2	no	—
5	C₄⋊C₄×H(3,𝔽₃)	C₄	1	2	yes	—
6	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×#6180	—	2	2	no	—
8	C₃²×#125	D₃	1	4	yes	—
9	C₃×#803	—	3	8	no	—
10	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
11	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
13	#6180	—	2	4	no	—
14	C₃×#330	—	9	1	no	—
15	#4543	—	3	4	no	—
16	C₃²×C₄⋊C₄	—	2	8	no	—
17	C₃²×C₄⋊C₄	Dic₃	1	8	yes	—
18	C₃²×C₄²	C₂×D₃	1	4	yes	—
19	C₂×C₃×C₃⋊C₈	—	6	8	no	—
20	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×H(3,𝔽₃)	—	2	1	no	—
22	C₄×H(3,𝔽₃)	—	2	2	no	—
23	C₄×H(3,𝔽₃)	—	4	2	no	—
24	C₃×#125	—	3	4	no	—
25	C₃×#125	C₃⋊D₃	1	1	yes	characteristic
26	C₃×#97	—	9	2	no	—
27	#803	—	3	8	no	—
28	C₂×C₃²×C₄	—	2	8	no	—
29	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
30	C₂×C₃²×C₄	#67	1	8	yes	—
31	C₃×C₃⋊C₈	—	6	16	no	—
32	#330	—	9	1	no	Sylow
33	C₂×H(3,𝔽₃)	#101	1	1	yes	characteristic
34	C₂×H(3,𝔽₃)	#104	1	2	yes	—
35	C₂×C₃×C₈	—	18	2	no	—
36	C₃×C₄⋊C₄	—	2	2	no	—
37	C₃×C₄⋊C₄	—	3	8	no	—
38	C₃×C₄⋊C₄	—	6	8	no	—
39	C₃×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
40	C₃×C₄²	—	3	4	no	—
41	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
42	C₂×C₃⋊C₈	—	6	8	no	—
43	C₂²×C₃²	#217	1	4	yes	—
44	C₃²×C₄	—	2	4	no	—
45	C₃²×C₄	—	2	8	no	—
46	C₃²×C₄	—	4	8	no	—
47	#125	#272	1	1	yes	characteristic
48	#97	—	9	2	no	—
49	H(3,𝔽₃)	#330	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	—	3	4	no	—
52	C₂×C₃×C₄	—	3	8	no	—
53	C₂×C₃×C₄	—	6	8	no	—
54	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
55	C₂×C₃×C₄	#638	1	2	yes	—
56	C₃×C₈	—	18	4	no	—
57	C₃⋊C₈	—	6	16	no	—
58	C₂×C₃²	#836	1	4	yes	—
59	C₂×C₃²	#837	1	8	yes	—
60	C₂×C₈	—	18	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	#1084	1	2	yes	—
63	C₄²	C₂×#272	1	1	yes	characteristic
64	C₂²×C₃	—	3	4	no	—
65	C₂²×C₃	#3757	1	1	yes	characteristic
66	C₃×C₄	—	2	1	no	—
67	C₃×C₄	—	2	2	no	—
68	C₃×C₄	—	4	2	no	—
69	C₃×C₄	—	6	8	no	—
70	C₃×C₄	—	6	4	no	—
71	C₃×C₄	—	12	8	no	—
72	C₃²	#4543	1	4	yes	complemented
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	C₂×#1084	1	1	yes	characteristic
75	C₂×C₄	#6235	1	2	yes	—
76	C₈	—	18	4	no	—
77	C₂×C₃	—	3	4	no	—
78	C₂×C₃	—	3	8	no	—
79	C₂×C₃	#63663	1	1	yes	characteristic
80	C₂×C₃	#63664	1	2	yes	—
81	C₂²	#88552	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	2	no	—
84	C₄	—	4	2	no	—
85	C₃	—	3	4	no	—
86	C₃	#10589409	1	1	yes	characteristic
87	C₂	#11720584	1	1	yes	characteristic
88	C₂	#11720586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	#3757
commutator	C₂×C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×#125	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂²⋊C₄
cosocle	C₄²×H(3,𝔽₃)	C₂²
hypercenter	C₃×#125	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#330
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
49. H(3,𝔽₃)	#330	semidirect
72. C₃²	#4543	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)×#125	2
#11720520	2
C₃×#4543	3