Subgroups — #13136909

Order 1728.

169 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88410	C₂	1	2	yes	maximal
2	C₈×#1084	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	C₄×#1084	C₄	1	1	yes	characteristic
5	C₂×#6180	C₂²	1	1	yes	characteristic
6	#88410	C₄	1	4	yes	complemented
7	C₃×C₈×Dic₃	—	6	2	no	—
8	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
9	C₂×#1084	C₈	1	2	yes	—
10	C₈×H(3,𝔽₃)	C₈	1	2	yes	—
11	#6180	C₂×C₄	1	2	yes	—
12	C₃×#346	—	9	1	no	maximal
13	C₂×C₃²×C₈	—	2	2	no	—
14	C₃×C₄×Dic₃	—	6	2	no	—
15	C₂×C₃×C₃⋊C₈	—	6	2	no	—
16	C₂²×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
17	C₄×H(3,𝔽₃)	C₄²	1	1	yes	characteristic
18	C₄×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
19	#1084	—	2	2	no	—
20	C₂×C₃×C₁₆	—	9	2	no	—
21	C₃×C₄×C₈	—	9	1	no	—
22	C₈×Dic₃	—	6	2	no	—
23	C₂×C₃²×C₄	—	2	2	no	—
24	C₂×C₃×Dic₃	—	6	4	no	—
25	C₃²×C₈	—	2	4	no	—
26	C₃×C₃⋊C₈	—	6	4	no	—
27	#346	—	9	1	no	Sylow
28	C₂×H(3,𝔽₃)	C₄×C₈	1	1	yes	characteristic
29	C₂×H(3,𝔽₃)	M₃₂	1	2	yes	—
30	C₂×C₃×C₈	—	6	2	no	—
31	C₂×C₃×C₈	—	9	1	no	—
32	C₂×C₃×C₈	#157	1	1	yes	characteristic
33	C₃×C₄²	—	9	1	no	—
34	C₄×Dic₃	—	6	2	no	—
35	C₂×C₃⋊C₈	—	6	2	no	—
36	C₃×C₁₆	—	9	4	no	—
37	C₂²×C₃²	—	2	2	no	—
38	C₃²×C₄	—	2	2	no	—
39	C₃²×C₄	—	2	2	no	—
40	C₃×Dic₃	—	6	8	no	—
41	C₂×C₁₆	—	9	2	no	—
42	C₄×C₈	—	9	1	no	—
43	H(3,𝔽₃)	#346	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₄	—	6	2	no	—
45	C₂×C₃×C₄	—	9	2	no	—
46	C₂×C₃×C₄	C₂×#157	1	1	yes	characteristic
47	C₂×Dic₃	—	6	4	no	—
48	C₃×C₈	—	6	4	no	—
49	C₃×C₈	—	9	2	no	—
50	C₃×C₈	#622	1	2	yes	—
51	C₃⋊C₈	—	6	4	no	—
52	C₂×C₃²	—	2	4	no	—
53	C₂×C₃²	—	2	2	no	—
54	C₂×C₈	—	9	1	no	—
55	C₂×C₈	#1088	1	1	yes	characteristic
56	C₄²	—	9	1	no	—
57	C₁₆	—	9	4	no	—
58	C₂²×C₃	—	6	2	no	—
59	C₂²×C₃	#3787	1	1	yes	characteristic
60	C₃×C₄	—	6	2	no	—
61	C₃×C₄	—	6	2	no	—
62	C₃×C₄	—	18	2	no	—
63	C₃×C₄	C₄×#157	1	1	yes	characteristic
64	C₃×C₄	C₂×#622	1	1	yes	characteristic
65	Dic₃	—	6	8	no	—
66	C₃²	—	2	2	no	—
67	C₂×C₄	—	9	2	no	—
68	C₂×C₄	C₂×#1088	1	1	yes	characteristic
69	C₈	—	9	2	no	—
70	C₈	#6188	1	2	yes	—
71	C₂×C₃	—	6	4	no	—
72	C₂×C₃	—	6	2	no	—
73	C₂×C₃	#63765	1	2	yes	—
74	C₂×C₃	C₄×#622	1	1	yes	characteristic
75	C₂²	#88626	1	1	yes	characteristic
76	C₄	—	18	2	no	—
77	C₄	C₄×#1088	1	1	yes	characteristic
78	C₄	C₂×#6188	1	1	yes	characteristic
79	C₃	—	6	2	no	—
80	C₃	#10590175	1	1	yes	characteristic
81	C₂	#11720771	1	2	yes	—
82	C₂	C₄×#6188	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃×C₄	C₂×#157
commutator	C₂×H(3,𝔽₃)	C₄×C₈
Fitting	C₂×C₈×H(3,𝔽₃)	C₄
socle	C₂²×C₃	#3787
prime radical	C₂²×H(3,𝔽₃)	C₂×C₈
cosocle	C₂×#6180	C₂²
hypercenter	C₂×C₃×C₈	#157
hypocenter	H(3,𝔽₃)	#346
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
6. #88410	C₄	semidirect
43. H(3,𝔽₃)	#346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88410	2
C₈×#1084	2
C₃×#346	9