Subgroups — #13136914

Order 1728.

157 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11720779	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	—	2	2	no	—
7	C₃×C₈×Dic₃	—	6	2	no	—
8	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
9	C₂×#1084	—	2	1	no	—
10	C₈×H(3,𝔽₃)	C₈	1	2	yes	—
11	#6180	D₄	1	1	yes	characteristic
12	#6180	Q₈	1	1	yes	characteristic
13	C₃×#364	—	9	1	no	maximal
14	C₂×C₃²×C₈	—	2	2	no	—
15	C₃×C₄×Dic₃	—	6	2	no	—
16	C₂×C₃×C₃⋊C₈	—	6	2	no	—
17	C₂²×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
18	C₄×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
19	C₄×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
20	#1084	—	2	2	no	—
21	C₃×M₃₂	—	9	2	no	—
22	C₃×C₄×C₈	—	9	1	no	—
23	C₈×Dic₃	—	6	2	no	—
24	C₂×C₃²×C₄	—	2	2	no	—
25	C₂×C₃×Dic₃	—	6	4	no	—
26	C₃²×C₈	—	2	4	no	—
27	C₃×C₃⋊C₈	—	6	2	no	—
28	C₃×C₃⋊C₈	—	6	2	no	—
29	#364	—	9	1	no	Sylow
30	C₂×H(3,𝔽₃)	—	2	1	no	—
31	C₂×H(3,𝔽₃)	C₄⋊C₈	1	1	yes	characteristic
32	C₂×C₃×C₈	—	6	2	no	—
33	C₂×C₃×C₈	—	9	1	no	—
34	C₂×C₃×C₈	#157	1	1	yes	characteristic
35	C₃×C₄²	—	9	1	no	—
36	C₄×Dic₃	—	6	2	no	—
37	C₂×C₃⋊C₈	—	6	2	no	—
38	C₃×C₁₆	—	18	2	no	—
39	C₂²×C₃²	—	2	2	no	—
40	C₃²×C₄	—	2	2	no	—
41	C₃²×C₄	—	2	2	no	—
42	C₃×Dic₃	—	6	8	no	—
43	M₃₂	—	9	2	no	—
44	C₄×C₈	—	9	1	no	—
45	H(3,𝔽₃)	#364	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃×C₄	—	6	2	no	—
47	C₂×C₃×C₄	—	18	1	no	—
48	C₂×C₃×C₄	C₂×#157	1	1	yes	characteristic
49	C₂×Dic₃	—	6	4	no	—
50	C₃×C₈	—	6	4	no	—
51	C₃×C₈	—	9	1	no	—
52	C₃×C₈	—	9	1	no	—
53	C₃×C₈	#622	1	2	yes	—
54	C₃⋊C₈	—	6	2	no	—
55	C₃⋊C₈	—	6	2	no	—
56	C₂×C₃²	—	2	4	no	—
57	C₂×C₃²	—	2	2	no	—
58	C₂×C₈	—	9	1	no	—
59	C₂×C₈	#1088	1	1	yes	characteristic
60	C₄²	—	9	1	no	—
61	C₁₆	—	18	2	no	—
62	C₂²×C₃	—	6	2	no	—
63	C₂²×C₃	#3788	1	1	yes	characteristic
64	C₃×C₄	—	6	2	no	—
65	C₃×C₄	—	6	2	no	—
66	C₃×C₄	—	18	2	no	—
67	C₃×C₄	C₄⋊#157	1	1	yes	characteristic
68	C₃×C₄	C₂×#622	1	1	yes	characteristic
69	Dic₃	—	6	8	no	—
70	C₃²	—	2	2	no	—
71	C₂×C₄	—	18	1	no	—
72	C₂×C₄	C₂×#1088	1	1	yes	characteristic
73	C₈	—	9	1	no	—
74	C₈	—	9	1	no	—
75	C₈	#6188	1	2	yes	—
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	—	6	4	no	—
78	C₂×C₃	—	6	2	no	—
79	C₂×C₃	C₄⋊#622	1	1	yes	characteristic
80	C₂²	#88627	1	1	yes	characteristic
81	C₄	—	18	2	no	—
82	C₄	C₄⋊#1088	1	1	yes	characteristic
83	C₄	C₂×#6188	1	1	yes	characteristic
84	C₃	—	6	2	no	—
85	C₃	#10590178	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	C₄⋊#6188	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₄⋊#157
commutator	C₄×H(3,𝔽₃)	C₂×C₈
Frattini	C₂×C₃×C₄	C₂×#157
Fitting	C₂×C₈×H(3,𝔽₃)	C₄
socle	C₂×C₃	C₄⋊#622
prime radical	C₂²×H(3,𝔽₃)	M₁₆
cosocle	C₂×#6180	C₂²
hypercenter	C₂×C₃×C₈	#157
hypocenter	H(3,𝔽₃)	#364
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
45. H(3,𝔽₃)	#364	semidirect

Maximal subgroups

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

subgroup	index
#11720779	2
C₈×#1084	2
C₃×#364	9