Subgroups — #11720794

Order 864.

177 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6235	C₂	1	1	yes	maximal, characteristic
2	#88631	C₂	1	2	yes	maximal, complemented
3	C₂³×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	C₂²×#272	C₄	1	1	yes	characteristic
5	#6235	—	2	2	no	—
6	C₂×#1084	C₂²	1	1	yes	characteristic
7	#6188	—	2	2	no	—
8	C₂×C₃×#67	—	6	2	no	—
9	C₂²×H(3,𝔽₃)	—	2	1	no	—
10	C₂²×H(3,𝔽₃)	—	4	1	no	—
11	C₂²×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₂×#272	—	2	1	no	—
13	C₂×#272	—	4	1	no	—
14	#1084	D₄	1	2	yes	—
15	C₃×#100	—	9	1	no	maximal
16	C₂³×C₃²	—	2	2	no	—
17	C₂²×C₃×D₃	—	6	2	no	—
18	C₃×#67	—	6	8	no	—
19	C₂×C₃×Dic₃	—	6	2	no	—
20	C₂×H(3,𝔽₃)	—	2	1	no	—
21	C₂×H(3,𝔽₃)	—	4	1	no	—
22	C₂×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
23	#272	—	4	1	no	—
24	C₂×C₃×D₄	—	9	1	no	—
25	C₂×#67	—	6	2	no	—
26	C₃×M₁₆	—	9	2	no	—
27	C₂²×C₃²	—	2	4	no	—
28	C₂²×C₃²	—	2	2	no	—
29	C₂²×C₃²	—	4	4	no	—
30	C₂×C₃×D₃	—	6	4	no	—
31	C₂×C₃×D₃	—	12	4	no	—
32	C₃×Dic₃	—	6	4	no	—
33	#100	—	9	1	no	Sylow
34	H(3,𝔽₃)	#100	1	1	yes	characteristic, complemented, Sylow
35	C₂³×C₃	—	6	2	no	—
36	C₂³×C₃	—	9	1	no	—
37	C₂³×C₃	#157	1	1	yes	characteristic
38	C₂²×D₃	—	6	2	no	—
39	C₃×D₄	—	18	2	no	—
40	C₂×C₃×C₄	—	9	1	no	—
41	#67	—	6	8	no	—
42	C₂×Dic₃	—	6	2	no	—
43	C₃×C₈	—	18	2	no	—
44	C₂×C₃²	—	2	4	no	—
45	C₂×C₃²	—	2	2	no	—
46	C₂×C₃²	—	4	4	no	—
47	C₃×D₃	—	12	4	no	—
48	C₂×D₄	—	9	1	no	—
49	M₁₆	—	9	2	no	—
50	C₂²×C₃	—	2	1	no	—
51	C₂²×C₃	—	4	1	no	—
52	C₂²×C₃	—	6	4	no	—
53	C₂²×C₃	—	6	2	no	—
54	C₂²×C₃	—	12	4	no	—
55	C₂²×C₃	—	18	1	no	—
56	C₂²×C₃	—	36	1	no	—
57	C₂²×C₃	C₂×#157	1	1	yes	characteristic
58	C₂×D₃	—	6	4	no	—
59	C₂×D₃	—	12	4	no	—
60	C₃×C₄	—	9	2	no	—
61	Dic₃	—	6	4	no	—
62	C₃²	—	2	2	no	—
63	C₂³	—	9	1	no	—
64	C₂³	#1088	1	1	yes	characteristic
65	D₄	—	18	2	no	—
66	C₂×C₄	—	9	1	no	—
67	C₈	—	18	2	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	4	1	no	—
70	C₂×C₃	—	6	4	no	—
71	C₂×C₃	—	6	2	no	—
72	C₂×C₃	—	12	4	no	—
73	C₂×C₃	—	36	1	no	—
74	C₂×C₃	C₂²⋊#157	1	1	yes	characteristic
75	D₃	—	12	4	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	18	1	no	—
79	C₂²	—	36	1	no	—
80	C₂²	C₂×#1088	1	1	yes	characteristic
81	C₄	—	9	2	no	—
82	C₃	—	6	2	no	—
83	C₃	#63788	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	—	36	1	no	—
87	C₂	C₂²⋊#1088	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂²⋊#157
commutator	C₂²×H(3,𝔽₃)	C₂×C₄
Frattini	C₂²×C₃	C₂×#157
Fitting	C₂³×H(3,𝔽₃)	C₄
prime radical	C₂×#6235	C₂
cosocle	C₂×#1084	C₂²
hypercenter	C₂³×C₃	#157
hypocenter	H(3,𝔽₃)	#100
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. #88631	C₂	semidirect
34. H(3,𝔽₃)	#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6235	2
#88631	2
C₃×#100	9