Subgroups — #11722449

Order 864.

173 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88744	C₂	1	1	yes	maximal, characteristic
2	#88631	C₂	1	2	yes	maximal, complemented
3	C₂×#6188	C₂	1	2	yes	maximal, complemented
4	#88627	C₂	1	1	yes	maximal, characteristic, complemented
5	#88626	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
7	#6235	C₄	1	2	yes	—
8	C₂×#1084	C₂²	1	2	yes	—
9	C₄×#272	C₂²	1	1	yes	characteristic
10	#6229	C₄	1	1	yes	characteristic
11	#6188	C₂²	1	1	yes	characteristic
12	#6188	C₂²	1	1	yes	characteristic
13	#6188	C₂²	1	2	yes	—
14	C₃×#237	—	6	2	no	—
15	C₂²×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
16	C₂×#272	C₂×C₄	1	1	yes	characteristic
17	C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
18	#1084	C₂×C₄	1	2	yes	—
19	#1084	C₂³	1	1	yes	characteristic
20	C₃×#131	—	9	1	no	maximal
21	C₃²×D₄	—	2	2	no	—
22	C₃×#67	—	6	4	no	—
23	C₂×C₃×Dic₃	—	6	4	no	—
24	C₃×C₄×D₃	—	6	2	no	—
25	C₃×Dic₆	—	6	2	no	—
26	C₂×H(3,𝔽₃)	—	2	2	no	—
27	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
28	#272	—	2	1	no	—
29	C₃×#41	—	9	1	no	—
30	#237	—	6	2	no	—
31	C₃×M₁₆	—	9	1	no	—
32	C₃×M₁₆	—	9	2	no	—
33	C₂×C₃×C₈	—	9	1	no	—
34	C₂×C₃×C₈	—	9	2	no	—
35	C₂²×C₃²	—	2	4	no	—
36	C₂×C₃×D₃	—	6	2	no	—
37	C₃²×C₄	—	2	2	no	—
38	C₃×Dic₃	—	6	4	no	—
39	C₃×Dic₃	—	6	2	no	—
40	#131	—	9	1	no	Sylow
41	H(3,𝔽₃)	#131	1	1	yes	characteristic, complemented, Sylow
42	C₃×Q₈	—	9	1	no	—
43	C₃×D₄	—	6	2	no	—
44	C₃×D₄	—	9	2	no	—
45	C₃×D₄	#157	1	1	yes	characteristic
46	C₂×C₃×C₄	—	9	2	no	—
47	C₂×C₃×C₄	—	9	1	no	—
48	#67	—	6	4	no	—
49	C₂×Dic₃	—	6	4	no	—
50	C₄×D₃	—	6	2	no	—
51	Dic₆	—	6	2	no	—
52	C₃×C₈	—	9	1	no	—
53	C₃×C₈	—	9	1	no	—
54	C₃×C₈	—	9	2	no	—
55	C₂×C₃²	—	2	2	no	—
56	C₂×C₃²	—	4	4	no	—
57	C₃×D₃	—	12	2	no	—
58	#41	—	9	1	no	—
59	M₁₆	—	9	2	no	—
60	M₁₆	—	9	1	no	—
61	C₂×C₈	—	9	2	no	—
62	C₂×C₈	—	9	1	no	—
63	C₂²×C₃	—	6	4	no	—
64	C₂²×C₃	—	9	1	no	—
65	C₂²×C₃	C₂×#157	1	2	yes	—
66	C₂×D₃	—	6	2	no	—
67	C₃×C₄	—	6	2	no	—
68	C₃×C₄	—	9	1	no	—
69	C₃×C₄	—	9	2	no	—
70	C₃×C₄	C₂×#157	1	1	yes	characteristic
71	Dic₃	—	6	4	no	—
72	Dic₃	—	6	2	no	—
73	C₃²	—	2	2	no	—
74	Q₈	—	9	1	no	—
75	D₄	—	9	2	no	—
76	D₄	#1088	1	1	yes	characteristic
77	C₂×C₄	—	9	2	no	—
78	C₂×C₄	—	9	1	no	—
79	C₈	—	9	1	no	—
80	C₈	—	9	2	no	—
81	C₈	—	9	1	no	—
82	C₂×C₃	—	2	2	no	—
83	C₂×C₃	—	6	2	no	—
84	C₂×C₃	—	12	4	no	—
85	C₂×C₃	—	18	1	no	—
86	C₂×C₃	C₂²×#157	1	1	yes	characteristic
87	D₃	—	12	2	no	—
88	C₂²	—	9	1	no	—
89	C₂²	C₂×#1088	1	2	yes	—
90	C₄	—	9	1	no	—
91	C₄	—	9	2	no	—
92	C₄	C₂×#1088	1	1	yes	characteristic
93	C₃	—	6	2	no	—
94	C₃	#64287	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	18	1	no	—
97	C₂	C₂²×#1088	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×#157
commutator	C₂×H(3,𝔽₃)	C₂²×C₄
Fitting	D₄×H(3,𝔽₃)	C₄
prime radical	#88744	C₂
cosocle	#1084	C₂³
hypercenter	C₃×D₄	#157
hypocenter	H(3,𝔽₃)	#131
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

Complements

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

N	H ≅ G/N	type
2. #88631	C₂	semidirect
3. C₂×#6188	C₂	semidirect
4. #88627	C₂	semidirect
5. #88626	C₂	semidirect
41. H(3,𝔽₃)	#131	semidirect

Maximal subgroups

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

subgroup	index
#88744	2
#88631	2
C₂×#6188	2
#88627	2
#88626	2
C₃×#131	9