Subgroups — C₂×#88631 #11722454

Order 864.

210 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1084	C₂	1	1	yes	maximal, characteristic
2	#88631	C₂	1	4	yes	maximal, direct product factor
3	C₂×#6188	C₂	1	2	yes	maximal, complemented
4	C₂³×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₂×#1084	C₄	1	2	yes	—
6	C₂×#1084	C₄	1	1	yes	characteristic
7	C₂×#1084	C₂²	1	1	yes	characteristic
8	C₂×#1084	C₂²	1	2	yes	—
9	#6188	C₂²	1	4	yes	complemented
10	C₂²×C₃×Dic₃	—	6	2	no	—
11	C₂²×H(3,𝔽₃)	—	2	2	no	—
12	C₂²×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
13	C₂²×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
14	#1084	C₂×C₄	1	1	yes	characteristic
15	#1084	C₂×C₄	1	2	yes	—
16	#1084	C₂³	1	1	yes	characteristic
17	C₂×C₃×M₁₆	—	9	1	no	maximal
18	C₂³×C₃²	—	2	2	no	—
19	C₂×C₃×Dic₃	—	6	4	no	—
20	C₂×C₃×Dic₃	—	6	2	no	—
21	C₂×C₃×Dic₃	—	6	4	no	—
22	C₂×C₃×Dic₃	—	6	2	no	—
23	C₂×H(3,𝔽₃)	—	2	2	no	—
24	C₂×H(3,𝔽₃)	M₁₆	1	2	yes	complemented
25	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	—	9	1	no	—
27	C₂²×Dic₃	—	6	2	no	—
28	C₃×M₁₆	—	9	4	no	—
29	C₂×C₃×C₈	—	9	2	no	—
30	C₂²×C₃²	—	2	8	no	—
31	C₂²×C₃²	—	2	4	no	—
32	C₂²×C₃²	—	2	2	no	—
33	C₃×Dic₃	—	6	4	no	—
34	C₃×Dic₃	—	6	2	no	—
35	C₃×Dic₃	—	6	2	no	—
36	C₂×M₁₆	—	9	1	no	Sylow
37	H(3,𝔽₃)	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
38	C₂³×C₃	—	6	2	no	—
39	C₂³×C₃	#157	1	1	yes	characteristic
40	C₂×C₃×C₄	—	9	2	no	—
41	C₂×C₃×C₄	—	9	2	no	—
42	C₂×C₃×C₄	—	9	1	no	—
43	C₂×C₃×C₄	—	9	1	no	—
44	C₂×Dic₃	—	6	2	no	—
45	C₂×Dic₃	—	6	4	no	—
46	C₂×Dic₃	—	6	4	no	—
47	C₂×Dic₃	—	6	2	no	—
48	C₃×C₈	—	9	4	no	—
49	C₂×C₃²	—	2	8	no	—
50	C₂×C₃²	—	2	2	no	—
51	C₂×C₃²	—	2	4	no	—
52	C₂²×C₄	—	9	1	no	—
53	M₁₆	—	9	4	no	—
54	C₂×C₈	—	9	2	no	—
55	C₂²×C₃	—	2	2	no	—
56	C₂²×C₃	—	6	8	no	—
57	C₂²×C₃	—	6	4	no	—
58	C₂²×C₃	—	6	2	no	—
59	C₂²×C₃	C₂×#157	1	2	yes	—
60	C₂²×C₃	C₂×#157	1	1	yes	characteristic
61	C₃×C₄	—	9	1	no	—
62	C₃×C₄	—	9	2	no	—
63	C₃×C₄	—	9	1	no	—
64	Dic₃	—	6	2	no	—
65	Dic₃	—	6	2	no	—
66	Dic₃	—	6	4	no	—
67	C₃²	—	2	2	no	—
68	C₂³	#1088	1	1	yes	characteristic
69	C₂×C₄	—	9	1	no	—
70	C₂×C₄	—	9	2	no	—
71	C₂×C₄	—	9	2	no	—
72	C₂×C₄	—	9	1	no	—
73	C₈	—	9	4	no	—
74	C₂×C₃	—	2	2	no	—
75	C₂×C₃	—	6	8	no	—
76	C₂×C₃	—	6	4	no	—
77	C₂×C₃	—	6	2	no	—
78	C₂×C₃	#3792	1	2	yes	—
79	C₂×C₃	C₂²×#157	1	1	yes	characteristic
80	C₂²	—	2	2	no	—
81	C₂²	C₂×#1088	1	2	yes	—
82	C₂²	C₂×#1088	1	1	yes	characteristic
83	C₄	—	9	1	no	—
84	C₄	—	9	1	no	—
85	C₄	—	9	2	no	—
86	C₃	—	6	2	no	—
87	C₃	C₂×#3792	1	1	yes	characteristic
88	C₂	—	2	2	no	—
89	C₂	#88631	1	2	yes	direct product factor
90	C₂	C₂²×#1088	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#157
commutator	C₂×H(3,𝔽₃)	C₂²×C₄
Frattini	C₂×C₃	C₂²×#157
Fitting, prime radical	C₂³×H(3,𝔽₃)	C₄
cosocle	#1084	C₂³
hypercenter	C₂³×C₃	#157
hypocenter	H(3,𝔽₃)	C₂×M₁₆
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. #88631	C₂	direct (N × H)
3. C₂×#6188	C₂	semidirect
9. #6188	C₂²	semidirect
24. C₂×H(3,𝔽₃)	M₁₆	semidirect
37. H(3,𝔽₃)	C₂×M₁₆	semidirect
89. C₂	#88631	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#1084	2
#88631	2
C₂×#6188	2
C₂×C₃×M₁₆	9

Subgroups — C2×#88631 #11722454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88631 #11722454