Subgroups — C₂²×#6196 #11722536

Order 864.

600 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	C₂²×(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal
3	C₂×#6196	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#627	—	3	2	no	maximal
5	C₂³×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×#1084	C₂²	1	6	yes	—
7	C₂×(C₃⋊Dic₃)⋊C₃	C₂²	1	12	yes	—
8	#6196	C₂²	1	16	yes	direct product factor
9	C₂²×C₃⋊Dic₃	D₃	1	2	yes	—
10	C₂²×C₃×Dic₃	—	3	2	no	—
11	C₂²×C₃×Dic₃	—	3	2	no	—
12	C₂²×C₃×Dic₃	—	6	1	no	—
13	C₂×#627	—	3	24	no	—
14	C₂²×H(3,𝔽₃)	Q₈	1	4	yes	complemented
15	C₂²×H(3,𝔽₃)	C₂³	1	3	yes	—
16	#1084	C₂³	1	4	yes	—
17	(C₃⋊Dic₃)⋊C₃	C₂³	1	8	yes	—
18	C₂²×Dic₆	—	9	1	no	—
19	C₂²×Dic₆	—	9	2	no	—
20	C₂³×C₃²	—	2	1	no	—
21	C₂³×C₃²	C₂×D₃	1	2	yes	—
22	C₂×C₃⋊Dic₃	C₂×D₃	1	12	yes	—
23	C₂×C₃×Dic₃	—	3	12	no	—
24	C₂×C₃×Dic₃	—	3	12	no	—
25	C₂×C₃×Dic₃	—	6	6	no	—
26	#627	—	3	32	no	—
27	C₂×H(3,𝔽₃)	C₂×Q₈	1	6	yes	complemented
28	C₂×H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic
29	C₂²×C₃×C₄	—	9	2	no	—
30	C₂²×C₃×C₄	—	9	1	no	—
31	C₂²×Dic₃	—	3	2	no	—
32	C₂²×Dic₃	—	3	2	no	—
33	C₂²×Dic₃	—	6	1	no	—
34	C₂²×Dic₃	—	9	2	no	—
35	C₂×Dic₆	—	9	12	no	—
36	C₂×Dic₆	—	9	24	no	—
37	C₂²×C₃²	—	2	4	no	—
38	C₂²×C₃²	—	2	3	no	—
39	C₂²×C₃²	Dic₆	1	8	yes	complemented
40	C₂²×C₃²	C₂²×D₃	1	6	yes	—
41	C₃⋊Dic₃	C₂²×D₃	1	8	yes	—
42	C₃×Dic₃	—	3	8	no	—
43	C₃×Dic₃	—	3	8	no	—
44	C₃×Dic₃	—	6	4	no	—
45	C₂²×Q₈	—	27	1	no	Sylow
46	H(3,𝔽₃)	C₂²×Q₈	1	1	yes	characteristic, complemented, Sylow
47	C₂³×C₃	—	3	2	no	—
48	C₂³×C₃	—	6	1	no	—
49	C₂³×C₃	D₃²	1	1	yes	characteristic
50	C₂×C₃×C₄	—	9	12	no	—
51	C₂×C₃×C₄	—	9	6	no	—
52	C₂×Dic₃	—	3	12	no	—
53	C₂×Dic₃	—	3	12	no	—
54	C₂×Dic₃	—	6	6	no	—
55	C₂×Dic₃	—	9	12	no	—
56	Dic₆	—	9	16	no	—
57	Dic₆	—	9	32	no	—
58	C₂×C₃²	—	2	1	no	—
59	C₂×C₃²	—	2	6	no	—
60	C₂×C₃²	C₂×Dic₆	1	12	yes	complemented
61	C₂×C₃²	C₂³×D₃	1	2	yes	—
62	C₂×Q₈	—	27	12	no	—
63	C₂²×C₄	—	9	1	no	—
64	C₂²×C₄	—	9	2	no	—
65	C₂²×C₃	—	3	8	no	—
66	C₂²×C₃	—	3	6	no	—
67	C₂²×C₃	—	6	4	no	—
68	C₂²×C₃	—	6	3	no	—
69	C₂²×C₃	#627	1	4	yes	—
70	C₂²×C₃	C₂×D₃²	1	3	yes	—
71	C₃×C₄	—	9	8	no	—
72	C₃×C₄	—	9	4	no	—
73	Dic₃	—	3	8	no	—
74	Dic₃	—	3	8	no	—
75	Dic₃	—	6	4	no	—
76	Dic₃	—	9	8	no	—
77	C₃²	—	2	1	no	—
78	C₃²	C₂²×Dic₆	1	2	yes	complemented
79	C₂³	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
80	Q₈	—	27	16	no	—
81	C₂×C₄	—	9	6	no	—
82	C₂×C₄	—	9	12	no	—
83	C₂×C₃	—	3	2	no	—
84	C₂×C₃	—	3	12	no	—
85	C₂×C₃	—	6	1	no	—
86	C₂×C₃	—	6	6	no	—
87	C₂×C₃	C₂×#627	1	6	yes	—
88	C₂×C₃	C₂²×D₃²	1	1	yes	characteristic
89	C₂²	#6196	1	4	yes	direct product factor
90	C₂²	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	1	3	yes	—
91	C₄	—	9	4	no	—
92	C₄	—	9	8	no	—
93	C₃	—	3	2	no	—
94	C₃	—	6	1	no	—
95	C₃	C₂²×#627	1	1	yes	characteristic
96	C₂	C₂×#6196	1	6	yes	direct product factor
97	C₂	C₂²×((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle	C₂×H(3,𝔽₃)	C₂⁴
Frattini	C₂×C₃	C₂²×D₃²
Fitting, prime radical	C₂³×H(3,𝔽₃)	C₂²
socle	C₂³×C₃	D₃²
hypocenter	H(3,𝔽₃)	C₂²×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
3. C₂×#6196	C₂	direct (N × H)
8. #6196	C₂²	direct (N × H)
14. C₂²×H(3,𝔽₃)	Q₈	semidirect
27. C₂×H(3,𝔽₃)	C₂×Q₈	semidirect
39. C₂²×C₃²	Dic₆	semidirect
46. H(3,𝔽₃)	C₂²×Q₈	semidirect
60. C₂×C₃²	C₂×Dic₆	semidirect
78. C₃²	C₂²×Dic₆	semidirect
89. C₂²	#6196	direct (N × H)
96. C₂	C₂×#6196	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
C₂²×(C₃⋊Dic₃)⋊C₃	2
C₂×#6196	2
C₂²×#627	3

Subgroups — C22×#6196 #11722536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#6196 #11722536