Subgroups — C₂²×(C₂²⋊C₉)⋊C₂ #64187

Order 288.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂²⋊C₉)⋊C₂	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#67	—	3	1	no	maximal
4	C₂²×D₉	—	4	1	no	maximal
5	C₂×C₂²⋊C₉	C₂²	1	3	yes	complemented
6	(C₂²⋊C₉)⋊C₂	C₂²	1	4	yes	direct product factor
7	C₂⁴×C₃	D₃	1	1	yes	characteristic
8	C₂³×D₃	—	3	1	no	—
9	C₂×#67	—	3	6	no	—
10	C₂×#67	—	3	6	no	—
11	C₂²×Dic₃	—	3	1	no	—
12	C₂²×C₉	—	4	1	no	—
13	C₂×D₉	—	4	6	no	—
14	C₂²⋊C₉	C₂³	1	1	yes	characteristic, complemented
15	C₂²×D₄	—	9	1	no	Sylow
16	C₂³×C₃	—	3	1	no	—
17	C₂³×C₃	—	3	3	no	—
18	C₂³×C₃	C₂×D₃	1	3	yes	—
19	C₂²×D₃	—	3	6	no	—
20	C₂²×D₃	—	6	3	no	—
21	C₂²×D₃	—	6	1	no	—
22	#67	—	3	12	no	—
23	#67	—	3	4	no	—
24	C₂×Dic₃	—	3	6	no	—
25	C₂×C₉	—	4	3	no	—
26	D₉	—	4	4	no	—
27	C₂⁴	—	9	1	no	—
28	C₂⁴	D₉	1	1	yes	characteristic, complemented
29	C₂×D₄	—	9	6	no	—
30	C₂×D₄	—	9	6	no	—
31	C₂²×C₄	—	9	1	no	—
32	C₂²×C₃	—	3	3	no	—
33	C₂²×C₃	—	3	3	no	—
34	C₂²×C₃	—	3	3	no	—
35	C₂²×C₃	—	6	1	no	—
36	C₂²×C₃	S₄	1	1	yes	characteristic
37	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
38	C₂×D₃	—	3	4	no	—
39	C₂×D₃	—	6	6	no	—
40	C₂×D₃	—	6	6	no	—
41	Dic₃	—	3	4	no	—
42	C₉	—	4	1	no	Sylow
43	C₂³	—	3	1	no	—
44	C₂³	—	3	3	no	—
45	C₂³	—	9	6	no	—
46	C₂³	—	18	3	no	—
47	C₂³	—	18	1	no	—
48	C₂³	C₂×D₉	1	3	yes	complemented
49	D₄	—	9	12	no	—
50	D₄	—	9	4	no	—
51	C₂×C₄	—	9	6	no	—
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	3	3	no	—
54	C₂×C₃	C₂×S₄	1	3	yes	—
55	D₃	—	6	4	no	—
56	C₂²	—	3	3	no	—
57	C₂²	—	3	3	no	—
58	C₂²	—	3	3	no	—
59	C₂²	—	6	1	no	—
60	C₂²	—	9	4	no	—
61	C₂²	—	18	6	no	—
62	C₂²	—	18	6	no	—
63	C₂²	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic, direct product factor
64	C₂²	C₂²×D₉	1	1	yes	characteristic, complemented
65	C₄	—	9	4	no	—
66	C₃	C₂²×S₄	1	1	yes	characteristic
67	C₂	—	3	1	no	—
68	C₂	—	3	3	no	—
69	C₂	—	18	4	no	—
70	C₂	C₂×(C₂²⋊C₉)⋊C₂	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂³
Frattini	C₃	C₂²×S₄
Fitting, socle	C₂⁴×C₃	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊C₉	C₂	semidirect
2. C₂×(C₂²⋊C₉)⋊C₂	C₂	direct (N × H)
5. C₂×C₂²⋊C₉	C₂²	semidirect
6. (C₂²⋊C₉)⋊C₂	C₂²	direct (N × H)
14. C₂²⋊C₉	C₂³	semidirect
28. C₂⁴	D₉	semidirect
48. C₂³	C₂×D₉	semidirect
63. C₂²	(C₂²⋊C₉)⋊C₂	direct (N × H)
64. C₂²	C₂²×D₉	semidirect
70. C₂	C₂×(C₂²⋊C₉)⋊C₂	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×C₂²⋊C₉	2
C₂×(C₂²⋊C₉)⋊C₂	2
C₂²×#67	3
C₂²×D₉	4

Subgroups — C22×(C22⋊C9)⋊C2 #64187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×(C₂²⋊C₉)⋊C₂ #64187