Subgroups — C₂×#3689 #63688

Order 288.

81 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#3689	C₂	1	2	yes	maximal, direct product factor
3	C₂×#215	—	3	1	no	maximal
4	C₂²×D₉	—	4	1	no	maximal
5	Q₈⋊C₉	C₂²	1	1	yes	characteristic, complemented
6	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
7	C₂×D₁₂	—	3	1	no	—
8	#215	—	3	2	no	—
9	#215	—	3	2	no	—
10	C₂×C₃⋊C₈	—	3	1	no	—
11	C₂²×C₉	—	4	1	no	—
12	C₂×D₉	—	4	2	no	—
13	C₂×D₉	—	4	4	no	—
14	C₂×SD₁₆	—	9	1	no	Sylow
15	C₂²×D₃	—	6	1	no	—
16	C₃×Q₈	—	3	1	no	—
17	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
18	C₂×C₃×C₄	—	3	1	no	—
19	D₁₂	—	3	2	no	—
20	D₁₂	—	6	1	no	—
21	C₃⋊C₈	—	3	2	no	—
22	C₂×C₉	—	4	2	no	—
23	C₂×C₉	—	4	1	no	—
24	D₉	—	4	4	no	—
25	C₂×Q₈	D₉	1	1	yes	characteristic, complemented
26	C₂×D₄	—	9	1	no	—
27	SD₁₆	—	9	2	no	—
28	SD₁₆	—	9	2	no	—
29	C₂×C₈	—	9	1	no	—
30	C₂²×C₃	S₄	1	1	yes	characteristic
31	C₂×D₃	—	6	2	no	—
32	C₂×D₃	—	12	2	no	—
33	C₃×C₄	—	3	1	no	—
34	C₃×C₄	—	3	1	no	—
35	C₉	—	4	1	no	Sylow
36	C₂³	—	18	1	no	—
37	Q₈	—	3	1	no	—
38	Q₈	C₂×D₉	1	1	yes	characteristic, complemented
39	D₄	—	9	2	no	—
40	D₄	—	18	1	no	—
41	C₂×C₄	—	3	1	no	—
42	C₈	—	9	2	no	—
43	C₂×C₃	GL(2,𝔽₃)	1	2	yes	—
44	C₂×C₃	C₂×S₄	1	1	yes	characteristic
45	D₃	—	12	2	no	—
46	C₂²	—	18	2	no	—
47	C₂²	—	36	2	no	—
48	C₂²	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic
49	C₄	—	3	1	no	—
50	C₄	—	3	1	no	—
51	C₃	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
52	C₂	—	36	2	no	—
53	C₂	#3689	1	2	yes	direct product factor
54	C₂	C₂×(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	Q₈⋊C₉	C₂²
Frattini	C₂×C₃	C₂×S₄
Fitting	C₂×C₃×Q₈	D₃
socle	C₂²×C₃	S₄
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈⋊C₉	C₂	semidirect
2. #3689	C₂	direct (N × H)
5. Q₈⋊C₉	C₂²	semidirect
25. C₂×Q₈	D₉	semidirect
38. Q₈	C₂×D₉	semidirect
53. C₂	#3689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈⋊C₉	2
#3689	2
C₂×#215	3
C₂²×D₉	4

Subgroups — C2×#3689 #63688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3689 #63688