Subgroups — #63689

Order 288.

65 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	1	yes	maximal, characteristic, complemented
3	#3688	C₂	1	1	yes	maximal, characteristic, complemented
4	#942	—	3	1	no	maximal
5	#611	—	4	1	no	maximal
6	Q₈⋊C₉	C₂²	1	1	yes	characteristic
7	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
8	#235	—	3	1	no	—
9	#216	—	3	1	no	—
10	#216	—	3	1	no	—
11	#215	—	3	1	no	—
12	#215	—	3	1	no	—
13	#208	—	3	1	no	—
14	C₂²×C₉	—	4	1	no	—
15	C₂×D₉	—	4	1	no	—
16	Dic₉	—	4	1	no	—
17	#137	—	9	1	no	Sylow
18	C₃×Q₈	—	3	1	no	—
19	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃×C₄	—	3	1	no	—
21	#67	—	6	1	no	—
22	D₁₂	—	3	1	no	—
23	C₄×D₃	—	6	1	no	—
24	Dic₆	—	3	1	no	—
25	C₃⋊C₈	—	3	1	no	—
26	C₃⋊C₈	—	3	1	no	—
27	C₂×C₉	—	4	1	no	—
28	C₂×C₉	—	8	1	no	—
29	D₉	—	8	1	no	—
30	#41	—	9	1	no	—
31	C₂×Q₈	D₉	1	1	yes	characteristic, complemented
32	Q₁₆	—	9	1	no	—
33	Q₁₆	—	9	1	no	—
34	SD₁₆	—	9	1	no	—
35	SD₁₆	—	9	1	no	—
36	M₁₆	—	9	1	no	—
37	C₂²×C₃	S₄	1	1	yes	characteristic
38	C₂×D₃	—	6	1	no	—
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	3	1	no	—
41	Dic₃	—	6	1	no	—
42	C₉	—	4	1	no	Sylow
43	Q₈	—	3	1	no	—
44	Q₈	—	9	1	no	—
45	Q₈	C₂×D₉	1	1	yes	characteristic
46	D₄	—	9	1	no	—
47	D₄	—	18	1	no	—
48	C₂×C₄	—	3	1	no	—
49	C₂×C₄	—	18	1	no	—
50	C₈	—	9	1	no	—
51	C₈	—	9	1	no	—
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	C₂×S₄	1	1	yes	characteristic
54	D₃	—	12	1	no	—
55	C₂²	—	18	1	no	—
56	C₂²	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic
57	C₄	—	3	1	no	—
58	C₄	—	3	1	no	—
59	C₄	—	18	1	no	—
60	C₃	#983	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	36	1	no	—
63	C₂	C₂×(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	Q₈⋊C₉	C₂²
Frattini, socle	C₂×C₃	C₂×S₄
Fitting	C₂×C₃×Q₈	D₃
hypercenter	C₂²	(C₂²⋊C₉)⋊C₂
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂ ◁ 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	semidirect
3. #3688	C₂	semidirect
31. C₂×Q₈	D₉	semidirect

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
#3688	2
#942	3
#611	4