Subgroups — C₂×#3688 #63687

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
2	#3688	C₂	1	2	yes	maximal, direct product factor
3	C₂×#216	—	3	1	no	maximal
4	C₂×Dic₉	—	4	1	no	maximal
5	Q₈⋊C₉	C₂²	1	1	yes	characteristic
6	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
7	C₂×Dic₆	—	3	1	no	—
8	#216	—	3	2	no	—
9	#216	—	3	2	no	—
10	C₂×C₃⋊C₈	—	3	1	no	—
11	C₂²×C₉	—	4	1	no	—
12	Dic₉	—	4	2	no	—
13	C₂×Q₁₆	—	9	1	no	Sylow
14	C₃×Q₈	—	3	1	no	—
15	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
16	C₂×C₃×C₄	—	3	1	no	—
17	C₂×Dic₃	—	6	1	no	—
18	Dic₆	—	3	2	no	—
19	Dic₆	—	6	1	no	—
20	C₃⋊C₈	—	3	2	no	—
21	C₂×C₉	—	4	2	no	—
22	C₂×C₉	—	4	1	no	—
23	C₂×Q₈	—	9	1	no	—
24	C₂×Q₈	D₉	1	1	yes	characteristic
25	Q₁₆	—	9	2	no	—
26	Q₁₆	—	9	2	no	—
27	C₂×C₈	—	9	1	no	—
28	C₂²×C₃	S₄	1	1	yes	characteristic
29	C₃×C₄	—	3	1	no	—
30	C₃×C₄	—	3	1	no	—
31	Dic₃	—	6	2	no	—
32	C₉	—	4	1	no	Sylow
33	Q₈	—	3	1	no	—
34	Q₈	—	9	2	no	—
35	Q₈	—	18	1	no	—
36	Q₈	C₂×D₉	1	1	yes	characteristic
37	C₂×C₄	—	3	1	no	—
38	C₂×C₄	—	18	1	no	—
39	C₈	—	9	2	no	—
40	C₂×C₃	#226	1	2	yes	—
41	C₂×C₃	C₂×S₄	1	1	yes	characteristic
42	C₂²	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic
43	C₄	—	3	1	no	—
44	C₄	—	3	1	no	—
45	C₄	—	18	2	no	—
46	C₃	C₂×#226	1	1	yes	characteristic
47	C₂	#3688	1	2	yes	direct product factor
48	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, prime radical	C₂²×C₃	S₄
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂²

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

N	H ≅ G/N	type
2. #3688	C₂	direct (N × H)
47. C₂	#3688	direct (N × H)

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