Subgroups — C₂×#3718 #63834

Order 288.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Dic₆	C₂	1	2	yes	maximal
2	C₂×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#3718	C₂	1	4	yes	maximal, direct product factor
4	C₂×#216	—	3	2	no	maximal
5	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃×Dic₃	—	2	2	no	—
7	C₃×Dic₆	—	2	2	no	—
8	C₃×Dic₆	C₂²	1	4	yes	—
9	C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₃×Q₈	—	3	2	no	—
11	C₂×Dic₆	D₃	1	2	yes	—
12	#216	—	3	8	no	—
13	C₂×C₃⋊C₈	—	3	2	no	—
14	C₂×C₃⋊C₈	—	6	1	no	—
15	C₂²×C₃²	D₄	1	1	yes	characteristic
16	C₃²×C₄	D₄	1	1	yes	characteristic
17	C₃²×C₄	C₂³	1	1	yes	characteristic
18	C₃×Dic₃	—	2	4	no	—
19	C₂×Q₁₆	—	9	1	no	Sylow
20	C₃×Q₈	—	3	4	no	—
21	C₃×Q₈	—	6	2	no	—
22	C₂×C₃×C₄	—	2	1	no	—
23	C₂×C₃×C₄	—	6	2	no	—
24	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
25	C₂×Dic₃	—	2	2	no	—
26	Dic₆	—	2	2	no	—
27	Dic₆	C₂×D₃	1	4	yes	—
28	C₃⋊C₈	—	3	4	no	—
29	C₃⋊C₈	—	6	2	no	—
30	C₂×C₃²	Q₁₆	1	2	yes	complemented
31	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
32	C₂×Q₈	—	3	2	no	—
33	Q₁₆	—	9	4	no	—
34	C₂×C₈	—	9	1	no	—
35	C₂²×C₃	—	2	1	no	—
36	C₂²×C₃	#67	1	2	yes	—
37	C₃×C₄	—	2	1	no	—
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	—	6	4	no	—
40	C₃×C₄	#67	1	2	yes	—
41	C₃×C₄	C₂²×D₃	1	2	yes	—
42	Dic₃	—	2	4	no	—
43	C₃²	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈	—	3	4	no	—
45	Q₈	—	6	2	no	—
46	C₂×C₄	—	6	2	no	—
47	C₂×C₄	D₃²	1	1	yes	characteristic
48	C₈	—	9	2	no	—
49	C₂×C₃	—	2	2	no	—
50	C₂×C₃	—	2	1	no	—
51	C₂×C₃	#216	1	4	yes	complemented
52	C₂×C₃	C₂×#67	1	2	yes	—
53	C₂²	#625	1	1	yes	characteristic
54	C₄	—	6	4	no	—
55	C₄	#625	1	1	yes	characteristic
56	C₄	C₂×D₃²	1	1	yes	characteristic
57	C₃	—	2	1	no	—
58	C₃	C₂×#216	1	2	yes	complemented
59	C₂	#3718	1	2	yes	direct product factor
60	C₂	C₂×#625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#625
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄	C₂²
socle, prime radical	C₂²×C₃²	D₄
hypercenter	C₂×C₄	D₃²
hypocenter	C₃²	C₂×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. #3718	C₂	direct (N × H)
30. C₂×C₃²	Q₁₆	semidirect
43. C₃²	C₂×Q₁₆	semidirect
51. C₂×C₃	#216	semidirect
58. C₃	C₂×#216	semidirect
59. C₂	#3718	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×C₃×Dic₆	2
C₂×C₃⋊(C₃⋊C₈)	2
#3718	2
C₂×#216	3

Subgroups — C2×#3718 #63834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3718 #63834