Subgroups — C₂×#3752 #64136

Order 288.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#3752	C₂	1	4	yes	maximal, direct product factor
5	C₂×(C₂×D₃)⋊C₄	—	3	4	no	maximal
6	C₂³×C₃²	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊D₃	—	2	2	no	—
8	C₂²×C₃⋊D₃	—	2	2	no	—
9	C₂²×C₃⋊D₃	C₄	1	4	yes	complemented
10	C₂²×C₃⋊D₃	C₂²	1	2	yes	—
11	C₂×C₃²×C₄	—	2	2	no	—
12	C₂×C₃²×C₄	C₂²	1	2	yes	complemented
13	C₂×C₃⋊Dic₃	—	2	2	no	—
14	C₂×C₃⋊Dic₃	C₂²	1	2	yes	complemented
15	C₂³×D₃	—	3	4	no	—
16	C₂²×C₃×C₄	D₃	1	4	yes	complemented
17	C₂²×Dic₃	—	3	4	no	—
18	(C₂×D₃)⋊C₄	—	3	16	no	—
19	C₂²×C₃²	C₂×C₄	1	2	yes	—
20	C₂²×C₃²	D₄	1	2	yes	—
21	C₂²×C₃²	D₄	1	2	yes	—
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃⋊D₃	—	2	8	no	—
24	C₂×C₃⋊D₃	—	2	2	no	—
25	C₂×C₃⋊D₃	—	2	2	no	—
26	C₂×C₃⋊D₃	C₂×C₄	1	4	yes	complemented
27	C₃²×C₄	—	2	2	no	—
28	C₃⋊Dic₃	—	2	2	no	—
29	C₂×C₂²⋊C₄	—	9	1	no	Sylow
30	C₂³×C₃	C₂×D₃	1	4	yes	—
31	C₂²×D₃	—	3	16	no	—
32	C₂²×D₃	—	3	8	no	—
33	C₂²×D₃	—	6	8	no	—
34	C₂²×D₃	—	6	8	no	—
35	C₂×C₃×C₄	—	2	8	no	—
36	C₂×C₃×C₄	C₂×D₃	1	8	yes	complemented
37	C₂×Dic₃	—	3	8	no	—
38	C₂×Dic₃	—	6	8	no	—
39	C₂×C₃²	C₂²⋊C₄	1	4	yes	complemented
40	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
42	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
43	C₃⋊D₃	—	2	4	no	—
44	C₂⁴	—	9	1	no	—
45	C₂²×C₄	—	9	1	no	—
46	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
47	C₂²⋊C₄	—	9	4	no	—
48	C₂²×C₃	C₄×D₃	1	8	yes	—
49	C₂²×C₃	D₁₂	1	8	yes	—
50	C₂²×C₃	#67	1	8	yes	—
51	C₂²×C₃	C₂²×D₃	1	4	yes	—
52	C₂×D₃	—	3	16	no	—
53	C₂×D₃	—	6	32	no	—
54	C₂×D₃	—	6	8	no	—
55	C₂×D₃	—	6	8	no	—
56	C₃×C₄	—	2	8	no	—
57	Dic₃	—	6	8	no	—
58	C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
59	C₂³	—	9	4	no	—
60	C₂³	—	9	2	no	—
61	C₂³	—	18	2	no	—
62	C₂³	—	18	2	no	—
63	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	9	2	no	—
66	C₂×C₄	—	18	2	no	—
67	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	complemented
68	C₂×C₃	(C₂×D₃)⋊C₄	1	16	yes	complemented
69	C₂×C₃	C₂×C₄×D₃	1	4	yes	—
70	C₂×C₃	C₂×D₁₂	1	4	yes	—
71	C₂×C₃	C₂×#67	1	4	yes	—
72	D₃	—	6	16	no	—
73	C₂²	—	9	4	no	—
74	C₂²	—	18	8	no	—
75	C₂²	—	18	2	no	—
76	C₂²	—	18	2	no	—
77	C₂²	C₄×C₃⋊D₃	1	2	yes	—
78	C₂²	C₄⋊(C₃⋊D₃)	1	2	yes	—
79	C₂²	#638	1	2	yes	—
80	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	18	2	no	—
83	C₃	C₂×(C₂×D₃)⋊C₄	1	4	yes	complemented
84	C₂	—	18	4	no	—
85	C₂	#3752	1	4	yes	direct product factor
86	C₂	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
87	C₂	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
88	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄
Frattini	C₂²	C₂²×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle	C₂³×C₃²	C₂²
prime radical	C₂³×C₃⋊D₃	C₂
cosocle	C₂²×C₃²	C₂³
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄	C₂	semidirect
3. C₂²×C₃⋊Dic₃	C₂	semidirect
4. #3752	C₂	direct (N × H)
9. C₂²×C₃⋊D₃	C₄	semidirect
12. C₂×C₃²×C₄	C₂²	semidirect
14. C₂×C₃⋊Dic₃	C₂²	semidirect
16. C₂²×C₃×C₄	D₃	semidirect
26. C₂×C₃⋊D₃	C₂×C₄	semidirect
36. C₂×C₃×C₄	C₂×D₃	semidirect
39. C₂×C₃²	C₂²⋊C₄	semidirect
46. C₂²×C₄	C₃⋊D₃	semidirect
58. C₃²	C₂×C₂²⋊C₄	semidirect
67. C₂×C₄	C₂×C₃⋊D₃	semidirect
68. C₂×C₃	(C₂×D₃)⋊C₄	semidirect
83. C₃	C₂×(C₂×D₃)⋊C₄	semidirect
85. C₂	#3752	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⋊D₃	2
C₂²×C₃²×C₄	2
C₂²×C₃⋊Dic₃	2
#3752	2
C₂×(C₂×D₃)⋊C₄	3

Subgroups — C2×#3752 #64136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3752 #64136