Subgroups — #64158

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#3755	C₂	1	2	yes	maximal, complemented
4	#3753	C₂	1	2	yes	maximal, complemented
5	#3748	C₂	1	1	yes	maximal, characteristic, complemented
6	#949	—	3	4	no	maximal
7	C₂²×C₃⋊D₃	—	2	1	no	—
8	C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₃²×D₄	—	2	1	no	—
10	C₃²×D₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₃²×C₄	—	2	1	no	—
12	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊D₃)	—	2	1	no	—
14	C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
15	C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
16	C₃×#41	D₃	1	4	yes	complemented
17	C₂×D₁₂	—	3	4	no	—
18	#215	—	3	8	no	—
19	#213	—	3	8	no	—
20	#208	—	3	4	no	—
21	C₂²×C₃²	—	2	1	no	—
22	C₂²×C₃²	D₄	1	1	yes	characteristic
23	C₂×C₃⋊D₃	—	2	2	no	—
24	C₂×C₃⋊D₃	—	4	2	no	—
25	C₃²×C₄	—	2	1	no	—
26	C₃²×C₄	D₄	1	1	yes	characteristic
27	C₃²×C₄	C₂³	1	1	yes	characteristic
28	AGL(1,ℤ/8)	—	9	1	no	Sylow
29	C₂²×D₃	—	6	4	no	—
30	C₃×Q₈	C₂×D₃	1	4	yes	complemented
31	C₃×D₄	—	2	4	no	—
32	C₃×D₄	C₂×D₃	1	4	yes	complemented
33	C₂×C₃×C₄	—	2	4	no	—
34	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
35	D₁₂	—	3	8	no	—
36	D₁₂	—	6	4	no	—
37	C₃⋊C₈	—	3	8	no	—
38	C₂×C₃²	—	2	1	no	—
39	C₂×C₃²	—	4	1	no	—
40	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
41	C₃⋊D₃	—	4	2	no	—
42	#41	C₃⋊D₃	1	1	yes	characteristic, complemented
43	C₂×D₄	—	9	1	no	—
44	SD₁₆	—	9	2	no	—
45	D₈	—	9	2	no	—
46	M₁₆	—	9	1	no	—
47	C₂²×C₃	—	2	4	no	—
48	C₂²×C₃	#67	1	4	yes	—
49	C₂×D₃	—	6	8	no	—
50	C₂×D₃	—	12	8	no	—
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	#67	1	4	yes	—
53	C₃×C₄	C₂²×D₃	1	4	yes	—
54	C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
55	C₂³	—	18	1	no	—
56	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
57	D₄	—	2	1	no	—
58	D₄	—	9	2	no	—
59	D₄	—	18	1	no	—
60	D₄	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
63	C₈	—	9	2	no	—
64	C₂×C₃	—	2	4	no	—
65	C₂×C₃	—	4	4	no	—
66	C₂×C₃	C₂×#67	1	4	yes	—
67	D₃	—	12	8	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	18	2	no	—
70	C₂²	—	36	2	no	—
71	C₂²	#638	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	#638	1	1	yes	characteristic
74	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
75	C₃	#949	1	4	yes	complemented
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	36	2	no	—
79	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#638
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#41	C₂
socle	C₂×C₃²	C₂×D₄
hypercenter	#41	C₃⋊D₃
hypocenter	C₃²	AGL(1,ℤ/8)
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₃²×#41	C₂	semidirect
2. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
3. #3755	C₂	semidirect
4. #3753	C₂	semidirect
5. #3748	C₂	semidirect
8. C₃²×Q₈	C₂²	semidirect
10. C₃²×D₄	C₂²	semidirect
15. C₃⋊(C₃⋊C₈)	C₂²	semidirect
16. C₃×#41	D₃	semidirect
30. C₃×Q₈	C₂×D₃	semidirect
32. C₃×D₄	C₂×D₃	semidirect
42. #41	C₃⋊D₃	semidirect
54. C₃²	AGL(1,ℤ/8)	semidirect
56. Q₈	C₂×C₃⋊D₃	semidirect
60. D₄	C₂×C₃⋊D₃	semidirect
75. C₃	#949	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#41	2
C₂×C₄⋊(C₃⋊D₃)	2
#3755	2
#3753	2
#3748	2
#949	3