Subgroups — #64117

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3828	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3745	C₂	1	2	yes	maximal, complemented
5	#3744	C₂	1	2	yes	maximal, complemented
6	#908	—	3	4	no	maximal
7	C₂²×C₃⋊D₃	—	2	1	no	—
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	#638	—	2	1	no	—
10	C₄⋊(C₃⋊D₃)	—	2	1	no	—
11	C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
12	C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
13	C₄×C₃⋊D₃	—	2	1	no	—
14	#634	C₂²	1	1	yes	characteristic, complemented
15	C₃²×C₈	C₂²	1	2	yes	complemented
16	#235	—	3	4	no	—
17	C₂×D₁₂	—	3	4	no	—
18	C₃×M₁₆	D₃	1	4	yes	complemented
19	D₂₄	—	3	8	no	—
20	#204	—	3	8	no	—
21	C₂²×C₃²	D₄	1	1	yes	characteristic
22	C₂×C₃⋊D₃	—	2	1	no	—
23	C₂×C₃⋊D₃	—	2	2	no	—
24	C₂×C₃⋊D₃	—	4	2	no	—
25	C₃²×C₄	D₄	1	1	yes	characteristic
26	C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₃⋊Dic₃	—	2	1	no	—
28	AGL(1,ℤ/8)	—	9	1	no	Sylow
29	C₂²×D₃	—	6	4	no	—
30	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
31	#67	—	6	4	no	—
32	D₁₂	—	3	8	no	—
33	D₁₂	—	3	4	no	—
34	D₁₂	—	6	4	no	—
35	C₄×D₃	—	6	4	no	—
36	Dic₆	—	3	4	no	—
37	C₃×C₈	C₂×D₃	1	8	yes	complemented
38	C₂×C₃²	—	2	1	no	—
39	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
40	C₃⋊D₃	—	4	1	no	—
41	C₃⋊D₃	—	4	2	no	—
42	#41	—	9	1	no	—
43	C₂×D₄	—	9	1	no	—
44	SD₁₆	—	9	2	no	—
45	D₈	—	9	2	no	—
46	M₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
47	C₂²×C₃	D₁₂	1	4	yes	—
48	C₂×D₃	—	6	4	no	—
49	C₂×D₃	—	6	8	no	—
50	C₂×D₃	—	12	8	no	—
51	C₃×C₄	D₁₂	1	4	yes	—
52	C₃×C₄	C₂²×D₃	1	4	yes	—
53	Dic₃	—	6	4	no	—
54	C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
55	C₂³	—	18	1	no	—
56	Q₈	—	9	1	no	—
57	D₄	—	9	1	no	—
58	D₄	—	9	2	no	—
59	D₄	—	18	1	no	—
60	D₄	—	18	1	no	—
61	C₂×C₄	—	18	1	no	—
62	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
63	C₈	C₂×C₃⋊D₃	1	2	yes	complemented
64	C₂×C₃	—	2	4	no	—
65	C₂×C₃	C₂×D₁₂	1	4	yes	—
66	D₃	—	12	8	no	—
67	D₃	—	12	4	no	—
68	C₂²	—	18	1	no	—
69	C₂²	—	18	2	no	—
70	C₂²	—	36	2	no	—
71	C₂²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
72	C₄	—	18	1	no	—
73	C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
74	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
75	C₃	#908	1	4	yes	complemented
76	C₂	—	2	1	no	—
77	C₂	—	36	2	no	—
78	C₂	—	36	1	no	—
79	C₂	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×M₁₆	C₂
socle	C₂×C₃²	C₂×D₄
hypercenter	M₁₆	C₃⋊D₃
hypocenter	C₃²	AGL(1,ℤ/8)
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #3828	C₂	semidirect
2. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
3. C₃²×M₁₆	C₂	semidirect
4. #3745	C₂	semidirect
5. #3744	C₂	semidirect
12. C₄⋊(C₃⋊D₃)	C₂²	semidirect
14. #634	C₂²	semidirect
15. C₃²×C₈	C₂²	semidirect
18. C₃×M₁₆	D₃	semidirect
37. C₃×C₈	C₂×D₃	semidirect
46. M₁₆	C₃⋊D₃	semidirect
54. C₃²	AGL(1,ℤ/8)	semidirect
63. C₈	C₂×C₃⋊D₃	semidirect
75. C₃	#908	semidirect

Maximal subgroups

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

subgroup	index
#3828	2
C₂×C₄⋊(C₃⋊D₃)	2
C₃²×M₁₆	2
#3745	2
#3744	2
#908	3