Subgroups — #64123

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3832	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3755	C₂	1	1	yes	maximal, characteristic, complemented
5	#3753	C₂	1	1	yes	maximal, characteristic, complemented
6	#3745	C₂	1	1	yes	maximal, characteristic, complemented
7	#3743	C₂	1	1	yes	maximal, characteristic, complemented
8	#914	—	3	4	no	maximal
9	C₂²×C₃⋊D₃	—	2	1	no	—
10	C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₃²×D₄	C₂²	1	1	yes	characteristic
12	#638	—	2	1	no	—
13	C₄⋊(C₃⋊D₃)	—	2	1	no	—
14	C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₃⋊D₃	—	2	1	no	—
17	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
18	C₃²×C₈	C₂²	1	1	yes	characteristic, complemented
19	C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
20	#239	—	3	4	no	—
21	D₃×D₄	—	3	4	no	—
22	C₃×SD₁₆	D₃	1	4	yes	complemented
23	#215	—	3	4	no	—
24	#213	—	3	4	no	—
25	D₂₄	—	3	4	no	—
26	ΓL(1,𝔽₂₅)	—	3	4	no	—
27	C₂²×C₃²	—	2	1	no	—
28	C₂×C₃⋊D₃	—	2	1	no	—
29	C₂×C₃⋊D₃	—	2	1	no	—
30	C₂×C₃⋊D₃	—	4	2	no	—
31	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
32	C₃²×C₄	—	2	1	no	—
33	C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₃⋊Dic₃	D₄	1	1	yes	characteristic
35	AGL(1,ℤ/8)	—	9	1	no	Sylow
36	C₂²×D₃	—	6	4	no	—
37	C₃×Q₈	C₂×D₃	1	4	yes	complemented
38	C₃×D₄	C₂×D₃	1	4	yes	—
39	#67	—	6	4	no	—
40	D₁₂	—	3	4	no	—
41	D₁₂	—	3	4	no	—
42	D₁₂	—	6	4	no	—
43	C₄×D₃	—	3	4	no	—
44	C₄×D₃	—	6	4	no	—
45	C₃×C₈	C₂×D₃	1	4	yes	complemented
46	C₃⋊C₈	—	3	4	no	—
47	C₂×C₃²	—	4	1	no	—
48	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
49	C₃⋊D₃	—	2	1	no	—
50	C₃⋊D₃	—	4	1	no	—
51	C₃⋊D₃	—	4	1	no	—
52	#41	—	9	1	no	—
53	C₂×D₄	—	9	1	no	—
54	SD₁₆	—	9	1	no	—
55	SD₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
56	D₈	—	9	1	no	—
57	D₈	—	9	1	no	—
58	M₁₆	—	9	1	no	—
59	C₂²×C₃	—	2	4	no	—
60	C₂×D₃	—	3	4	no	—
61	C₂×D₃	—	6	4	no	—
62	C₂×D₃	—	6	4	no	—
63	C₂×D₃	—	12	8	no	—
64	C₃×C₄	—	2	4	no	—
65	C₃×C₄	C₂²×D₃	1	4	yes	—
66	Dic₃	—	3	4	no	—
67	C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
68	C₂³	—	18	1	no	—
69	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
70	D₄	—	9	1	no	—
71	D₄	—	9	1	no	—
72	D₄	—	18	1	no	—
73	D₄	—	18	1	no	—
74	D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
75	C₂×C₄	—	9	1	no	—
76	C₂×C₄	—	18	1	no	—
77	C₈	—	9	1	no	—
78	C₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
79	C₂×C₃	—	4	4	no	—
80	C₂×C₃	D₃×D₄	1	4	yes	—
81	D₃	—	6	4	no	—
82	D₃	—	12	4	no	—
83	D₃	—	12	4	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	9	1	no	—
86	C₂²	—	18	1	no	—
87	C₂²	—	18	1	no	—
88	C₂²	—	36	2	no	—
89	C₄	—	2	1	no	—
90	C₄	—	9	1	no	—
91	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
92	C₃	#914	1	4	yes	complemented
93	C₂	—	4	1	no	—
94	C₂	—	18	1	no	—
95	C₂	—	36	1	no	—
96	C₂	—	36	1	no	—
97	C₂	D₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×SD₁₆	C₂
socle	C₂×C₃²	C₂×D₄
hypercenter	SD₁₆	C₃⋊D₃
hypocenter	C₃²	AGL(1,ℤ/8)
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. #3832	C₂	semidirect
2. D₄×C₃⋊D₃	C₂	semidirect
3. C₃²×SD₁₆	C₂	semidirect
4. #3755	C₂	semidirect
5. #3753	C₂	semidirect
6. #3745	C₂	semidirect
7. #3743	C₂	semidirect
10. C₃²×Q₈	C₂²	semidirect
15. C₄⋊(C₃⋊D₃)	C₂²	semidirect
18. C₃²×C₈	C₂²	semidirect
19. C₃⋊(C₃⋊C₈)	C₂²	semidirect
22. C₃×SD₁₆	D₃	semidirect
37. C₃×Q₈	C₂×D₃	semidirect
45. C₃×C₈	C₂×D₃	semidirect
55. SD₁₆	C₃⋊D₃	semidirect
67. C₃²	AGL(1,ℤ/8)	semidirect
69. Q₈	C₂×C₃⋊D₃	semidirect
78. C₈	C₂×C₃⋊D₃	semidirect
92. C₃	#914	semidirect

Maximal subgroups

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

subgroup	index
#3832	2
D₄×C₃⋊D₃	2
C₃²×SD₁₆	2
#3755	2
#3753	2
#3745	2
#3743	2
#914	3