Subgroups — #10593641

Order 576.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64267	C₂	1	2	yes	maximal, complemented
2	C₃×#131⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#5407	—	3	3	no	maximal
4	#5142	—	3	1	no	maximal
5	C₃×#41⋊C₃	C₂²	1	1	yes	characteristic
6	C₃⋊GL(2,𝔽₃)	—	2	2	no	—
7	#3745	—	4	1	no	maximal
8	#986	—	3	6	no	—
9	C₃×#131	D₃	1	1	yes	characteristic, complemented
10	#949	—	3	2	no	—
11	C₂×D₂₄	—	3	1	no	—
12	#131⋊C₃	D₃	1	3	yes	complemented
13	#823	—	3	2	no	—
14	#819	—	3	1	no	—
15	C₄⋊(C₃⋊D₃)	—	4	2	no	—
16	C₃×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
17	C₃²×C₈	—	4	1	no	—
18	#472	—	9	1	no	Sylow
19	C₃×#41	C₂×D₃	1	1	yes	characteristic
20	C₂×D₁₂	—	3	2	no	—
21	#41⋊C₃	C₂×D₃	1	3	yes	—
22	GL(2,𝔽₃)	—	6	6	no	—
23	C₃×M₁₆	—	3	1	no	—
24	C₂×C₃×C₈	—	3	1	no	—
25	#215	—	6	2	no	—
26	#213	—	6	2	no	—
27	#208	—	3	2	no	—
28	D₂₄	—	6	1	no	—
29	D₂₄	—	6	1	no	—
30	D₂₄	—	12	3	no	—
31	C₂×C₃⋊D₃	—	8	2	no	—
32	C₃²×C₄	—	4	1	no	—
33	AGL(1,ℤ/8)	—	9	2	no	—
34	C₂×D₈	—	9	1	no	—
35	#131	C₃⋊D₃	1	1	yes	characteristic, complemented
36	#108	—	9	1	no	—
37	#100	—	9	2	no	—
38	C₂²×D₃	—	6	2	no	—
39	C₃×Q₈	D₁₂	1	1	yes	characteristic
40	C₃×D₄	—	3	1	no	—
41	C₂×C₃×C₄	—	3	1	no	—
42	D₁₂	—	6	2	no	—
43	D₁₂	—	6	2	no	—
44	D₁₂	—	12	6	no	—
45	SL(2,𝔽₃)	D₁₂	1	3	yes	—
46	C₃×C₈	—	3	1	no	—
47	C₃×C₈	—	4	3	no	—
48	C₃×C₈	S₄	1	1	yes	characteristic
49	C₃⋊C₈	—	6	2	no	—
50	C₂×C₃²	—	4	1	no	—
51	C₃⋊D₃	—	16	2	no	—
52	#41	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₂×D₄	—	9	2	no	—
54	SD₁₆	—	18	2	no	—
55	D₈	—	18	1	no	—
56	D₈	—	18	2	no	—
57	D₈	—	18	1	no	—
58	M₁₆	—	3	1	no	—
59	M₁₆	—	9	2	no	—
60	C₂×C₈	—	3	1	no	—
61	C₂²×C₃	—	3	1	no	—
62	C₂×D₃	—	12	2	no	—
63	C₂×D₃	—	24	6	no	—
64	C₂×D₃	—	24	2	no	—
65	C₃×C₄	—	3	1	no	—
66	C₃×C₄	—	4	3	no	—
67	C₃×C₄	C₂×S₄	1	1	yes	characteristic
68	C₃²	—	4	1	no	Sylow
69	C₂³	—	18	2	no	—
70	Q₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
71	D₄	—	3	1	no	—
72	D₄	—	18	2	no	—
73	D₄	—	18	2	no	—
74	C₂×C₄	—	3	1	no	—
75	C₈	—	3	1	no	—
76	C₈	—	18	2	no	—
77	C₈	C₃⋊S₄	1	1	yes	characteristic
78	C₂×C₃	—	4	3	no	—
79	C₂×C₃	—	6	1	no	—
80	C₂×C₃	C₄⋊S₄	1	1	yes	characteristic
81	D₃	—	24	2	no	—
82	D₃	—	48	6	no	—
83	C₂²	—	3	1	no	—
84	C₂²	—	36	2	no	—
85	C₂²	—	72	2	no	—
86	C₄	—	3	1	no	—
87	C₄	C₂×C₃⋊S₄	1	1	yes	characteristic
88	C₃	—	4	3	no	—
89	C₃	#5407	1	1	yes	characteristic, complemented
90	C₂	—	6	1	no	—
91	C₂	—	72	2	no	—
92	C₂	C₄⋊(C₃⋊S₄)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊(C₃⋊S₄)
commutator, cosocle	C₃×#41⋊C₃	C₂²
Frattini	C₄	C₂×C₃⋊S₄
Fitting	C₃×#131	D₃
socle	C₂×C₃	C₄⋊S₄
hypercenter	C₈	C₃⋊S₄
hypocenter	C₃×SL(2,𝔽₃)	D₄
LCS	G ▷ C₃×#41⋊C₃ ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. #64267	C₂	semidirect
2. C₃×#131⋊C₃	C₂	semidirect
9. C₃×#131	D₃	semidirect
12. #131⋊C₃	D₃	semidirect
35. #131	C₃⋊D₃	semidirect
89. C₃	#5407	semidirect

Maximal subgroups

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

subgroup	index
#64267	2
C₃×#131⋊C₃	2
#5407	3
#5142	3
#3745	4