Subgroups — #10589361

Order 576.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3745	C₂	1	1	yes	maximal, characteristic
2	C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#63639	C₂	1	1	yes	maximal, characteristic, complemented
4	#4495	—	3	4	no	maximal
5	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
6	C₃²×M₁₆	—	2	1	no	—
7	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
8	#3745	—	2	1	no	—
9	#3745	C₄	1	2	yes	—
10	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
11	C₂×D₂₄	—	3	4	no	—
12	C₃×#108	D₃	1	4	yes	complemented
13	#812	—	3	4	no	—
14	C₂²×C₃⋊D₃	—	4	1	no	—
15	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₄⋊(C₃⋊D₃)	—	2	2	no	—
17	C₄⋊(C₃⋊D₃)	—	4	1	no	—
18	C₃²×C₈	—	4	1	no	—
19	C₃²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃²×C₈	D₄	1	1	yes	characteristic
21	#361	—	9	1	no	Sylow
22	C₂×D₁₂	—	6	4	no	—
23	C₃×M₁₆	—	2	4	no	—
24	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
25	D₂₄	—	3	8	no	—
26	D₂₄	—	6	4	no	—
27	C₃⋊C₁₆	—	6	4	no	—
28	C₂²×C₃²	SD₁₆	1	1	yes	characteristic
29	C₂×C₃⋊D₃	—	4	2	no	—
30	C₂×C₃⋊D₃	—	8	2	no	—
31	C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₃²×C₄	D₈	1	1	yes	characteristic
33	C₂×D₈	—	9	1	no	—
34	M₃₂	—	9	1	no	—
35	#108	C₃⋊D₃	1	1	yes	characteristic, complemented
36	C₂²×D₃	—	12	4	no	—
37	C₂×C₃×C₄	#67	1	4	yes	—
38	D₁₂	—	6	8	no	—
39	D₁₂	—	12	4	no	—
40	C₃×C₈	—	4	4	no	—
41	C₃×C₈	C₄×D₃	1	4	yes	—
42	C₃×C₈	D₁₂	1	4	yes	—
43	C₂×C₃²	—	2	1	no	—
44	C₂×C₃²	#102	1	1	yes	characteristic
45	C₃⋊D₃	—	8	2	no	—
46	C₂×D₄	—	18	1	no	—
47	D₈	—	9	2	no	—
48	D₈	—	18	1	no	—
49	M₁₆	—	2	1	no	—
50	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₁₆	—	18	1	no	—
52	C₂²×C₃	#215	1	4	yes	—
53	C₂×D₃	—	12	8	no	—
54	C₂×D₃	—	24	8	no	—
55	C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
56	C₃×C₄	#213	1	4	yes	—
57	C₃²	#361	1	1	yes	characteristic, complemented, Sylow
58	C₂³	—	36	1	no	—
59	D₄	—	18	2	no	—
60	D₄	—	36	1	no	—
61	C₂×C₄	#638	1	1	yes	characteristic
62	C₈	—	4	1	no	—
63	C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
64	C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
65	C₂×C₃	—	2	4	no	—
66	C₂×C₃	#809	1	4	yes	—
67	D₃	—	24	8	no	—
68	C₂²	—	36	2	no	—
69	C₂²	—	72	2	no	—
70	C₂²	#3755	1	1	yes	characteristic
71	C₄	#3752	1	1	yes	characteristic
72	C₄	#3753	1	1	yes	characteristic
73	C₃	#4495	1	4	yes	complemented
74	C₂	—	2	1	no	—
75	C₂	—	72	2	no	—
76	C₂	#63636	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#63636
commutator	C₃²×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×C₃⋊D₃
Fitting	C₃²×#108	C₂
socle	C₂×C₃²	#102
prime radical	C₂×#3745	C₂
cosocle	C₂×C₃²×C₈	C₂²
hypercenter	#108	C₃⋊D₃
hypocenter	C₃²	#361
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₃²×#108	C₂	semidirect
3. #63639	C₂	semidirect
12. C₃×#108	D₃	semidirect
35. #108	C₃⋊D₃	semidirect
57. C₃²	#361	semidirect
73. C₃	#4495	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3745	2
C₃²×#108	2
#63639	2
#4495	3