Subgroups — #10589362

Order 576.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3746	C₂	1	1	yes	maximal, characteristic
2	C₃²×#108	C₂	1	1	yes	maximal, characteristic
3	#63639	C₂	1	1	yes	maximal, characteristic
4	#4496	—	3	4	no	maximal
5	C₂×#634	—	2	1	no	—
6	C₃²×M₁₆	—	2	1	no	—
7	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
8	#3746	—	2	1	no	—
9	#3746	C₄	1	2	yes	—
10	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
11	C₂×Dic₁₂	—	3	4	no	—
12	C₃×#108	D₃	1	4	yes	—
13	#812	—	3	4	no	—
14	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
15	C₂×C₃⋊Dic₃	—	4	1	no	—
16	#634	—	2	2	no	—
17	#634	—	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	#362	—	9	1	no	Sylow
22	C₂×Dic₆	—	6	4	no	—
23	C₃×M₁₆	—	2	4	no	—
24	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
25	Dic₁₂	—	3	8	no	—
26	Dic₁₂	—	6	4	no	—
27	C₃⋊C₁₆	—	6	4	no	—
28	C₂²×C₃²	SD₁₆	1	1	yes	characteristic
29	C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
30	C₃²×C₄	D₈	1	1	yes	characteristic
31	C₃⋊Dic₃	—	4	2	no	—
32	C₂×Q₁₆	—	9	1	no	—
33	M₃₂	—	9	1	no	—
34	#108	C₃⋊D₃	1	1	yes	characteristic
35	C₂×C₃×C₄	#67	1	4	yes	—
36	C₂×Dic₃	—	12	4	no	—
37	Dic₆	—	6	8	no	—
38	Dic₆	—	12	4	no	—
39	C₃×C₈	—	4	4	no	—
40	C₃×C₈	C₄×D₃	1	4	yes	—
41	C₃×C₈	D₁₂	1	4	yes	—
42	C₂×C₃²	—	2	1	no	—
43	C₂×C₃²	#102	1	1	yes	characteristic
44	C₂×Q₈	—	18	1	no	—
45	Q₁₆	—	9	2	no	—
46	Q₁₆	—	18	1	no	—
47	M₁₆	—	2	1	no	—
48	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
49	C₁₆	—	18	1	no	—
50	C₂²×C₃	#215	1	4	yes	—
51	C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
52	C₃×C₄	#213	1	4	yes	—
53	Dic₃	—	12	8	no	—
54	C₃²	#362	1	1	yes	characteristic, complemented, Sylow
55	Q₈	—	18	2	no	—
56	Q₈	—	36	1	no	—
57	C₂×C₄	—	36	1	no	—
58	C₂×C₄	#638	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
61	C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
62	C₂×C₃	—	2	4	no	—
63	C₂×C₃	#809	1	4	yes	—
64	C₂²	#3755	1	1	yes	characteristic
65	C₄	—	36	2	no	—
66	C₄	#3752	1	1	yes	characteristic
67	C₄	#3753	1	1	yes	characteristic
68	C₃	#4496	1	4	yes	complemented
69	C₂	—	2	1	no	—
70	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	C₂²×C₃²	SD₁₆
cosocle	C₂×C₃²×C₈	C₂²
hypercenter	#108	C₃⋊D₃
hypocenter	C₃²	#362
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
54. C₃²	#362	semidirect
68. C₃	#4496	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3746	2
C₃²×#108	2
#63639	2
#4496	3