Subgroups — #10592592

Order 576.

306 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×C₃⋊D₃	C₂	1	2	yes	maximal
2	C₂×#3743	C₂	1	1	yes	maximal, characteristic
3	C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	#63649	C₂	1	2	yes	maximal, complemented
5	#63646	C₂	1	1	yes	maximal, characteristic, complemented
6	#4893	—	3	4	no	maximal
7	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₃²×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
10	#3748	C₂²	1	2	yes	—
11	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	#3743	—	2	2	no	—
13	#3743	C₄	1	4	yes	—
14	C₈×C₃⋊D₃	—	2	2	no	—
15	D₃×M₁₆	—	3	8	no	—
16	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
17	C₃×#108	D₃	1	4	yes	complemented
18	#822	—	3	8	no	—
19	#819	—	3	4	no	—
20	C₂²×C₃⋊D₃	Q₈	1	1	yes	characteristic
21	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
23	C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
24	C₄×C₃⋊D₃	D₄	1	2	yes	—
25	C₃²×C₈	—	2	2	no	—
26	C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₃⋊(C₃⋊C₈)	—	2	2	no	—
28	C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
29	#430	—	9	1	no	Sylow
30	C₂×C₄×D₃	—	3	4	no	—
31	C₃×M₁₆	C₂×D₃	1	8	yes	—
32	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
33	#208	—	3	8	no	—
34	C₂×C₃⋊C₈	—	3	4	no	—
35	ΓL(1,𝔽₂₅)	—	3	16	no	—
36	ΓL(1,𝔽₂₅)	—	6	8	no	—
37	D₃×C₈	—	6	8	no	—
38	C₂²×C₃²	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₃⋊D₃	—	4	1	no	—
40	C₂×C₃⋊D₃	C₄⋊C₄	1	2	yes	—
41	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
43	C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
44	C₂×M₁₆	—	9	1	no	—
45	C₂×M₁₆	—	9	2	no	—
46	#108	—	9	2	no	—
47	#108	—	9	1	no	—
48	#108	C₃⋊D₃	1	1	yes	characteristic, complemented
49	C₂²×D₃	—	3	4	no	—
50	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
51	C₂×Dic₃	—	3	4	no	—
52	C₄×D₃	—	3	8	no	—
53	C₄×D₃	—	3	8	no	—
54	C₃×C₈	—	2	8	no	—
55	C₃×C₈	C₄×D₃	1	8	yes	—
56	C₃⋊C₈	—	3	8	no	—
57	C₃⋊C₈	—	6	8	no	—
58	C₂×C₃²	—	2	1	no	—
59	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₃⋊D₃	—	2	2	no	—
61	C₂²×C₄	—	9	1	no	—
62	M₁₆	—	9	4	no	—
63	M₁₆	—	9	2	no	—
64	M₁₆	—	18	2	no	—
65	M₁₆	C₂×C₃⋊D₃	1	2	yes	—
66	C₂×C₈	—	9	1	no	—
67	C₂×C₈	—	18	2	no	—
68	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
69	C₂²×C₃	D₃×Q₈	1	4	yes	—
70	C₂×D₃	—	3	8	no	—
71	C₂×D₃	—	12	4	no	—
72	C₃×C₄	C₂×C₄×D₃	1	4	yes	—
73	C₃×C₄	D₃×D₄	1	4	yes	—
74	Dic₃	—	3	8	no	—
75	C₃²	#430	1	1	yes	characteristic, complemented, Sylow
76	C₂³	—	9	1	no	—
77	C₂×C₄	—	9	1	no	—
78	C₂×C₄	—	9	2	no	—
79	C₂×C₄	—	9	2	no	—
80	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	9	2	no	—
83	C₈	—	18	2	no	—
84	C₈	C₄×C₃⋊D₃	1	2	yes	—
85	C₂×C₃	—	2	4	no	—
86	C₂×C₃	D₃×C₄⋊C₄	1	4	yes	—
87	D₃	—	6	8	no	—
88	C₂²	—	9	2	no	—
89	C₂²	—	36	1	no	—
90	C₂²	Q₈×C₃⋊D₃	1	1	yes	characteristic
91	C₄	—	9	2	no	—
92	C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
93	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
94	C₃	#4893	1	4	yes	complemented
95	C₂	—	2	1	no	—
96	C₂	—	18	2	no	—
97	C₂	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#108	C₂
socle	C₂×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	Q₈
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	#108	C₃⋊D₃
hypocenter	C₃²	#430
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
3. C₃²×#108	C₂	semidirect
4. #63649	C₂	semidirect
5. #63646	C₂	semidirect
17. C₃×#108	D₃	semidirect
48. #108	C₃⋊D₃	semidirect
75. C₃²	#430	semidirect
94. C₃	#4893	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×C₃⋊D₃	2
C₂×#3743	2
C₃²×#108	2
#63649	2
#63646	2
#4893	3