Subgroups — #10592606

Order 576.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64110	C₂	1	1	yes	maximal, characteristic
2	C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#63625	C₂	1	2	yes	maximal, complemented
5	C₁₆×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
6	#4907	—	3	4	no	maximal
7	#3828	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
9	#3748	C₄	1	1	yes	characteristic
10	#3743	C₄	1	2	yes	—
11	C₈×C₃⋊D₃	C₂²	1	2	yes	—
12	C₃²×C₁₆	C₂²	1	2	yes	—
13	C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	#901	—	3	4	no	—
16	C₃×M₃₂	D₃	1	4	yes	complemented
17	C₂×C₃⋊C₁₆	—	3	4	no	—
18	#798	—	3	8	no	—
19	D₃×C₁₆	—	3	8	no	—
20	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	#638	C₈	1	2	yes	—
22	C₄⋊(C₃⋊D₃)	C₈	1	1	yes	characteristic
23	C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
24	#634	C₈	1	1	yes	characteristic
25	C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₈	C₂³	1	1	yes	characteristic
27	C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
28	#504	—	9	1	no	Sylow
29	#235	—	3	4	no	—
30	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
31	#208	—	3	4	no	—
32	ΓL(1,𝔽₂₅)	—	3	8	no	—
33	D₃×C₈	—	3	8	no	—
34	C₃×C₁₆	C₂×D₃	1	8	yes	—
35	C₃⋊C₁₆	—	3	4	no	—
36	C₃⋊C₁₆	—	3	4	no	—
37	C₂²×C₃²	C₂×C₈	1	1	yes	characteristic
38	C₂×C₃⋊D₃	C₂×C₈	1	2	yes	—
39	C₃²×C₄	C₂×C₈	1	1	yes	characteristic
40	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₃⋊Dic₃	C₂×C₈	1	2	yes	—
42	#131	—	9	1	no	—
43	M₃₂	—	9	2	no	—
44	M₃₂	C₃⋊D₃	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	9	1	no	—
46	C₂×C₁₆	—	9	2	no	—
47	C₂×C₃×C₄	C₄×D₃	1	4	yes	—
48	#67	—	3	8	no	—
49	D₁₂	—	3	4	no	—
50	C₄×D₃	—	3	8	no	—
51	Dic₆	—	3	4	no	—
52	C₃×C₈	C₄×D₃	1	4	yes	—
53	C₃×C₈	C₂²×D₃	1	4	yes	—
54	C₃⋊C₈	—	3	8	no	—
55	C₂×C₃²	—	2	1	no	—
56	C₂×C₃²	C₂²×C₈	1	1	yes	characteristic
57	C₃⋊D₃	—	2	2	no	—
58	#41	—	9	1	no	—
59	M₁₆	—	9	1	no	—
60	M₁₆	—	9	2	no	—
61	C₂×C₈	—	9	2	no	—
62	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
63	C₁₆	—	9	1	no	—
64	C₁₆	—	9	1	no	—
65	C₁₆	C₂×C₃⋊D₃	1	2	yes	—
66	C₂²×C₃	D₃×C₈	1	4	yes	—
67	C₂×D₃	—	3	8	no	—
68	C₃×C₄	D₃×C₈	1	4	yes	—
69	C₃×C₄	C₂×C₄×D₃	1	4	yes	—
70	Dic₃	—	3	8	no	—
71	C₃²	#504	1	1	yes	characteristic, complemented, Sylow
72	Q₈	—	9	1	no	—
73	D₄	—	9	2	no	—
74	D₄	—	9	1	no	—
75	C₂×C₄	—	9	2	no	—
76	C₂×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
77	C₈	—	9	2	no	—
78	C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
79	C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
80	C₂×C₃	—	2	4	no	—
81	C₂×C₃	C₂×D₃×C₈	1	4	yes	—
82	D₃	—	6	8	no	—
83	C₂²	—	9	2	no	—
84	C₂²	C₈×C₃⋊D₃	1	1	yes	characteristic
85	C₄	—	9	2	no	—
86	C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
87	C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
88	C₃	#4907	1	4	yes	complemented
89	C₂	—	2	1	no	—
90	C₂	—	18	2	no	—
91	C₂	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×C₃⋊D₃
commutator, socle	C₂×C₃²	C₂²×C₈
Fitting	C₃²×M₃₂	C₂
prime radical	#3828	C₄
cosocle	C₃²×C₈	C₂³
hypercenter	M₃₂	C₃⋊D₃
hypocenter	C₃²	#504
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₃²×M₃₂	C₂	semidirect
3. C₂×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
4. #63625	C₂	semidirect
5. C₁₆×C₃⋊D₃	C₂	semidirect
16. C₃×M₃₂	D₃	semidirect
44. M₃₂	C₃⋊D₃	semidirect
71. C₃²	#504	semidirect
88. C₃	#4907	semidirect

Maximal subgroups

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

subgroup	index
#64110	2
C₃²×M₃₂	2
C₂×C₃⋊(C₃⋊C₁₆)	2
#63625	2
C₁₆×C₃⋊D₃	2
#4907	3