Subgroups — #10596644

Order 576.

422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3832	C₂	1	1	yes	maximal, characteristic
2	#64289	C₂	1	8	yes	maximal, complemented
3	#64283	C₂	1	6	yes	maximal, complemented
4	C₂×C₃²×Q₈	C₄	1	1	yes	characteristic
5	#3832	C₄	1	4	yes	—
6	#3832	C₂²	1	4	yes	—
7	C₂×C₄⋊(C₃⋊D₃)	C₄	1	3	yes	—
8	C₂×C₄×C₃⋊D₃	C₂²	1	3	yes	—
9	#3792	C₂²	1	4	yes	—
10	#3788	C₂²	1	12	yes	complemented
11	#3787	C₂²	1	12	yes	complemented
12	C₂×#239	—	6	2	no	—
13	C₂²×C₃⋊D₃	C₂×C₄	1	3	yes	—
14	C₃²×Q₈	C₂×C₄	1	4	yes	—
15	C₂×C₃²×C₄	C₂×C₄	1	3	yes	—
16	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
17	C₄⋊(C₃⋊D₃)	C₂×C₄	1	12	yes	—
18	C₄×C₃⋊D₃	C₂×C₄	1	6	yes	—
19	C₄×C₃⋊D₃	C₂³	1	6	yes	—
20	#622	C₂³	1	8	yes	—
21	#568	—	9	1	no	maximal, Sylow
22	C₂×C₃×Q₈	—	2	2	no	—
23	#239	—	6	8	no	—
24	#239	—	6	8	no	—
25	C₂×D₁₂	—	6	6	no	—
26	C₂×C₄×D₃	—	6	6	no	—
27	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₃⋊D₃	—	4	3	no	—
29	C₂×C₃⋊D₃	C₂²×C₄	1	6	yes	—
30	C₃²×C₄	C₂²×C₄	1	6	yes	—
31	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
32	C₃⋊Dic₃	C₂⁴	1	1	yes	characteristic
33	C₂×#41	—	9	1	no	—
34	#131	—	9	8	no	—
35	C₂×M₁₆	—	9	6	no	—
36	C₂²×D₃	—	6	6	no	—
37	C₃×Q₈	—	2	8	no	—
38	C₂×C₃×C₄	—	2	6	no	—
39	C₂×Dic₃	—	6	2	no	—
40	D₁₂	—	6	24	no	—
41	C₄×D₃	—	6	12	no	—
42	C₄×D₃	—	6	12	no	—
43	C₂×C₃²	—	2	1	no	—
44	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
45	C₃⋊D₃	—	2	6	no	—
46	#41	—	9	4	no	—
47	#41	—	9	4	no	—
48	C₂×Q₈	#157	1	1	yes	characteristic
49	C₂×D₄	—	9	3	no	—
50	C₂²×C₄	—	9	3	no	—
51	M₁₆	—	9	4	no	—
52	M₁₆	—	9	12	no	—
53	C₂×C₈	—	9	12	no	—
54	C₂²×C₃	—	2	2	no	—
55	C₂×D₃	—	6	12	no	—
56	C₂×D₃	—	12	12	no	—
57	C₃×C₄	—	2	12	no	—
58	Dic₃	—	6	2	no	—
59	Dic₃	—	6	2	no	—
60	C₃²	#568	1	1	yes	characteristic, complemented, Sylow
61	C₂³	—	9	3	no	—
62	Q₈	C₂×#157	1	4	yes	—
63	D₄	—	9	12	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	9	6	no	—
66	C₂×C₄	—	9	6	no	—
67	C₂×C₄	C₂×#157	1	3	yes	—
68	C₈	—	9	8	no	—
69	C₂×C₃	—	2	4	no	—
70	C₂×C₃	—	2	2	no	—
71	D₃	—	12	12	no	—
72	C₂²	—	9	6	no	—
73	C₂²	—	36	3	no	—
74	C₂²	C₂²×#157	1	1	yes	characteristic
75	C₄	—	9	1	no	—
76	C₄	—	9	1	no	—
77	C₄	C₂²×#157	1	6	yes	—
78	C₃	—	2	2	no	—
79	C₂	—	2	1	no	—
80	C₂	—	18	6	no	—
81	C₂	C₂³×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#157
commutator, socle	C₂×C₃²	C₂³×C₄
Fitting	C₂×C₃²×Q₈	C₄
prime radical	C₂×#3832	C₂
cosocle	C₃⋊Dic₃	C₂⁴
hypercenter	C₂×Q₈	#157
hypocenter	C₃²	#568
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. #64289	C₂	semidirect
3. #64283	C₂	semidirect
10. #3788	C₂²	semidirect
11. #3787	C₂²	semidirect
60. C₃²	#568	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3832	2
#64289	2
#64283	2
#568	9