Subgroups — #10593387

Order 576.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3787	C₂	1	1	yes	maximal, characteristic
2	#63900	C₂	1	2	yes	maximal
3	#63742	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂×#624	—	2	2	no	—
6	C₂×#622	C₂²	1	2	yes	—
7	#3787	C₄	1	4	yes	complemented
8	C₃×C₄⋊Dic₃	—	2	2	no	—
9	#3724	C₂²	1	4	yes	—
10	D₃×C₄⋊C₄	—	6	2	no	—
11	C₂²×C₃⋊D₃	D₄	1	1	yes	characteristic
12	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
13	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
14	C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
15	C₄×C₃⋊D₃	Q₈	1	2	yes	—
16	C₂×C₃×Dic₃	—	2	4	no	—
17	#624	—	4	4	no	—
18	#622	C₂×C₄	1	4	yes	complemented
19	C₂×#106	—	9	1	no	maximal, Sylow
20	C₂×C₄×D₃	—	6	2	no	—
21	C₂×C₄×D₃	—	6	4	no	—
22	C₃×C₄⋊C₄	—	6	2	no	—
23	C₄⋊Dic₃	—	2	2	no	—
24	Dic₃⋊C₄	—	6	4	no	—
25	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
26	C₂×C₃⋊D₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₃⋊D₃	SD₁₆	1	4	yes	—
28	C₃²×C₄	C₄⋊C₄	1	2	yes	—
29	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
30	C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
31	C₃×Dic₃	—	4	4	no	—
32	C₂²×C₈	—	9	1	no	—
33	C₂×C₄⋊C₄	—	9	2	no	—
34	#106	—	9	4	no	—
35	C₂²×D₃	—	6	2	no	—
36	C₂×C₃×C₄	—	2	2	no	—
37	C₂×C₃×C₄	—	6	4	no	—
38	C₂×Dic₃	—	2	4	no	—
39	C₂×Dic₃	—	6	2	no	—
40	C₄×D₃	—	6	4	no	—
41	C₄×D₃	—	6	4	no	—
42	C₄×D₃	—	12	8	no	—
43	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃²	C₂×SD₁₆	1	2	yes	—
45	C₃⋊D₃	#106	1	4	yes	complemented
46	C₂²×C₄	—	9	1	no	—
47	C₂²×C₄	—	18	2	no	—
48	C₂×C₈	—	9	2	no	—
49	C₂×C₈	—	9	4	no	—
50	C₄⋊C₄	—	6	2	no	—
51	C₄⋊C₄	—	9	4	no	—
52	C₂²×C₃	—	2	2	no	—
53	C₂×D₃	—	6	4	no	—
54	C₂×D₃	—	6	4	no	—
55	C₂×D₃	—	6	4	no	—
56	C₃×C₄	—	2	4	no	—
57	C₃×C₄	—	12	4	no	—
58	Dic₃	—	4	4	no	—
59	Dic₃	—	6	2	no	—
60	Dic₃	—	6	2	no	—
61	C₃²	C₂×#106	1	1	yes	characteristic, complemented, Sylow
62	C₂³	—	9	1	no	—
63	C₂×C₄	—	6	4	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	9	2	no	—
66	C₂×C₄	—	9	2	no	—
67	C₂×C₄	—	36	4	no	—
68	C₂×C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
69	C₈	—	9	4	no	—
70	C₂×C₃	—	2	2	no	—
71	C₂×C₃	—	2	2	no	—
72	C₂×C₃	—	2	2	no	—
73	D₃	—	6	8	no	—
74	C₂²	—	9	4	no	—
75	C₂²	—	9	2	no	—
76	C₂²	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
77	C₄	—	9	1	no	—
78	C₄	—	9	1	no	—
79	C₄	—	12	4	no	—
80	C₄	#3773	1	2	yes	—
81	C₃	—	2	2	no	—
82	C₂	—	9	4	no	—
83	C₂	#64227	1	2	yes	—
84	C₂	C₂×#3773	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×GO⁺(4,𝔽₂)
commutator	C₃⋊Dic₃	C₂²×C₄
Fitting	C₂×C₃²×C₄	D₄
socle	C₂²×C₃²	C₂×D₄
prime radical	C₂²×C₃⋊D₃	D₄
cosocle	C₂×C₃⋊Dic₃	C₂³
hypercenter	C₂×C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	C₂×#106
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. #63742	C₂	semidirect
7. #3787	C₄	semidirect
18. #622	C₂×C₄	semidirect
45. C₃⋊D₃	#106	semidirect
61. C₃²	C₂×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3787	2
#63900	2
#63742	2
C₂×#106	9