Subgroups — #10593450

Order 576.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3776	C₂	1	4	yes	maximal
2	#63919	C₂	1	2	yes	maximal
3	C₄×#622	C₂	1	1	yes	maximal, characteristic
4	C₂×#627	C₂²	1	4	yes	—
5	C₂×#622	C₂²	1	2	yes	—
6	#3776	—	2	8	no	—
7	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₃×C₄⋊Dic₃	—	2	2	no	—
9	#3724	—	2	2	no	—
10	#888	—	6	2	no	—
11	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
12	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
13	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
14	C₂×C₃×Dic₃	—	2	4	no	—
15	#627	—	2	8	no	—
16	#622	D₄	1	4	yes	—
17	#494	—	9	1	no	maximal, Sylow
18	C₂×Dic₆	—	6	4	no	—
19	C₃×C₄⋊C₄	—	6	2	no	—
20	C₄⋊Dic₃	—	2	2	no	—
21	Dic₃⋊C₄	—	6	4	no	—
22	C₄×Dic₃	—	6	2	no	—
23	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
24	C₃²×C₄	Q₁₆	1	2	yes	complemented
25	C₃⋊Dic₃	Q₁₆	1	2	yes	complemented
26	C₃⋊Dic₃	C₂×D₄	1	2	yes	—
27	C₃×Dic₃	—	4	4	no	—
28	C₂×Q₁₆	—	9	4	no	—
29	C₄⋊Q₈	—	9	2	no	—
30	C₄×C₈	—	9	1	no	—
31	C₂×C₃×C₄	—	2	2	no	—
32	C₂×C₃×C₄	—	6	4	no	—
33	C₂×Dic₃	—	2	4	no	—
34	C₂×Dic₃	—	6	2	no	—
35	C₂×Dic₃	—	6	2	no	—
36	Dic₆	—	12	8	no	—
37	C₂×C₃²	#127	1	1	yes	characteristic
38	C₂×C₃²	C₂×Q₁₆	1	1	yes	characteristic
39	C₂×C₃²	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×Q₈	—	9	4	no	—
41	Q₁₆	—	18	8	no	—
42	C₂×C₈	—	9	2	no	—
43	C₄⋊C₄	—	6	2	no	—
44	C₄⋊C₄	—	18	2	no	—
45	C₄²	—	9	1	no	—
46	C₂²×C₃	—	2	2	no	—
47	C₃×C₄	—	2	4	no	—
48	C₃×C₄	—	12	4	no	—
49	Dic₃	—	4	4	no	—
50	Dic₃	—	6	4	no	—
51	Dic₃	—	6	4	no	—
52	C₃²	#494	1	1	yes	characteristic, complemented, Sylow
53	Q₈	—	18	8	no	—
54	C₂×C₄	—	6	4	no	—
55	C₂×C₄	—	9	1	no	—
56	C₂×C₄	—	9	1	no	—
57	C₂×C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
58	C₈	—	9	4	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	—	2	2	no	—
61	C₂×C₃	—	2	2	no	—
62	C₂²	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
63	C₄	—	9	2	no	—
64	C₄	—	9	2	no	—
65	C₄	—	12	4	no	—
66	C₄	#3776	1	2	yes	complemented
67	C₃	—	2	2	no	—
68	C₂	#64228	1	1	yes	characteristic
69	C₂	#64231	1	1	yes	characteristic
70	C₂	C₂×#3776	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×GO⁺(4,𝔽₂)
commutator, cosocle	C₂×C₃⋊Dic₃	C₂³
Fitting	C₂×C₃²×C₄	D₄
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	C₂×C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	#494
LCS	G ▷ C₂×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
24. C₃²×C₄	Q₁₆	semidirect
25. C₃⋊Dic₃	Q₁₆	semidirect
52. C₃²	#494	semidirect
66. C₄	#3776	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3776	2
#63919	2
C₄×#622	2
#494	9