Subgroups — #10593390

Order 576.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63870	C₂	1	2	yes	maximal
2	C₄×#622	C₂	1	1	yes	maximal, characteristic
3	#63743	C₂	1	1	yes	maximal, characteristic
4	#63743	C₂	1	1	yes	maximal, characteristic
5	#63742	C₂	1	2	yes	maximal
6	C₂×#622	C₂²	1	1	yes	characteristic
7	C₂×#622	C₂²	1	1	yes	characteristic
8	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
9	C₃×Dic₃⋊C₄	—	2	2	no	—
10	#3724	C₂²	1	2	yes	—
11	#3724	C₂²	1	2	yes	—
12	Dic₃⋊Dic₃	—	2	2	no	—
13	#889	—	6	2	no	—
14	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
15	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
16	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
17	C₂×C₃×Dic₃	—	2	2	no	—
18	C₂×C₃×Dic₃	—	2	2	no	—
19	#622	Q₈	1	2	yes	—
20	#622	Q₈	1	2	yes	—
21	#499	—	9	1	no	maximal, Sylow
22	C₃×C₄⋊C₄	—	6	2	no	—
23	C₄⋊Dic₃	—	6	2	no	—
24	Dic₃⋊C₄	—	2	2	no	—
25	Dic₃⋊C₄	—	6	2	no	—
26	Dic₃⋊C₄	—	6	2	no	—
27	Dic₃⋊C₄	—	6	2	no	—
28	C₄×Dic₃	—	6	2	no	—
29	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
30	C₃²×C₄	—	2	1	no	—
31	C₃⋊Dic₃	—	2	1	no	—
32	C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
33	C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
34	C₃×Dic₃	—	4	2	no	—
35	C₃×Dic₃	—	4	2	no	—
36	#125	—	9	2	no	—
37	#107	—	9	1	no	—
38	#107	—	9	1	no	—
39	#106	—	9	2	no	—
40	C₄×C₈	—	9	1	no	—
41	C₂×C₃×C₄	—	2	2	no	—
42	C₂×C₃×C₄	—	6	2	no	—
43	C₂×C₃×C₄	—	6	2	no	—
44	C₂×Dic₃	—	2	2	no	—
45	C₂×Dic₃	—	2	2	no	—
46	C₂×Dic₃	—	6	2	no	—
47	C₂×Dic₃	—	6	2	no	—
48	C₂×C₃²	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₃²	#135	1	1	yes	characteristic
50	C₂×C₃²	#135	1	1	yes	characteristic
51	C₂×C₈	—	9	1	no	—
52	C₂×C₈	—	9	1	no	—
53	C₄⋊C₄	—	6	2	no	—
54	C₄⋊C₄	—	9	2	no	—
55	C₄⋊C₄	—	9	2	no	—
56	C₄⋊C₄	—	18	2	no	—
57	C₄²	—	9	1	no	—
58	C₂²×C₃	—	2	2	no	—
59	C₃×C₄	—	4	2	no	—
60	C₃×C₄	—	12	2	no	—
61	C₃×C₄	—	12	2	no	—
62	Dic₃	—	4	2	no	—
63	Dic₃	—	4	2	no	—
64	Dic₃	—	6	2	no	—
65	Dic₃	—	6	2	no	—
66	Dic₃	—	12	2	no	—
67	C₃²	#499	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄	—	6	2	no	—
69	C₂×C₄	—	6	2	no	—
70	C₂×C₄	—	9	1	no	—
71	C₂×C₄	—	9	1	no	—
72	C₂×C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
73	C₈	—	9	2	no	—
74	C₈	—	9	2	no	—
75	C₂×C₃	—	2	2	no	—
76	C₂×C₃	—	2	2	no	—
77	C₂×C₃	—	2	2	no	—
78	C₂²	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	9	1	no	—
81	C₄	—	9	1	no	—
82	C₄	—	12	2	no	—
83	C₄	—	12	2	no	—
84	C₄	—	18	1	no	—
85	C₃	—	2	2	no	—
86	C₂	#64222	1	1	yes	characteristic
87	C₂	#64223	1	1	yes	characteristic
88	C₂	#64237	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	#499
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
67. C₃²	#499	semidirect

Maximal subgroups

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

subgroup	index
#63870	2
C₄×#622	2
#63743	2
#63742	2
#499	9