Subgroups — #64239

Order 288.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#627	C₂	1	1	yes	maximal, characteristic, complemented
2	#3805	C₂	1	1	yes	maximal, characteristic
3	#3792	C₂	1	1	yes	maximal, characteristic, complemented
4	#3776	C₂	1	2	yes	maximal, complemented
5	#3775	C₂	1	2	yes	maximal, complemented
6	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
7	C₃×#67	—	2	1	no	—
8	C₂×C₃×Dic₃	—	2	1	no	—
9	#627	—	2	1	no	—
10	#627	C₂²	1	2	yes	—
11	#627	C₂²	1	1	yes	characteristic
12	#625	C₂²	1	1	yes	characteristic
13	D₃×Dic₃	—	2	1	no	—
14	#622	C₂²	1	2	yes	—
15	#237	—	6	1	no	—
16	C₂×Dic₆	—	6	1	no	—
17	C₂²×C₃²	D₄	1	1	yes	characteristic
18	C₂×C₃×D₃	—	2	1	no	—
19	C₃⋊Dic₃	D₄	1	1	yes	characteristic
20	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
21	C₃×Dic₃	—	2	1	no	—
22	C₃×Dic₃	—	2	2	no	—
23	#137	—	9	1	no	maximal, Sylow
24	C₃×D₄	—	6	1	no	—
25	C₂×C₃×C₄	—	6	1	no	—
26	#67	—	2	1	no	—
27	#67	—	6	1	no	—
28	C₂×Dic₃	—	2	1	no	—
29	C₂×Dic₃	—	6	1	no	—
30	C₂×Dic₃	—	6	1	no	—
31	C₂×Dic₃	—	6	1	no	—
32	C₄×D₃	—	6	1	no	—
33	Dic₆	—	6	2	no	—
34	Dic₆	—	6	1	no	—
35	Dic₆	—	6	2	no	—
36	C₂×C₃²	—	2	1	no	—
37	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
38	C₃×D₃	—	4	1	no	—
39	#41	—	9	1	no	—
40	C₂×Q₈	—	9	1	no	—
41	Q₁₆	—	9	2	no	—
42	SD₁₆	—	9	2	no	—
43	M₁₆	—	9	1	no	—
44	C₂²×C₃	—	2	1	no	—
45	C₂²×C₃	—	2	1	no	—
46	C₂²×C₃	—	6	1	no	—
47	C₂×D₃	—	2	1	no	—
48	C₃×C₄	—	6	2	no	—
49	C₃×C₄	—	6	1	no	—
50	Dic₃	—	2	2	no	—
51	Dic₃	—	2	1	no	—
52	Dic₃	—	6	1	no	—
53	Dic₃	—	6	1	no	—
54	Dic₃	—	6	1	no	—
55	Dic₃	—	6	1	no	—
56	C₃²	#137	1	1	yes	characteristic, complemented, Sylow
57	Q₈	—	9	2	no	—
58	Q₈	—	9	1	no	—
59	Q₈	—	18	1	no	—
60	D₄	—	6	1	no	—
61	D₄	—	9	1	no	—
62	C₂×C₄	—	6	1	no	—
63	C₂×C₄	—	9	1	no	—
64	C₂×C₄	—	18	1	no	—
65	C₈	—	9	2	no	—
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	4	1	no	—
70	C₂×C₃	—	12	1	no	—
71	D₃	—	4	1	no	—
72	C₂²	—	6	1	no	—
73	C₂²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
74	C₄	—	6	2	no	—
75	C₄	—	6	1	no	—
76	C₄	—	9	1	no	—
77	C₄	—	9	1	no	—
78	C₃	—	2	1	no	—
79	C₃	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	12	1	no	—
82	C₂	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂³
Fitting	C₂²×C₃²	D₄
socle	C₂×C₃²	C₂×D₄
prime radical	#3805	C₂
hypercenter	C₂²	GO⁺(4,𝔽₂)
hypocenter	C₃²	#137
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×#627	C₂	semidirect
3. #3792	C₂	semidirect
4. #3776	C₂	semidirect
5. #3775	C₂	semidirect
56. C₃²	#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#627	2
#3805	2
#3792	2
#3776	2
#3775	2
#137	9