Subgroups — #64223

Order 288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3798	C₂	1	2	yes	maximal, complemented
2	#3787	C₂	1	1	yes	maximal, characteristic, complemented
3	#3776	C₂	1	1	yes	maximal, characteristic, complemented
4	#3775	C₂	1	2	yes	maximal, complemented
5	#3774	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃×C₄×D₃	—	2	2	no	—
8	#627	C₂²	1	2	yes	—
9	#626	—	2	2	no	—
10	#625	C₂²	1	2	yes	—
11	#622	C₂²	1	1	yes	characteristic
12	#622	C₂²	1	1	yes	characteristic
13	#235	—	6	2	no	—
14	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
15	C₂×C₃×D₃	—	2	2	no	—
16	C₃²×C₄	D₄	1	1	yes	characteristic
17	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
18	C₃×Dic₃	—	2	2	no	—
19	#135	—	9	1	no	maximal, Sylow
20	C₂×C₃×C₄	—	6	2	no	—
21	#67	—	6	2	no	—
22	#67	—	6	2	no	—
23	D₁₂	—	6	2	no	—
24	C₄×D₃	—	2	2	no	—
25	C₄×D₃	—	6	2	no	—
26	Dic₆	—	6	2	no	—
27	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
28	C₃⋊D₃	—	2	1	no	—
29	C₃×D₃	—	4	2	no	—
30	#41	—	9	2	no	—
31	Q₁₆	—	9	1	no	—
32	SD₁₆	—	9	2	no	—
33	D₈	—	9	1	no	—
34	C₂×C₈	—	9	1	no	—
35	C₂²×C₃	—	6	2	no	—
36	C₂×D₃	—	2	2	no	—
37	C₂×D₃	—	6	2	no	—
38	C₃×C₄	—	2	2	no	—
39	C₃×C₄	—	6	2	no	—
40	Dic₃	—	2	2	no	—
41	Dic₃	—	6	2	no	—
42	C₃²	#135	1	1	yes	characteristic, complemented, Sylow
43	Q₈	—	9	2	no	—
44	D₄	—	9	2	no	—
45	D₄	—	18	2	no	—
46	C₂×C₄	—	6	2	no	—
47	C₂×C₄	—	9	1	no	—
48	C₈	—	9	1	no	—
49	C₈	—	9	1	no	—
50	C₂×C₃	—	2	2	no	—
51	C₂×C₃	—	12	2	no	—
52	D₃	—	4	2	no	—
53	D₃	—	12	2	no	—
54	C₂²	—	6	2	no	—
55	C₂²	—	9	1	no	—
56	C₄	—	6	2	no	—
57	C₄	—	9	1	no	—
58	C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
59	C₃	—	2	2	no	—
60	C₂	—	12	2	no	—
61	C₂	—	18	1	no	—
62	C₂	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂³
Frattini	C₂	C₂×GO⁺(4,𝔽₂)
Fitting	C₃²×C₄	D₄
socle	C₂×C₃²	C₂×D₄
hypocenter	C₃²	#135
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. #3798	C₂	semidirect
2. #3787	C₂	semidirect
3. #3776	C₂	semidirect
4. #3775	C₂	semidirect
5. #3774	C₂	semidirect
42. C₃²	#135	semidirect

Maximal subgroups

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

subgroup	index
#3798	2
#3787	2
#3776	2
#3775	2
#3774	2
#135	9