Subgroups — #64227

Order 288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3802	C₂	1	1	yes	maximal, characteristic
2	#3797	C₂	1	1	yes	maximal, characteristic, complemented
3	#3787	C₂	1	1	yes	maximal, characteristic, complemented
4	#3775	C₂	1	4	yes	maximal, complemented
5	C₂×D₃²	—	2	1	no	—
6	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃×D₁₂	—	2	1	no	—
8	C₃×Dic₆	—	2	1	no	—
9	#627	C₂²	1	2	yes	complemented
10	#625	C₂²	1	2	yes	—
11	#624	—	2	1	no	—
12	#622	C₂²	1	2	yes	complemented
13	D₃×Q₈	—	6	1	no	—
14	D₃×D₄	—	6	1	no	—
15	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
16	C₂×C₃×D₃	—	2	2	no	—
17	D₃²	—	4	2	no	—
18	C₃²×C₄	D₄	1	1	yes	characteristic
19	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
20	C₃×Dic₃	—	2	2	no	—
21	C₂×SD₁₆	—	9	1	no	maximal, Sylow
22	C₂²×D₃	—	6	2	no	—
23	C₃×Q₈	—	6	1	no	—
24	C₃×D₄	—	6	1	no	—
25	#67	—	6	2	no	—
26	D₁₂	—	2	1	no	—
27	C₄×D₃	—	6	1	no	—
28	C₄×D₃	—	6	1	no	—
29	C₄×D₃	—	6	2	no	—
30	Dic₆	—	2	1	no	—
31	Dic₆	—	6	2	no	—
32	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
33	C₃⋊D₃	SD₁₆	1	2	yes	complemented
34	C₃×D₃	—	4	2	no	—
35	C₂×Q₈	—	9	1	no	—
36	C₂×D₄	—	9	1	no	—
37	SD₁₆	—	9	4	no	—
38	C₂×C₈	—	9	1	no	—
39	C₂²×C₃	—	6	2	no	—
40	C₂×D₃	—	2	2	no	—
41	C₂×D₃	—	6	1	no	—
42	C₂×D₃	—	6	1	no	—
43	C₂×D₃	—	12	4	no	—
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	6	2	no	—
47	Dic₃	—	2	2	no	—
48	Dic₃	—	6	1	no	—
49	Dic₃	—	6	1	no	—
50	C₃²	C₂×SD₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂³	—	18	1	no	—
52	Q₈	—	6	1	no	—
53	Q₈	—	9	2	no	—
54	D₄	—	6	1	no	—
55	D₄	—	9	2	no	—
56	C₂×C₄	—	9	1	no	—
57	C₂×C₄	—	18	1	no	—
58	C₈	—	9	2	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	12	2	no	—
62	D₃	—	4	2	no	—
63	D₃	—	6	2	no	—
64	D₃	—	6	2	no	—
65	C₂²	—	6	2	no	—
66	C₂²	—	9	1	no	—
67	C₂²	—	36	2	no	—
68	C₄	—	6	2	no	—
69	C₄	—	9	1	no	—
70	C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
71	C₃	—	2	1	no	—
72	C₃	—	2	1	no	—
73	C₂	—	9	2	no	—
74	C₂	—	12	2	no	—
75	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	C₃²×C₄	D₄
socle	C₂×C₃²	C₂×D₄
prime radical	#3802	C₂
hypercenter	C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	C₂×SD₁₆
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. #3797	C₂	semidirect
3. #3787	C₂	semidirect
4. #3775	C₂	semidirect
9. #627	C₂²	semidirect
12. #622	C₂²	semidirect
33. C₃⋊D₃	SD₁₆	semidirect
50. C₃²	C₂×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3802	2
#3797	2
#3787	2
#3775	2
C₂×SD₁₆	9