Subgroups — C₄×GO⁺(4,𝔽₂) #64229

Order 288.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×GO⁺(4,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃²	C₂	1	2	yes	maximal, complemented
3	C₄×#157	C₂	1	1	yes	maximal, characteristic, complemented
4	#3773	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂×D₃²	C₂²	1	2	yes	—
7	C₂×#157	C₂²	1	1	yes	characteristic
8	C₂×#157	C₂²	1	1	yes	characteristic
9	GO⁺(4,𝔽₂)	C₄	1	4	yes	direct product factor
10	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₃×C₄×D₃	—	2	2	no	—
12	#624	C₂²	1	2	yes	—
13	D₃×Dic₃	—	2	2	no	—
14	C₂×C₄×D₃	—	6	2	no	—
15	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
16	C₂×C₃×D₃	—	2	2	no	—
17	D₃²	—	2	2	no	—
18	D₃²	C₂×C₄	1	4	yes	complemented
19	#157	—	2	1	no	—
20	#157	C₂×C₄	1	2	yes	complemented
21	C₃²×C₄	D₄	1	1	yes	characteristic, complemented
22	C₃⋊Dic₃	D₄	1	1	yes	characteristic, complemented
23	C₃×Dic₃	—	2	2	no	—
24	C₄×D₄	—	9	1	no	maximal, Sylow
25	C₂²×D₃	—	6	2	no	—
26	C₂×C₃×C₄	—	6	2	no	—
27	C₂×Dic₃	—	6	2	no	—
28	C₄×D₃	—	2	2	no	—
29	C₄×D₃	—	6	2	no	—
30	C₄×D₃	—	6	2	no	—
31	C₄×D₃	—	6	2	no	—
32	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
33	C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
34	C₃⋊D₃	#41	1	1	yes	characteristic
35	C₃×D₃	—	2	4	no	—
36	C₂×D₄	—	9	1	no	—
37	C₂²×C₄	—	9	2	no	—
38	C₄⋊C₄	—	9	1	no	—
39	C₂²⋊C₄	—	9	2	no	—
40	C₄²	—	9	1	no	—
41	C₂²×C₃	—	6	2	no	—
42	C₂×D₃	—	2	2	no	—
43	C₂×D₃	—	6	4	no	—
44	C₂×D₃	—	6	2	no	—
45	C₂×D₃	—	6	4	no	—
46	C₃×C₄	—	2	2	no	—
47	C₃×C₄	—	6	2	no	—
48	Dic₃	—	2	2	no	—
49	Dic₃	—	6	2	no	—
50	C₃²	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
51	C₂³	—	9	2	no	—
52	D₄	—	9	4	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	—	9	2	no	—
55	C₂×C₄	—	9	1	no	—
56	C₂×C₄	—	9	1	no	—
57	C₂×C₄	—	9	1	no	—
58	C₂×C₄	—	18	2	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	—	6	4	no	—
61	D₃	—	2	4	no	—
62	D₃	—	6	2	no	—
63	D₃	—	6	2	no	—
64	C₂²	—	6	2	no	—
65	C₂²	—	9	4	no	—
66	C₂²	—	9	1	no	—
67	C₂²	—	18	2	no	—
68	C₄	—	6	2	no	—
69	C₄	—	9	2	no	—
70	C₄	—	9	1	no	—
71	C₄	—	18	1	no	—
72	C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic, direct product factor
73	C₃	—	2	2	no	—
74	C₂	—	6	4	no	—
75	C₂	—	9	1	no	—
76	C₂	—	9	1	no	—
77	C₂	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₂²×C₄
Frattini	C₂	C₂×GO⁺(4,𝔽₂)
Fitting	C₃²×C₄	D₄
socle	C₂×C₃²	C₂×D₄
prime radical	C₂×GO⁺(4,𝔽₂)	C₂
cosocle	C₂×C₃⋊D₃	C₂³
hypocenter	C₃²	C₄×D₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₄×D₃²	C₂	semidirect
3. C₄×#157	C₂	semidirect
4. #3773	C₂	semidirect
5. D₃²⋊C₄	C₂	semidirect
9. GO⁺(4,𝔽₂)	C₄	direct (N × H)
18. D₃²	C₂×C₄	semidirect
20. #157	C₂×C₄	semidirect
21. C₃²×C₄	D₄	semidirect
22. C₃⋊Dic₃	D₄	semidirect
50. C₃²	C₄×D₄	semidirect
72. C₄	GO⁺(4,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×GO⁺(4,𝔽₂)	2
C₄×D₃²	2
C₄×#157	2
#3773	2
D₃²⋊C₄	2
C₄×D₄	9

Subgroups — C4×GO+(4,𝔽2) #64229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×GO⁺(4,𝔽₂) #64229