Subgroups — C₂×#3777 #64246

Order 288.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#157	C₂	1	2	yes	maximal
2	C₂²×#157	C₂	1	1	yes	maximal, characteristic
3	#3777	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂×#157	—	2	4	no	—
6	C₂×#157	C₄	1	4	yes	complemented
7	C₂×#157	C₂²	1	4	yes	—
8	C₂×#157	C₂²	1	2	yes	—
9	C₂²×C₃²	Q₈	1	1	yes	characteristic
10	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	—
11	C₂×C₃⋊D₃	D₄	1	2	yes	—
12	C₂×C₃⋊D₃	Q₈	1	1	yes	characteristic
13	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
14	#157	—	2	4	no	—
15	#157	C₂×C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄	—	9	1	no	maximal, Sylow
17	C₂²×D₃	—	12	1	no	—
18	C₂×C₃²	C₄⋊C₄	1	2	yes	complemented
19	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
20	C₃⋊D₃	C₄⋊C₄	1	2	yes	complemented
21	C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
22	C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄	—	9	1	no	—
24	C₂²×C₄	—	9	2	no	—
25	C₄⋊C₄	—	9	4	no	—
26	C₂²×C₃	—	4	1	no	—
27	C₂×D₃	—	12	2	no	—
28	C₂×D₃	—	12	2	no	—
29	C₂×D₃	—	12	1	no	—
30	C₂×D₃	—	12	1	no	—
31	C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
32	C₂³	—	9	1	no	—
33	C₂×C₄	—	9	2	no	—
34	C₂×C₄	—	9	4	no	—
35	C₂×C₄	—	9	4	no	—
36	C₂×C₄	—	18	4	no	—
37	C₂×C₃	—	4	1	no	—
38	C₂×C₃	—	4	2	no	—
39	D₃	—	12	2	no	—
40	D₃	—	12	1	no	—
41	D₃	—	12	1	no	—
42	C₂²	—	9	1	no	—
43	C₂²	—	9	1	no	—
44	C₂²	—	9	2	no	—
45	C₂²	—	9	2	no	—
46	C₂²	PSU(3,𝔽₂)	1	1	yes	characteristic
47	C₄	—	9	4	no	—
48	C₄	—	18	4	no	—
49	C₃	—	4	1	no	—
50	C₂	—	9	1	no	—
51	C₂	—	9	1	no	—
52	C₂	—	9	2	no	—
53	C₂	#3777	1	2	yes	direct product factor
54	C₂	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	PSU(3,𝔽₂)
commutator	C₃⋊D₃	C₂²×C₄
Frattini	C₂	C₂×PSU(3,𝔽₂)
Fitting, socle	C₂²×C₃²	Q₈
prime radical	C₂²×C₃⋊D₃	C₂²
cosocle	C₂×C₃⋊D₃	C₂³
hypocenter	C₃²	C₂×C₄⋊C₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
3. #3777	C₂	direct (N × H)
6. C₂×#157	C₄	semidirect
15. #157	C₂×C₄	semidirect
18. C₂×C₃²	C₄⋊C₄	semidirect
20. C₃⋊D₃	C₄⋊C₄	semidirect
31. C₃²	C₂×C₄⋊C₄	semidirect
53. C₂	#3777	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#157	2
#3777	2
C₂×C₄⋊C₄	9

Subgroups — C2×#3777 #64246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3777 #64246