Subgroups — #64287

Order 288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3830	C₂	1	1	yes	maximal, characteristic
2	#3792	C₂	1	2	yes	maximal, complemented
3	C₂×#622	C₂	1	2	yes	maximal, complemented
4	#3788	C₂	1	1	yes	maximal, characteristic, complemented
5	#3787	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×D₄	C₄	1	1	yes	characteristic
7	#638	C₄	1	2	yes	—
8	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
9	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	#634	C₄	1	1	yes	characteristic
11	#622	C₂²	1	1	yes	characteristic
12	#622	C₂²	1	1	yes	characteristic
13	#622	C₂²	1	2	yes	—
14	#237	—	6	2	no	—
15	C₂²×C₃²	C₂×C₄	1	2	yes	—
16	C₂×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
17	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
18	C₃⋊Dic₃	C₂×C₄	1	2	yes	—
19	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
20	#131	—	9	1	no	maximal, Sylow
21	C₃×D₄	—	2	2	no	—
22	#67	—	6	4	no	—
23	C₂×Dic₃	—	6	4	no	—
24	C₄×D₃	—	6	2	no	—
25	Dic₆	—	6	2	no	—
26	C₂×C₃²	—	2	2	no	—
27	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
28	C₃⋊D₃	—	2	1	no	—
29	#41	—	9	1	no	—
30	M₁₆	—	9	1	no	—
31	M₁₆	—	9	2	no	—
32	C₂×C₈	—	9	1	no	—
33	C₂×C₈	—	9	2	no	—
34	C₂²×C₃	—	2	4	no	—
35	C₂×D₃	—	6	2	no	—
36	C₃×C₄	—	2	2	no	—
37	Dic₃	—	6	4	no	—
38	Dic₃	—	6	2	no	—
39	C₃²	#131	1	1	yes	characteristic, complemented, Sylow
40	Q₈	—	9	1	no	—
41	D₄	—	9	2	no	—
42	D₄	#157	1	1	yes	characteristic
43	C₂×C₄	—	9	2	no	—
44	C₂×C₄	—	9	1	no	—
45	C₈	—	9	1	no	—
46	C₈	—	9	1	no	—
47	C₈	—	9	2	no	—
48	C₂×C₃	—	2	2	no	—
49	C₂×C₃	—	4	4	no	—
50	D₃	—	12	2	no	—
51	C₂²	—	9	1	no	—
52	C₂²	C₂×#157	1	2	yes	—
53	C₄	—	9	1	no	—
54	C₄	—	9	2	no	—
55	C₄	C₂×#157	1	1	yes	characteristic
56	C₃	—	2	2	no	—
57	C₂	—	2	2	no	—
58	C₂	—	18	1	no	—
59	C₂	C₂²×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#157
commutator, socle	C₂×C₃²	C₂²×C₄
Fitting	C₃²×D₄	C₄
prime radical	#3830	C₂
cosocle	C₃⋊Dic₃	C₂³
hypercenter	D₄	#157
hypocenter	C₃²	#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. #3792	C₂	semidirect
3. C₂×#622	C₂	semidirect
4. #3788	C₂	semidirect
5. #3787	C₂	semidirect
39. C₃²	#131	semidirect

Maximal subgroups

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

subgroup	index
#3830	2
#3792	2
C₂×#622	2
#3788	2
#3787	2
#131	9