Subgroups — #63650

Order 288.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#3748	C₂	1	1	yes	maximal, characteristic, complemented
4	#823	—	3	4	no	maximal
5	C₂²×C₃⋊D₃	C₄	1	2	yes	—
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₄⋊(C₃⋊D₃)	—	2	1	no	—
8	C₄⋊(C₃⋊D₃)	—	2	1	no	—
9	C₃²×C₈	—	2	1	no	—
10	C₃⋊(C₃⋊C₈)	—	2	1	no	—
11	C₂×D₁₂	—	3	4	no	—
12	C₃×M₁₆	D₃	1	4	yes	complemented
13	#208	—	3	4	no	—
14	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃⋊D₃	—	2	2	no	—
16	C₂×C₃⋊D₃	—	4	2	no	—
17	C₃²×C₄	D₄	1	1	yes	characteristic
18	C₃²×C₄	D₄	1	1	yes	characteristic
19	#100	—	9	1	no	Sylow
20	C₂²×D₃	—	3	8	no	—
21	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
22	D₁₂	—	6	4	no	—
23	D₁₂	—	6	4	no	—
24	C₃×C₈	—	2	4	no	—
25	C₃⋊C₈	—	6	4	no	—
26	C₂×C₃²	—	2	1	no	—
27	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
28	C₃⋊D₃	—	4	2	no	—
29	C₂×D₄	—	9	1	no	—
30	M₁₆	—	9	1	no	—
31	M₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
32	C₂²×C₃	C₄×D₃	1	4	yes	—
33	C₂×D₃	—	6	8	no	—
34	C₂×D₃	—	12	8	no	—
35	C₃×C₄	D₁₂	1	4	yes	—
36	C₃×C₄	#67	1	4	yes	—
37	C₃²	#100	1	1	yes	characteristic, complemented, Sylow
38	C₂³	—	9	2	no	—
39	D₄	—	18	1	no	—
40	D₄	—	18	1	no	—
41	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
42	C₈	—	2	1	no	—
43	C₈	—	18	1	no	—
44	C₂×C₃	—	2	4	no	—
45	C₂×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
46	D₃	—	12	8	no	—
47	C₂²	—	18	2	no	—
48	C₂²	—	36	2	no	—
49	C₂²	C₄×C₃⋊D₃	1	1	yes	characteristic
50	C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
51	C₄	#638	1	1	yes	characteristic
52	C₃	#823	1	4	yes	complemented
53	C₂	—	2	1	no	—
54	C₂	—	36	2	no	—
55	C₂	#3752	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3752
commutator	C₂²×C₃²	C₂×C₄
Frattini	C₂×C₄	C₂×C₃⋊D₃
Fitting	C₃²×M₁₆	C₂
socle	C₂×C₃²	C₂²⋊C₄
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂
cosocle	C₂×C₃²×C₄	C₂²
hypercenter	M₁₆	C₃⋊D₃
hypocenter	C₃²	#100
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃²×M₁₆	C₂	semidirect
3. #3748	C₂	semidirect
12. C₃×M₁₆	D₃	semidirect
31. M₁₆	C₃⋊D₃	semidirect
37. C₃²	#100	semidirect
52. C₃	#823	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊D₃)	2
C₃²×M₁₆	2
#3748	2
#823	3