Subgroups — #63369

Order 288.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#809	—	3	1	no	maximal
5	C₂²×D₉	—	2	1	no	—
6	C₂×C₄×C₉	—	2	1	no	—
7	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
8	D₃₆	—	2	1	no	—
9	D₃₆	C₄	1	2	yes	complemented
10	C₉⋊C₈	—	2	1	no	—
11	C₂×D₁₂	—	3	1	no	—
12	C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
13	C₂×C₃⋊C₈	—	3	1	no	—
14	C₂²×C₉	D₄	1	1	yes	characteristic
15	C₂×D₉	—	2	2	no	—
16	C₂×D₉	—	4	1	no	—
17	C₂×D₉	—	4	1	no	—
18	C₄×C₉	—	4	1	no	—
19	C₄×C₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₉	D₄	1	1	yes	characteristic
21	#102	—	9	1	no	Sylow
22	C₂²×D₃	—	6	1	no	—
23	C₂×C₃×C₄	—	2	1	no	—
24	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
25	D₁₂	—	3	2	no	—
26	D₁₂	—	6	1	no	—
27	C₃⋊C₈	—	6	1	no	—
28	C₂×C₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₉	D₈	1	1	yes	characteristic
30	C₂×C₉	SD₁₆	1	1	yes	characteristic
31	D₉	—	4	2	no	—
32	C₂×D₄	—	9	1	no	—
33	C₂×C₈	—	9	1	no	—
34	C₄⋊C₄	D₉	1	1	yes	characteristic, complemented
35	C₂²×C₃	#67	1	1	yes	characteristic
36	C₂×D₃	—	6	2	no	—
37	C₂×D₃	—	12	1	no	—
38	C₂×D₃	—	12	1	no	—
39	C₃×C₄	—	4	1	no	—
40	C₃×C₄	C₄×D₃	1	1	yes	characteristic
41	C₃×C₄	D₁₂	1	1	yes	characteristic
42	C₉	#102	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	18	1	no	—
44	D₄	—	9	2	no	—
45	D₄	—	18	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	C₂×D₉	1	1	yes	characteristic
48	C₈	—	18	1	no	—
49	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
50	C₂×C₃	#213	1	1	yes	characteristic
51	C₂×C₃	#215	1	1	yes	characteristic
52	D₃	—	12	2	no	—
53	C₂²	—	18	2	no	—
54	C₂²	—	36	1	no	—
55	C₂²	—	36	1	no	—
56	C₂²	#611	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	C₄×D₉	1	1	yes	characteristic
59	C₄	D₃₆	1	1	yes	characteristic
60	C₃	#809	1	1	yes	characteristic
61	C₂	—	36	2	no	—
62	C₂	(C₂×D₉)⋊C₄	1	1	yes	characteristic
63	C₂	#3673	1	1	yes	characteristic
64	C₂	#3675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#611
commutator	C₄×C₉	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×D₃
Fitting	C₉×C₄⋊C₄	C₂
socle	C₂²×C₃	#67
prime radical	C₂×D₃₆	C₂
cosocle	C₂×C₄×C₉	C₂²
hypercenter	C₄⋊C₄	D₉
hypocenter	C₉	#102
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₉×C₄⋊C₄	C₂	semidirect
3. C₂×C₉⋊C₈	C₂	semidirect
9. D₃₆	C₄	semidirect
34. C₄⋊C₄	D₉	semidirect
42. C₉	#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₆	2
C₉×C₄⋊C₄	2
C₂×C₉⋊C₈	2
#809	3