Subgroups — #63450

Order 288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₈	C₂	1	2	yes	maximal
2	C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	Dic₉⋊C₄	C₂	1	2	yes	maximal
5	C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
6	#888	—	3	1	no	maximal
7	C₂×C₄×C₉	C₂²	1	2	yes	—
8	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₉	C₂²	1	2	yes	—
10	C₂×Dic₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₉	C₂²	1	1	yes	characteristic
12	Dic₁₈	—	2	4	no	—
13	C₂×Dic₆	—	3	2	no	—
14	C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
15	C₄⋊Dic₃	—	3	1	no	—
16	Dic₃⋊C₄	—	3	2	no	—
17	C₄×Dic₃	—	3	1	no	—
18	C₂²×C₉	C₂³	1	1	yes	characteristic
19	C₄×C₉	—	2	2	no	—
20	C₄×C₉	Q₈	1	2	yes	complemented
21	Dic₉	—	2	2	no	—
22	Dic₉	D₄	1	2	yes	—
23	Dic₉	Q₈	1	2	yes	complemented
24	C₄⋊Q₈	—	9	1	no	Sylow
25	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
26	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₃	—	3	1	no	—
28	C₂×Dic₃	—	3	1	no	—
29	C₂×Dic₃	—	3	2	no	—
30	Dic₆	—	6	4	no	—
31	C₂×C₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×Q₈	—	9	2	no	—
35	C₄⋊C₄	—	9	2	no	—
36	C₄⋊C₄	—	9	1	no	—
37	C₄⋊C₄	D₉	1	1	yes	characteristic
38	C₄²	—	9	1	no	—
39	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
40	C₃×C₄	—	2	2	no	—
41	C₃×C₄	Dic₆	1	2	yes	—
42	Dic₃	—	3	2	no	—
43	Dic₃	—	3	2	no	—
44	Dic₃	—	6	2	no	—
45	C₉	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
46	Q₈	—	18	4	no	—
47	C₂×C₄	—	9	1	no	—
48	C₂×C₄	—	9	1	no	—
49	C₂×C₄	—	9	2	no	—
50	C₂×C₄	C₂×D₉	1	2	yes	—
51	C₂×C₄	C₂×D₉	1	1	yes	characteristic
52	C₂×C₃	C₂×Dic₆	1	1	yes	characteristic
53	C₂×C₃	D₃×D₄	1	1	yes	characteristic
54	C₂×C₃	D₃×Q₈	1	1	yes	characteristic
55	C₂²	C₂²×D₉	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	9	2	no	—
58	C₄	—	9	2	no	—
59	C₄	—	18	2	no	—
60	C₄	Dic₁₈	1	2	yes	complemented
61	C₃	#888	1	1	yes	characteristic
62	C₂	C₂×Dic₁₈	1	1	yes	characteristic
63	C₂	D₄×D₉	1	1	yes	characteristic
64	C₂	Q₈×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂³
Frattini, socle, prime radical	C₂²×C₃	C₂²×D₃
Fitting	C₉×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₉
hypocenter	C₉	C₄⋊Q₈
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
20. C₄×C₉	Q₈	semidirect
23. Dic₉	Q₈	semidirect
45. C₉	C₄⋊Q₈	semidirect
60. C₄	Dic₁₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₈	2
C₉×C₄⋊C₄	2
C₄⋊Dic₉	2
Dic₉⋊C₄	2
C₄×Dic₉	2
#888	3