Subgroups — #63458

Order 288.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	(C₂×D₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#896	—	3	1	no	maximal
9	C₂²×D₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₉	C₂²	1	1	yes	characteristic
16	C₄×D₉	—	2	2	no	—
17	Dic₁₈	—	2	2	no	—
18	C₂×C₄×D₃	—	3	1	no	—
19	C₂×Dic₆	—	3	1	no	—
20	C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
21	(C₂×D₃)⋊C₄	—	3	1	no	—
22	(C₂×D₃)⋊C₄	—	3	1	no	—
23	Dic₃⋊C₄	—	3	1	no	—
24	Dic₃⋊C₄	—	3	1	no	—
25	C₂²×C₉	C₂³	1	1	yes	characteristic
26	C₂×D₉	—	2	1	no	—
27	C₂×D₉	—	2	1	no	—
28	C₂×D₉	Q₈	1	2	yes	complemented
29	C₄×C₉	—	2	1	no	—
30	C₄×C₉	—	2	1	no	—
31	C₄×C₉	—	2	1	no	—
32	Dic₉	—	2	1	no	—
33	Dic₉	—	2	1	no	—
34	Dic₉	D₄	1	2	yes	—
35	C₂²⋊Q₈	—	9	1	no	Sylow
36	C₂²×D₃	—	3	1	no	—
37	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂×Dic₃	—	3	1	no	—
41	C₂×Dic₃	—	3	1	no	—
42	C₂×Dic₃	—	3	1	no	—
43	C₄×D₃	—	6	2	no	—
44	Dic₆	—	6	2	no	—
45	C₂×C₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₉	#41	1	1	yes	characteristic
48	D₉	—	2	2	no	—
49	C₂×Q₈	—	9	1	no	—
50	C₂²×C₄	—	9	1	no	—
51	C₄⋊C₄	—	9	1	no	—
52	C₄⋊C₄	—	9	1	no	—
53	C₄⋊C₄	D₉	1	1	yes	characteristic, complemented
54	C₂²⋊C₄	—	9	1	no	—
55	C₂²⋊C₄	—	9	1	no	—
56	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
57	C₂×D₃	—	3	2	no	—
58	C₂×D₃	—	6	1	no	—
59	C₂×D₃	—	6	1	no	—
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	2	1	no	—
63	Dic₃	—	3	2	no	—
64	Dic₃	—	6	1	no	—
65	Dic₃	—	6	1	no	—
66	C₉	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
67	C₂³	—	9	1	no	—
68	Q₈	—	18	2	no	—
69	C₂×C₄	—	9	1	no	—
70	C₂×C₄	—	9	1	no	—
71	C₂×C₄	—	9	1	no	—
72	C₂×C₄	—	18	2	no	—
73	C₂×C₄	C₂×D₉	1	1	yes	characteristic
74	C₂×C₄	C₂×D₉	1	1	yes	characteristic
75	C₂×C₄	C₂×D₉	1	1	yes	characteristic
76	C₂×C₃	#235	1	1	yes	characteristic
77	C₂×C₃	D₃×D₄	1	1	yes	characteristic
78	C₂×C₃	D₃×Q₈	1	1	yes	characteristic
79	D₃	—	6	2	no	—
80	C₂²	—	9	2	no	—
81	C₂²	—	18	1	no	—
82	C₂²	—	18	1	no	—
83	C₂²	C₂²×D₉	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	9	2	no	—
88	C₄	—	18	1	no	—
89	C₄	—	18	1	no	—
90	C₃	#896	1	1	yes	characteristic
91	C₂	—	18	2	no	—
92	C₂	#3697	1	1	yes	characteristic
93	C₂	D₄×D₉	1	1	yes	characteristic
94	C₂	Q₈×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂³
Frattini, socle	C₂²×C₃	C₂²×D₃
Fitting	C₉×C₄⋊C₄	C₂
prime radical	C₂²×D₉	C₂²
hypercenter	C₄⋊C₄	D₉
hypocenter	C₉	C₂²⋊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
2. C₂×Dic₁₈	C₂	semidirect
3. C₉×C₄⋊C₄	C₂	semidirect
6. Dic₉⋊C₄	C₂	semidirect
7. Dic₉⋊C₄	C₂	semidirect
28. C₂×D₉	Q₈	semidirect
53. C₄⋊C₄	D₉	semidirect
66. C₉	C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₉	2
C₂×Dic₁₈	2
C₉×C₄⋊C₄	2
(C₂×D₉)⋊C₄	2
Dic₉⋊C₄	2
#896	3