Subgroups — C₉×#4874 #13142180

Order 1728.

330 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#888	C₂	1	2	yes	maximal
2	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic
3	C₉×#818	C₂	1	1	yes	maximal, characteristic
4	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₉×#807	C₂	1	2	yes	maximal
6	C₃×#4874	C₃	1	1	yes	maximal, characteristic
7	C₉×#500	—	3	1	no	maximal
8	C₂×C₉×Dic₆	—	2	2	no	—
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₉×Dic₃⋊C₄	—	2	2	no	—
13	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×#888	C₂×C₃	1	2	yes	—
16	C₃²×#107	C₂×C₃	1	1	yes	characteristic
17	C₃×#818	C₂×C₃	1	1	yes	characteristic
18	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₃×#807	C₂×C₃	1	2	yes	—
20	C₉×C₄⋊Q₈	—	3	2	no	—
21	C₉×#107	—	2	1	no	—
22	C₉×#107	—	3	1	no	—
23	C₉×#107	—	3	2	no	—
24	C₉×#107	D₃	1	1	yes	characteristic
25	C₄×C₈×C₉	—	3	1	no	—
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₉×Dic₃	—	2	2	no	—
29	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
30	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
31	C₉×Dic₆	—	4	4	no	—
32	C₃×C₈×C₉	Q₈	1	2	yes	complemented
33	C₉×C₃⋊C₈	Q₈	1	2	yes	complemented
34	C₃×#500	—	3	1	no	—
35	#4874	C₉	1	1	yes	characteristic, direct product factor
36	C₂×C₃×Dic₆	—	2	2	no	—
37	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
38	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
39	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
40	C₃×Dic₃⋊C₄	—	2	2	no	—
41	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×Q₈×C₉	—	6	2	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	—	3	1	no	—
46	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	3	2	no	—
49	C₉×C₄⋊C₄	—	6	2	no	—
50	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₄²×C₉	—	3	1	no	—
52	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
53	C₃×C₄×C₉	—	4	2	no	—
54	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
55	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₉×Dic₃	—	4	2	no	—
57	C₉×Dic₃	D₈	1	2	yes	—
58	C₉×Dic₃	Q₁₆	1	2	yes	—
59	C₃×C₄⋊Q₈	—	3	2	no	—
60	#888	C₂×C₉	1	2	yes	—
61	C₃×#107	—	2	1	no	—
62	C₃×#107	—	3	1	no	—
63	C₃×#107	—	3	2	no	—
64	C₃×#107	C₂×C₉	1	1	yes	characteristic
65	C₃×#107	C₃×D₃	1	1	yes	characteristic
66	C₃×C₄×C₈	—	3	1	no	—
67	#818	C₂×C₉	1	1	yes	characteristic
68	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic
69	#807	C₂×C₉	1	2	yes	—
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂×C₃×Dic₃	—	2	2	no	—
73	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
75	C₃×Dic₆	—	4	4	no	—
76	C₃²×C₈	C₃×Q₈	1	2	yes	—
77	C₃×C₃⋊C₈	C₃×Q₈	1	2	yes	—
78	Q₈×C₉	—	12	4	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	1	no	—
81	C₂×C₄×C₉	—	3	1	no	—
82	C₂×C₄×C₉	—	3	1	no	—
83	C₂×C₄×C₉	—	4	2	no	—
84	C₂×C₄×C₉	—	6	2	no	—
85	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
86	C₈×C₉	—	2	2	no	—
87	C₈×C₉	—	3	2	no	—
88	C₈×C₉	Dic₆	1	2	yes	complemented
89	#500	—	3	1	no	Sylow
90	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
92	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₃×Q₈	—	6	2	no	—
94	C₂×Dic₆	—	2	2	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	3	1	no	—
97	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
99	C₃×C₄⋊C₄	—	2	2	no	—
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₄⋊C₄	—	6	2	no	—
102	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
103	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
104	C₃×C₄²	—	3	1	no	—
105	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
106	Dic₃⋊C₄	—	2	2	no	—
107	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
108	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
109	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
110	C₃²×C₄	—	4	2	no	—
111	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
112	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
113	C₃×Dic₃	—	4	2	no	—
114	C₃×Dic₃	C₃×D₈	1	2	yes	—
115	C₃×Dic₃	C₃×Q₁₆	1	2	yes	—
116	C₂²×C₉	—	2	1	no	—
117	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
118	C₄×C₉	—	2	1	no	—
119	C₄×C₉	—	2	1	no	—
120	C₄×C₉	—	3	2	no	—
121	C₄×C₉	—	3	2	no	—
122	C₄×C₉	—	4	2	no	—
123	C₄×C₉	—	8	2	no	—
124	C₄×C₉	—	12	2	no	—
125	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
126	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
127	C₄⋊Q₈	—	3	2	no	—
128	#107	—	3	1	no	—
129	#107	—	3	2	no	—
130	#107	D₃×C₉	1	1	yes	characteristic
131	C₄×C₈	—	3	1	no	—
132	C₃×C₉	#500	1	1	yes	characteristic, complemented, Sylow
133	C₃×Q₈	—	12	4	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	3	1	no	—
138	C₂×C₃×C₄	—	3	1	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	6	2	no	—
141	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
143	C₂×Dic₃	—	2	2	no	—
144	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
145	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
146	Dic₆	—	4	4	no	—
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	—	3	2	no	—
149	C₃×C₈	Q₈×C₉	1	2	yes	complemented
150	C₃×C₈	C₃×Dic₆	1	2	yes	—
151	C₃⋊C₈	Q₈×C₉	1	2	yes	complemented
152	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
153	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
154	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
155	C₂×C₉	—	2	1	no	—
156	C₂×C₉	—	2	1	no	—
157	C₂×C₉	—	2	1	no	—
158	C₂×C₉	#888	1	1	yes	characteristic
159	C₂×C₉	D₃×D₈	1	1	yes	characteristic
160	C₂×C₉	D₃×Q₁₆	1	1	yes	characteristic
161	C₂×Q₈	—	6	2	no	—
162	C₂×C₈	—	3	1	no	—
163	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
164	C₄⋊C₄	—	3	2	no	—
165	C₄⋊C₄	—	6	2	no	—
166	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
167	C₄²	—	3	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
170	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	3	2	no	—
174	C₃×C₄	—	3	2	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	—	8	2	no	—
178	C₃×C₄	—	12	2	no	—
179	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
180	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
181	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
182	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
183	Dic₃	—	4	2	no	—
184	Dic₃	C₉×D₈	1	2	yes	—
185	Dic₃	C₉×Q₁₆	1	2	yes	—
186	C₃²	C₃×#500	1	1	yes	characteristic
187	C₉	—	2	1	no	—
188	C₉	#4874	1	1	yes	characteristic, direct product factor
189	Q₈	—	12	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	3	1	no	—
192	C₂×C₄	—	3	1	no	—
193	C₂×C₄	—	6	2	no	—
194	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
195	C₈	—	3	2	no	—
196	C₈	C₉×Dic₆	1	2	yes	complemented
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
202	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
203	C₂×C₃	C₃×#888	1	1	yes	characteristic
204	C₂×C₃	C₃×D₃×D₈	1	1	yes	characteristic
205	C₂×C₃	C₃×D₃×Q₁₆	1	1	yes	characteristic
206	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
207	C₄	—	3	2	no	—
208	C₄	—	3	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	12	2	no	—
211	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
212	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
213	C₃	—	2	1	no	—
214	C₃	C₉×#500	1	1	yes	characteristic, complemented
215	C₃	C₃×#4874	1	1	yes	characteristic
216	C₂	C₉×#888	1	1	yes	characteristic
217	C₂	D₃×C₉×D₈	1	1	yes	characteristic
218	C₂	D₃×C₉×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#107	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₉×#500
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#107

Complements

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

N	H ≅ G/N	type
32. C₃×C₈×C₉	Q₈	semidirect
33. C₉×C₃⋊C₈	Q₈	semidirect
35. #4874	C₉	direct (N × H)
88. C₈×C₉	Dic₆	semidirect
132. C₃×C₉	#500	semidirect
149. C₃×C₈	Q₈×C₉	semidirect
151. C₃⋊C₈	Q₈×C₉	semidirect
188. C₉	#4874	direct (N × H)
196. C₈	C₉×Dic₆	semidirect
214. C₃	C₉×#500	semidirect

Maximal subgroups

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

subgroup	index
C₉×#888	2
C₃×C₉×#107	2
C₉×#818	2
C₈×C₉×Dic₃	2
C₉×#807	2
C₃×#4874	3
C₉×#500	3

Subgroups — C9×#4874 #13142180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4874 #13142180