Subgroups — C₉×#4873 #13142211

Order 1728.

366 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₉×#887	C₂	1	2	yes	maximal
3	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic
4	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₉×#810	C₂	1	2	yes	maximal
6	C₃×#4873	C₃	1	1	yes	maximal, characteristic
7	C₄×C₉×Q₁₆	—	3	1	no	maximal
8	C₂×C₉×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₉×Dic₃⋊C₄	—	2	2	no	—
12	C₄×C₉×Dic₃	—	2	2	no	—
13	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₉×Dic₁₂	C₄	1	4	yes	complemented
16	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
17	C₃×#887	C₂×C₃	1	2	yes	—
18	C₃²×#107	C₂×C₃	1	1	yes	characteristic
19	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
20	C₃×#810	C₂×C₃	1	2	yes	—
21	C₂×C₉×Q₁₆	—	3	1	no	—
22	C₄×Q₈×C₉	—	3	2	no	—
23	C₉×#107	—	2	1	no	—
24	C₉×#107	D₃	1	1	yes	characteristic
25	C₉×#103	—	3	2	no	—
26	C₄×C₈×C₉	—	3	1	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₉×Dic₃	—	2	2	no	—
30	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
31	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
32	C₉×Dic₆	—	2	2	no	—
33	C₉×Dic₆	C₂×C₄	1	4	yes	—
34	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
35	C₉×C₃⋊C₈	—	2	1	no	—
36	C₃×C₄×Q₁₆	—	3	1	no	—
37	#4873	C₉	1	1	yes	characteristic, direct product factor
38	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
39	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
40	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
41	C₃×Dic₃⋊C₄	—	2	2	no	—
42	C₃×C₄×Dic₃	—	2	2	no	—
43	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
45	C₃×Dic₁₂	C₃×C₄	1	4	yes	—
46	C₂×Q₈×C₉	—	3	2	no	—
47	C₉×Q₁₆	—	3	4	no	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	—	3	1	no	—
50	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
51	C₉×C₄⋊C₄	—	2	2	no	—
52	C₉×C₄⋊C₄	—	6	2	no	—
53	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
54	C₄²×C₉	—	3	1	no	—
55	C₄²×C₉	—	6	2	no	—
56	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
57	C₃×C₄×C₉	—	4	2	no	—
58	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	#41	1	1	yes	characteristic
60	C₉×Dic₃	—	2	4	no	—
61	C₉×Dic₃	—	2	1	no	—
62	C₉×Dic₃	Q₁₆	1	2	yes	complemented
63	C₂×C₃×Q₁₆	—	3	1	no	—
64	C₃×C₄×Q₈	—	3	2	no	—
65	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic
66	#887	C₂×C₉	1	2	yes	—
67	C₃×#107	—	2	1	no	—
68	C₃×#107	C₂×C₉	1	1	yes	characteristic
69	C₃×#107	C₃×D₃	1	1	yes	characteristic
70	C₃×#103	—	3	2	no	—
71	C₃×C₄×C₈	—	3	1	no	—
72	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic
73	#810	C₂×C₉	1	2	yes	—
74	C₂×C₃²×C₄	—	2	2	no	—
75	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₃×Dic₃	—	2	2	no	—
77	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
78	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
79	C₃×Dic₆	—	2	2	no	—
80	C₃×Dic₆	C₂×C₃×C₄	1	4	yes	—
81	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
82	C₃×C₃⋊C₈	—	2	1	no	—
83	Q₈×C₉	—	3	4	no	—
84	Q₈×C₉	—	6	2	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	3	1	no	—
88	C₂×C₄×C₉	—	3	1	no	—
89	C₂×C₄×C₉	—	4	2	no	—
90	C₂×C₄×C₉	—	6	2	no	—
91	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
92	C₈×C₉	—	2	2	no	—
93	C₈×C₉	—	6	1	no	—
94	C₈×C₉	C₄×D₃	1	2	yes	—
95	C₄×Q₁₆	—	3	1	no	Sylow
96	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₃×C₉	#135	1	1	yes	characteristic
99	C₂×C₃×Q₈	—	3	2	no	—
100	C₂×Dic₆	C₂²×C₉	1	2	yes	—
101	C₃×Q₁₆	—	3	4	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	3	1	no	—
104	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	6	2	no	—
108	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
109	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
110	C₃×C₄²	—	3	1	no	—
111	C₃×C₄²	—	6	2	no	—
112	Dic₃⋊C₄	—	2	2	no	—
113	C₄×Dic₃	—	2	2	no	—
114	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
115	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
116	Dic₁₂	C₄×C₉	1	4	yes	complemented
117	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
118	C₃²×C₄	—	4	2	no	—
119	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₃²×C₄	C₃×#41	1	1	yes	characteristic
121	C₃×Dic₃	—	2	4	no	—
122	C₃×Dic₃	—	2	1	no	—
123	C₃×Dic₃	C₃×Q₁₆	1	2	yes	—
124	C₂²×C₉	—	2	1	no	—
125	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
126	C₄×C₉	—	2	1	no	—
127	C₄×C₉	—	2	1	no	—
128	C₄×C₉	—	3	2	no	—
129	C₄×C₉	—	4	2	no	—
130	C₄×C₉	—	6	4	no	—
131	C₄×C₉	—	6	1	no	—
132	C₄×C₉	—	8	2	no	—
133	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
134	C₄×C₉	#239	1	1	yes	characteristic
135	C₂×Q₁₆	—	3	1	no	—
136	C₄×Q₈	—	3	2	no	—
137	#107	D₃×C₉	1	1	yes	characteristic
138	#103	—	3	2	no	—
139	C₄×C₈	—	3	1	no	—
140	C₃×C₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
141	C₃×Q₈	—	3	4	no	—
142	C₃×Q₈	—	6	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	3	1	no	—
147	C₂×C₃×C₄	—	3	1	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	—	6	2	no	—
150	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
152	C₂×Dic₃	—	2	2	no	—
153	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
154	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
155	Dic₆	—	2	2	no	—
156	Dic₆	C₂×C₄×C₉	1	4	yes	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	6	1	no	—
159	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
160	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
161	C₃⋊C₈	—	2	1	no	—
162	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
163	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
164	C₂×C₃²	C₃×#135	1	1	yes	characteristic
165	C₂×C₉	—	2	1	no	—
166	C₂×C₉	—	2	1	no	—
167	C₂×C₉	—	2	1	no	—
168	C₂×C₉	#893	1	1	yes	characteristic
169	C₂×C₉	#912	1	1	yes	characteristic
170	C₂×C₉	D₃×Q₁₆	1	1	yes	characteristic
171	C₂×Q₈	—	3	2	no	—
172	Q₁₆	—	3	4	no	—
173	C₂×C₈	—	3	1	no	—
174	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
175	C₄⋊C₄	—	6	2	no	—
176	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
177	C₄²	—	3	1	no	—
178	C₄²	—	6	2	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
181	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	3	2	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	6	4	no	—
188	C₃×C₄	—	6	1	no	—
189	C₃×C₄	—	8	2	no	—
190	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
191	C₃×C₄	C₉×#41	1	1	yes	characteristic
192	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
193	C₃×C₄	C₃×#239	1	1	yes	characteristic
194	Dic₃	—	2	4	no	—
195	Dic₃	—	2	1	no	—
196	Dic₃	C₉×Q₁₆	1	2	yes	complemented
197	C₃²	C₃×C₄×Q₁₆	1	1	yes	characteristic
198	C₉	—	2	1	no	—
199	C₉	#4873	1	1	yes	characteristic, direct product factor
200	Q₈	—	3	4	no	—
201	Q₈	—	6	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	3	1	no	—
204	C₂×C₄	—	3	1	no	—
205	C₂×C₄	—	6	2	no	—
206	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
207	C₈	—	6	1	no	—
208	C₈	C₄×D₃×C₉	1	2	yes	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
213	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
214	C₂×C₃	C₉×#135	1	1	yes	characteristic
215	C₂×C₃	C₃×#893	1	1	yes	characteristic
216	C₂×C₃	C₃×#912	1	1	yes	characteristic
217	C₂×C₃	C₃×D₃×Q₁₆	1	1	yes	characteristic
218	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
219	C₄	—	3	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	6	1	no	—
222	C₄	—	6	4	no	—
223	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
224	C₄	C₉×#239	1	1	yes	characteristic
225	C₃	—	2	1	no	—
226	C₃	C₄×C₉×Q₁₆	1	1	yes	characteristic, complemented
227	C₃	C₃×#4873	1	1	yes	characteristic
228	C₂	C₉×#893	1	1	yes	characteristic
229	C₂	C₉×#912	1	1	yes	characteristic
230	C₂	D₃×C₉×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×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	C₄×C₉×Q₁₆
LCS	G ▷ 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
15. C₉×Dic₁₂	C₄	semidirect
37. #4873	C₉	direct (N × H)
62. C₉×Dic₃	Q₁₆	semidirect
116. Dic₁₂	C₄×C₉	semidirect
140. C₃×C₉	C₄×Q₁₆	semidirect
196. Dic₃	C₉×Q₁₆	semidirect
199. C₉	#4873	direct (N × H)
226. C₃	C₄×C₉×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×Dic₁₂	2
C₉×#887	2
C₃×C₉×#107	2
C₈×C₉×Dic₃	2
C₉×#810	2
C₃×#4873	3
C₄×C₉×Q₁₆	3

Subgroups — C9×#4873 #13142211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4873 #13142211