Subgroups — C₉×#4696 #13141437

Order 1728.

434 conjugacy classes of subgroups.

The subgroup lattice has 246 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, complemented
2	C₂×C₉×D₂₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#204	C₂	1	2	yes	maximal, complemented
4	C₉×#875	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4696	C₃	1	1	yes	maximal, characteristic
7	C₉×#495	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₂²	1	2	yes	—
9	C₂×C₉×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₉×(C₂×D₃)⋊C₄	—	2	4	no	—
14	C₉×Dic₁₂	—	2	2	no	—
15	C₉×D₂₄	—	2	2	no	—
16	C₉×#204	—	2	4	no	—
17	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×D₂₄	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#204	C₂×C₃	1	2	yes	—
20	C₃×#875	C₂×C₃	1	2	yes	—
21	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₉×Q₁₆	—	3	1	no	—
23	C₂×C₉×SD₁₆	—	3	2	no	—
24	C₂×C₉×D₈	—	3	1	no	—
25	C₉×#124	—	3	2	no	—
26	C₄×C₈×C₉	—	2	1	no	—
27	C₄×C₈×C₉	D₃	1	1	yes	characteristic, complemented
28	C₂²×D₃×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₉×Dic₃	—	2	2	no	—
32	C₉×D₁₂	—	2	2	no	—
33	C₉×D₁₂	—	2	2	no	—
34	C₉×Dic₆	—	2	2	no	—
35	C₉×Dic₆	—	2	2	no	—
36	C₃×C₈×C₉	D₄	1	2	yes	—
37	C₃×C₈×C₉	D₄	1	2	yes	—
38	C₃×#495	—	3	1	no	—
39	#4696	C₉	1	1	yes	characteristic, direct product factor
40	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
41	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
42	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
45	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
46	C₃×Dic₁₂	—	2	2	no	—
47	C₃×D₂₄	—	2	2	no	—
48	C₃×#204	—	2	4	no	—
49	C₂×Q₈×C₉	—	3	2	no	—
50	C₂×D₄×C₉	—	3	2	no	—
51	C₉×Q₁₆	—	6	2	no	—
52	C₉×SD₁₆	—	6	4	no	—
53	C₉×D₈	—	6	2	no	—
54	C₂×C₈×C₉	—	2	1	no	—
55	C₂×C₈×C₉	—	2	1	no	—
56	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
57	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×C₂²⋊C₄	—	6	4	no	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₂×D₃×C₉	—	4	2	no	—
63	C₂×D₃×C₉	—	4	4	no	—
64	C₃×C₄×C₉	—	2	2	no	—
65	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₉×Dic₃	—	4	2	no	—
68	C₂×C₃×Q₁₆	—	3	1	no	—
69	C₂×C₃×SD₁₆	—	3	2	no	—
70	C₂×C₃×D₈	—	3	1	no	—
71	C₃×#124	—	3	2	no	—
72	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
73	C₂×D₂₄	C₂×C₉	1	1	yes	characteristic
74	C₂×#204	C₂×C₉	1	2	yes	complemented
75	#875	C₂×C₉	1	2	yes	complemented
76	C₃×C₄×C₈	—	2	1	no	—
77	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
78	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
79	C₂²×C₃×D₃	—	2	2	no	—
80	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	2	2	no	—
83	C₃×D₁₂	—	2	2	no	—
84	C₃×D₁₂	—	2	2	no	—
85	C₃×Dic₆	—	2	2	no	—
86	C₃×Dic₆	—	2	2	no	—
87	C₂³×C₉	—	6	2	no	—
88	C₃²×C₈	C₃×D₄	1	2	yes	—
89	C₃²×C₈	C₃×D₄	1	2	yes	—
90	Q₈×C₉	—	6	2	no	—
91	Q₈×C₉	—	6	2	no	—
92	D₄×C₉	—	6	2	no	—
93	D₄×C₉	—	6	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	6	2	no	—
97	C₂×C₄×C₉	D₁₂	1	2	yes	—
98	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	2	2	no	—
101	C₈×C₉	D₁₂	1	2	yes	—
102	C₈×C₉	D₁₂	1	2	yes	—
103	#495	—	3	1	no	Sylow
104	C₂×C₃×C₉	#127	1	1	yes	characteristic
105	C₂×C₃×C₉	#135	1	2	yes	—
106	D₃×C₉	—	8	2	no	—
107	C₂×C₃×Q₈	—	3	2	no	—
108	C₂×C₃×D₄	—	3	2	no	—
109	C₂×D₁₂	C₂²×C₉	1	2	yes	—
110	C₂×Dic₆	C₂²×C₉	1	2	yes	—
111	C₃×Q₁₆	—	6	2	no	—
112	C₃×SD₁₆	—	6	4	no	—
113	C₃×D₈	—	6	2	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	2	1	no	—
116	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
120	C₃×C₂²⋊C₄	—	6	4	no	—
121	C₃×C₄²	—	2	1	no	—
122	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
123	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
124	(C₂×D₃)⋊C₄	—	2	4	no	—
125	Dic₁₂	—	2	2	no	—
126	D₂₄	—	2	2	no	—
127	#204	—	2	4	no	—
128	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×C₃×D₃	—	4	2	no	—
130	C₂×C₃×D₃	—	4	4	no	—
131	C₃²×C₄	—	2	2	no	—
132	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
133	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
134	C₃×Dic₃	—	4	2	no	—
135	C₂²×C₉	—	2	1	no	—
136	C₂²×C₉	—	12	2	no	—
137	C₂²×C₉	—	12	4	no	—
138	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
139	C₄×C₉	—	2	4	no	—
140	C₄×C₉	—	2	2	no	—
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	12	2	no	—
144	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
145	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
146	C₂×Q₁₆	—	3	1	no	—
147	C₂×SD₁₆	—	3	2	no	—
148	C₂×D₈	—	3	1	no	—
149	#124	—	3	2	no	—
150	C₄×C₈	D₃×C₉	1	1	yes	characteristic, complemented
151	C₃×C₉	#495	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₃	—	6	2	no	—
153	C₂²×D₃	—	2	2	no	—
154	C₃×Q₈	—	6	2	no	—
155	C₃×Q₈	—	6	2	no	—
156	C₃×D₄	—	6	2	no	—
157	C₃×D₄	—	6	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	6	2	no	—
161	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
162	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
163	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
164	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
165	C₂×Dic₃	—	2	2	no	—
166	D₁₂	—	2	2	no	—
167	D₁₂	—	2	2	no	—
168	Dic₆	—	2	2	no	—
169	Dic₆	—	2	2	no	—
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	D₄×C₉	1	2	yes	—
173	C₃×C₈	D₄×C₉	1	2	yes	—
174	C₃×C₈	C₃×D₁₂	1	2	yes	—
175	C₃×C₈	C₃×D₁₂	1	2	yes	—
176	C₂×C₃²	C₃×#127	1	1	yes	characteristic
177	C₂×C₃²	C₃×#135	1	2	yes	—
178	C₃×D₃	—	8	2	no	—
179	C₂×C₉	—	2	2	no	—
180	C₂×C₉	—	2	1	no	—
181	C₂×C₉	—	24	2	no	—
182	C₂×C₉	#874	1	1	yes	characteristic
183	C₂×C₉	#904	1	2	yes	—
184	C₂×Q₈	—	3	2	no	—
185	C₂×D₄	—	3	2	no	—
186	Q₁₆	—	6	2	no	—
187	SD₁₆	—	6	4	no	—
188	D₈	—	6	2	no	—
189	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
190	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
191	C₂²⋊C₄	—	6	4	no	—
192	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	12	4	no	—
195	C₂²×C₃	—	12	2	no	—
196	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
197	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
198	C₂×D₃	—	4	2	no	—
199	C₂×D₃	—	4	4	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	12	2	no	—
206	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
207	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
208	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
209	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
210	Dic₃	—	4	2	no	—
211	C₃²	C₃×#495	1	1	yes	characteristic
212	C₉	—	2	1	no	—
213	C₉	#4696	1	1	yes	characteristic, direct product factor
214	C₂³	—	6	2	no	—
215	Q₈	—	6	2	no	—
216	Q₈	—	6	2	no	—
217	D₄	—	6	2	no	—
218	D₄	—	6	2	no	—
219	C₂×C₄	—	6	2	no	—
220	C₂×C₄	C₉×D₁₂	1	2	yes	—
221	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
222	C₈	C₉×D₁₂	1	2	yes	—
223	C₈	C₉×D₁₂	1	2	yes	—
224	C₂×C₃	—	2	2	no	—
225	C₂×C₃	—	2	1	no	—
226	C₂×C₃	—	24	2	no	—
227	C₂×C₃	C₉×#127	1	1	yes	characteristic
228	C₂×C₃	C₉×#135	1	2	yes	—
229	C₂×C₃	C₃×#874	1	1	yes	characteristic
230	C₂×C₃	C₃×#904	1	2	yes	—
231	D₃	—	8	2	no	—
232	C₂²	—	12	4	no	—
233	C₂²	—	12	2	no	—
234	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	12	2	no	—
237	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
238	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
239	C₃	—	2	1	no	—
240	C₃	C₉×#495	1	1	yes	characteristic, complemented
241	C₃	C₃×#4696	1	1	yes	characteristic
242	C₂	—	24	2	no	—
243	C₂	C₉×#874	1	1	yes	characteristic
244	C₂	C₉×#904	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₂₄	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₉×#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×Dic₁₂	C₂	semidirect
3. C₂×C₉×#204	C₂	semidirect
4. C₉×#875	C₂	semidirect
5. C₃×C₄×C₈×C₉	C₂	semidirect
27. C₄×C₈×C₉	D₃	semidirect
39. #4696	C₉	direct (N × H)
72. C₂×Dic₁₂	C₂×C₉	semidirect
74. C₂×#204	C₂×C₉	semidirect
75. #875	C₂×C₉	semidirect
77. C₃×C₄×C₈	C₂×C₉	semidirect
150. C₄×C₈	D₃×C₉	semidirect
151. C₃×C₉	#495	semidirect
213. C₉	#4696	direct (N × H)
240. C₃	C₉×#495	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
C₂×C₉×D₂₄	2
C₂×C₉×#204	2
C₉×#875	2
C₃×C₄×C₈×C₉	2
C₃×#4696	3
C₉×#495	3

Subgroups — C9×#4696 #13141437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4696 #13141437