Subgroups — C₉×#4694 #13141435

Order 1728.

398 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	C₉×#874	C₂	1	1	yes	maximal, characteristic
2	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	4	yes	maximal, complemented
5	C₃×#4694	C₃	1	1	yes	maximal, characteristic
6	C₉×#486	—	3	1	no	maximal
7	C₂×C₉×D₁₂	—	2	1	no	—
8	C₂×C₉×D₁₂	—	2	1	no	—
9	C₂×C₉×D₁₂	C₂²	1	2	yes	—
10	C₂×C₉×Dic₆	—	2	1	no	—
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊Dic₃	—	2	1	no	—
14	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₃×#874	C₂×C₃	1	1	yes	characteristic
16	C₃×#869	C₂×C₃	1	1	yes	characteristic
17	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×#821	C₂×C₃	1	4	yes	—
19	C₉×C₄⋊Q₈	—	3	1	no	—
20	C₉×#127	—	3	1	no	—
21	C₉×#102	—	3	4	no	—
22	C₄×C₈×C₉	—	2	1	no	—
23	C₄×C₈×C₉	D₃	1	1	yes	characteristic, complemented
24	C₂²×D₃×C₉	—	2	2	no	—
25	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₉×Dic₃	—	2	2	no	—
29	C₉×D₁₂	—	2	4	no	—
30	C₉×D₁₂	—	4	2	no	—
31	C₉×D₁₂	—	4	2	no	—
32	C₉×Dic₆	—	4	2	no	—
33	C₃×C₈×C₉	—	2	2	no	—
34	C₃×#486	—	3	1	no	—
35	#4694	C₉	1	1	yes	characteristic, direct product factor
36	C₂×C₃×D₁₂	—	2	1	no	—
37	C₂×C₃×D₁₂	—	2	1	no	—
38	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
39	C₂×C₃×Dic₆	—	2	1	no	—
40	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
41	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
42	C₃×C₄⋊Dic₃	—	2	1	no	—
43	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
44	C₂×Q₈×C₉	—	6	1	no	—
45	C₂×D₄×C₉	—	3	2	no	—
46	C₂×D₄×C₉	—	6	1	no	—
47	C₂×D₄×C₉	—	6	1	no	—
48	C₂×C₈×C₉	—	2	2	no	—
49	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
50	C₉×C₄⋊C₄	—	3	2	no	—
51	C₉×C₄⋊C₄	—	6	1	no	—
52	C₄²×C₉	—	2	1	no	—
53	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
55	C₂×D₃×C₉	—	4	2	no	—
56	C₂×D₃×C₉	—	4	2	no	—
57	C₂×D₃×C₉	—	4	2	no	—
58	C₃×C₄×C₉	D₈	1	2	yes	—
59	C₃×C₄×C₉	SD₁₆	1	2	yes	—
60	C₃×C₄×C₉	#41	1	2	yes	—
61	C₉×Dic₃	—	4	2	no	—
62	C₃×C₄⋊Q₈	—	3	1	no	—
63	C₃×#127	—	3	1	no	—
64	#874	C₂×C₉	1	1	yes	characteristic
65	#869	C₂×C₉	1	1	yes	characteristic, complemented
66	C₃×#102	—	3	4	no	—
67	C₃×C₄×C₈	—	2	1	no	—
68	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
69	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
70	#821	C₂×C₉	1	4	yes	complemented
71	C₂²×C₃×D₃	—	2	2	no	—
72	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₃×Dic₃	—	2	2	no	—
76	C₃×D₁₂	—	2	4	no	—
77	C₃×D₁₂	—	4	2	no	—
78	C₃×D₁₂	—	4	2	no	—
79	C₃×Dic₆	—	4	2	no	—
80	C₂³×C₉	—	6	2	no	—
81	C₃²×C₈	—	2	2	no	—
82	Q₈×C₉	—	12	2	no	—
83	D₄×C₉	—	6	4	no	—
84	D₄×C₉	—	12	2	no	—
85	D₄×C₉	—	12	2	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	1	no	—
89	C₂×C₄×C₉	—	6	2	no	—
90	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
91	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
93	C₈×C₉	—	2	4	no	—
94	C₈×C₉	—	2	2	no	—
95	#486	—	3	1	no	Sylow
96	C₂×C₃×C₉	#124	1	1	yes	characteristic
97	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
99	D₃×C₉	—	8	2	no	—
100	C₂×C₃×Q₈	—	6	1	no	—
101	C₂×C₃×D₄	—	3	2	no	—
102	C₂×C₃×D₄	—	6	1	no	—
103	C₂×C₃×D₄	—	6	1	no	—
104	C₂×D₁₂	—	2	1	no	—
105	C₂×D₁₂	—	2	1	no	—
106	C₂×D₁₂	C₂²×C₉	1	2	yes	—
107	C₂×Dic₆	—	2	1	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
110	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
111	C₃×C₄⋊C₄	—	3	2	no	—
112	C₃×C₄⋊C₄	—	6	1	no	—
113	C₃×C₄²	—	2	1	no	—
114	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
115	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
116	C₄⋊Dic₃	—	2	1	no	—
117	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
118	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
119	C₂×C₃×D₃	—	4	2	no	—
120	C₂×C₃×D₃	—	4	2	no	—
121	C₂×C₃×D₃	—	4	2	no	—
122	C₃²×C₄	C₃×D₈	1	2	yes	—
123	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
124	C₃²×C₄	C₃×#41	1	2	yes	—
125	C₃×Dic₃	—	4	2	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	—	12	2	no	—
128	C₂²×C₉	—	12	2	no	—
129	C₂²×C₉	—	12	2	no	—
130	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	12	2	no	—
135	C₄×C₉	#204	1	2	yes	—
136	C₄×C₉	D₂₄	1	2	yes	—
137	C₄×C₉	#235	1	2	yes	—
138	C₄⋊Q₈	—	3	1	no	—
139	#127	—	3	1	no	—
140	#102	—	3	4	no	—
141	C₄×C₈	D₃×C₉	1	1	yes	characteristic, complemented
142	C₃×C₉	#486	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₃	—	6	2	no	—
144	C₂²×D₃	—	2	2	no	—
145	C₃×Q₈	—	12	2	no	—
146	C₃×D₄	—	6	4	no	—
147	C₃×D₄	—	12	2	no	—
148	C₃×D₄	—	12	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	6	2	no	—
153	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
154	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
156	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
157	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
159	C₂×Dic₃	—	2	2	no	—
160	D₁₂	—	2	4	no	—
161	D₁₂	—	4	2	no	—
162	D₁₂	—	4	2	no	—
163	Dic₆	—	4	2	no	—
164	C₃×C₈	—	2	4	no	—
165	C₃×C₈	—	2	2	no	—
166	C₃×C₈	—	2	2	no	—
167	C₂×C₃²	C₃×#124	1	1	yes	characteristic
168	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
169	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
170	C₃×D₃	—	8	2	no	—
171	C₂×C₉	—	2	1	no	—
172	C₂×C₉	—	2	1	no	—
173	C₂×C₉	—	2	1	no	—
174	C₂×C₉	—	24	2	no	—
175	C₂×C₉	#875	1	1	yes	characteristic
176	C₂×C₉	C₂×#204	1	1	yes	characteristic
177	C₂×C₉	C₂×D₂₄	1	1	yes	characteristic
178	C₂×Q₈	—	6	1	no	—
179	C₂×D₄	—	3	2	no	—
180	C₂×D₄	—	6	1	no	—
181	C₂×D₄	—	6	1	no	—
182	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
183	C₄⋊C₄	—	3	2	no	—
184	C₄⋊C₄	—	6	1	no	—
185	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	12	2	no	—
188	C₂²×C₃	—	12	2	no	—
189	C₂²×C₃	—	12	2	no	—
190	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
191	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
192	C₂×D₃	—	4	2	no	—
193	C₂×D₃	—	4	2	no	—
194	C₂×D₃	—	4	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	12	2	no	—
199	C₃×C₄	C₉×D₈	1	2	yes	—
200	C₃×C₄	C₉×SD₁₆	1	2	yes	—
201	C₃×C₄	C₉×#41	1	2	yes	—
202	C₃×C₄	C₃×#204	1	2	yes	—
203	C₃×C₄	C₃×D₂₄	1	2	yes	—
204	C₃×C₄	C₃×#235	1	2	yes	—
205	Dic₃	—	4	2	no	—
206	C₃²	C₃×#486	1	1	yes	characteristic
207	C₉	—	2	1	no	—
208	C₉	#4694	1	1	yes	characteristic, direct product factor
209	C₂³	—	6	2	no	—
210	Q₈	—	12	2	no	—
211	D₄	—	6	4	no	—
212	D₄	—	12	2	no	—
213	D₄	—	12	2	no	—
214	C₂×C₄	—	6	2	no	—
215	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
216	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	24	2	no	—
223	C₂×C₃	C₉×#124	1	1	yes	characteristic
224	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
225	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
226	C₂×C₃	C₃×#875	1	1	yes	characteristic
227	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
228	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
229	D₃	—	8	2	no	—
230	C₂²	—	12	2	no	—
231	C₂²	—	12	2	no	—
232	C₂²	—	12	2	no	—
233	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
234	C₄	—	12	2	no	—
235	C₄	C₉×#204	1	2	yes	—
236	C₄	C₉×D₂₄	1	2	yes	—
237	C₄	C₉×#235	1	2	yes	—
238	C₃	—	2	1	no	—
239	C₃	C₉×#486	1	1	yes	characteristic, complemented
240	C₃	C₃×#4694	1	1	yes	characteristic
241	C₂	—	24	2	no	—
242	C₂	C₉×#875	1	1	yes	characteristic
243	C₂	C₂×C₉×#204	1	1	yes	characteristic
244	C₂	C₂×C₉×D₂₄	1	1	yes	characteristic

	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	C₃×#874	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₉×#486
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
2. C₉×#869	C₂	semidirect
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₉×#821	C₂	semidirect
23. C₄×C₈×C₉	D₃	semidirect
35. #4694	C₉	direct (N × H)
65. #869	C₂×C₉	semidirect
68. C₃×C₄×C₈	C₂×C₉	semidirect
70. #821	C₂×C₉	semidirect
141. C₄×C₈	D₃×C₉	semidirect
142. C₃×C₉	#486	semidirect
208. C₉	#4694	direct (N × H)
239. C₃	C₉×#486	semidirect

Maximal subgroups

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

subgroup	index
C₉×#874	2
C₉×#869	2
C₃×C₄×C₈×C₉	2
C₉×#821	2
C₃×#4694	3
C₉×#486	3

Subgroups — C9×#4694 #13141435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4694 #13141435