Subgroups — C₉×#4849 #13142043

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 266 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	2	yes	maximal
2	C₉×#869	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₉×#816	C₂	1	2	yes	maximal
6	C₃×#4849	C₃	1	1	yes	maximal, characteristic
7	C₉×#462	—	3	1	no	maximal
8	C₂×C₉×Dic₆	C₂²	1	2	yes	—
9	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Dic₃	—	2	1	no	—
13	C₉×C₄⋊Dic₃	—	2	1	no	—
14	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₉×Dic₃⋊C₄	—	2	1	no	—
16	C₄×C₉×Dic₃	—	2	1	no	—
17	C₉×Dic₁₂	—	2	4	no	—
18	C₂×C₃×Dic₁₂	C₂×C₃	1	2	yes	—
19	C₃×#869	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×#816	C₂×C₃	1	2	yes	—
23	C₂×C₉×Q₁₆	—	3	2	no	—
24	C₉×C₄⋊Q₈	—	3	1	no	—
25	C₄×Q₈×C₉	—	3	1	no	—
26	C₉×C₄⋊C₈	—	2	1	no	—
27	C₉×C₄⋊C₈	D₃	1	1	yes	characteristic
28	C₉×#103	—	3	2	no	—
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₉×Dic₃	—	2	1	no	—
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₉×Dic₃	—	2	1	no	—
35	C₉×Dic₆	—	2	1	no	—
36	C₉×Dic₆	—	2	2	no	—
37	C₉×Dic₆	—	2	2	no	—
38	C₉×Dic₆	D₄	1	2	yes	—
39	C₃×C₈×C₉	—	2	2	no	—
40	C₃×#462	—	3	1	no	—
41	#4849	C₉	1	1	yes	characteristic, direct product factor
42	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
43	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
45	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₃×C₄⋊Dic₃	—	2	1	no	—
47	C₃×C₄⋊Dic₃	—	2	1	no	—
48	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
49	C₃×Dic₃⋊C₄	—	2	1	no	—
50	C₃×C₄×Dic₃	—	2	1	no	—
51	C₃×Dic₁₂	—	2	4	no	—
52	C₂×Q₈×C₉	—	3	1	no	—
53	C₂×Q₈×C₉	—	3	2	no	—
54	C₉×Q₁₆	—	6	4	no	—
55	C₂×C₈×C₉	—	2	2	no	—
56	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
57	C₉×C₄⋊C₄	—	3	1	no	—
58	C₉×C₄⋊C₄	—	6	1	no	—
59	C₉×C₄⋊C₄	—	6	1	no	—
60	C₉×C₄⋊C₄	—	6	1	no	—
61	C₄²×C₉	—	2	1	no	—
62	C₄²×C₉	—	6	1	no	—
63	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₃×C₄×C₉	—	2	1	no	—
66	C₃×C₄×C₉	Q₁₆	1	2	yes	complemented
67	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×C₉	#41	1	1	yes	characteristic
69	C₉×Dic₃	—	2	2	no	—
70	C₉×Dic₃	—	4	2	no	—
71	C₉×Dic₃	—	4	1	no	—
72	C₂×C₃×Q₁₆	—	3	2	no	—
73	C₃×C₄⋊Q₈	—	3	1	no	—
74	C₃×C₄×Q₈	—	3	1	no	—
75	C₂×Dic₁₂	C₂×C₉	1	2	yes	—
76	#869	C₂×C₉	1	1	yes	characteristic
77	C₄×Dic₆	C₂×C₉	1	1	yes	characteristic
78	C₃×C₄⋊C₈	—	2	1	no	—
79	C₃×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
80	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
81	C₃×#103	—	3	2	no	—
82	#816	C₂×C₉	1	2	yes	—
83	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₃×Dic₃	—	2	1	no	—
87	C₂×C₃×Dic₃	—	2	1	no	—
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₃×Dic₆	—	2	1	no	—
90	C₃×Dic₆	—	2	2	no	—
91	C₃×Dic₆	—	2	2	no	—
92	C₃×Dic₆	C₃×D₄	1	2	yes	—
93	C₃²×C₈	—	2	2	no	—
94	Q₈×C₉	—	3	2	no	—
95	Q₈×C₉	—	6	1	no	—
96	Q₈×C₉	—	6	2	no	—
97	Q₈×C₉	—	6	2	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	1	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	6	1	no	—
102	C₂×C₄×C₉	—	6	1	no	—
103	C₂×C₄×C₉	—	6	2	no	—
104	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
105	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
107	C₈×C₉	—	2	2	no	—
108	C₈×C₉	—	4	2	no	—
109	#462	—	3	1	no	Sylow
110	C₂×C₃×C₉	#121	1	1	yes	characteristic
111	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₃×C₉	#137	1	1	yes	characteristic
113	C₂×C₃×Q₈	—	3	1	no	—
114	C₂×C₃×Q₈	—	3	2	no	—
115	C₂×Dic₆	C₂²×C₉	1	2	yes	—
116	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
117	C₃×Q₁₆	—	6	4	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
121	C₃×C₄⋊C₄	—	3	1	no	—
122	C₃×C₄⋊C₄	—	6	1	no	—
123	C₃×C₄⋊C₄	—	6	1	no	—
124	C₃×C₄⋊C₄	—	6	1	no	—
125	C₃×C₄²	—	2	1	no	—
126	C₃×C₄²	—	6	1	no	—
127	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
128	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
129	C₄⋊Dic₃	—	2	1	no	—
130	C₄⋊Dic₃	—	2	1	no	—
131	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
132	Dic₃⋊C₄	—	2	1	no	—
133	C₄×Dic₃	—	2	1	no	—
134	Dic₁₂	—	2	4	no	—
135	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
136	C₃²×C₄	—	2	1	no	—
137	C₃²×C₄	C₃×Q₁₆	1	2	yes	—
138	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
139	C₃²×C₄	C₃×#41	1	1	yes	characteristic
140	C₃×Dic₃	—	2	2	no	—
141	C₃×Dic₃	—	4	2	no	—
142	C₃×Dic₃	—	4	1	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	2	2	no	—
147	C₄×C₉	—	2	1	no	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	4	1	no	—
150	C₄×C₉	—	6	2	no	—
151	C₄×C₉	—	12	2	no	—
152	C₄×C₉	—	12	1	no	—
153	C₄×C₉	Dic₁₂	1	2	yes	complemented
154	C₄×C₉	D₃×D₄	1	1	yes	characteristic
155	C₄×C₉	#239	1	1	yes	characteristic
156	C₂×Q₁₆	—	3	2	no	—
157	C₄⋊Q₈	—	3	1	no	—
158	C₄×Q₈	—	3	1	no	—
159	C₄⋊C₈	D₃×C₉	1	1	yes	characteristic
160	#103	—	3	2	no	—
161	C₃×C₉	#462	1	1	yes	characteristic, complemented, Sylow
162	C₃×Q₈	—	3	2	no	—
163	C₃×Q₈	—	6	1	no	—
164	C₃×Q₈	—	6	2	no	—
165	C₃×Q₈	—	6	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	6	1	no	—
170	C₂×C₃×C₄	—	6	1	no	—
171	C₂×C₃×C₄	—	6	2	no	—
172	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
173	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
175	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	2	no	—
181	Dic₆	—	2	1	no	—
182	Dic₆	—	2	2	no	—
183	Dic₆	—	2	2	no	—
184	Dic₆	D₄×C₉	1	2	yes	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	—	4	2	no	—
188	C₂×C₃²	C₃×#121	1	1	yes	characteristic
189	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
190	C₂×C₃²	C₃×#137	1	1	yes	characteristic
191	C₂×C₉	—	2	1	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	#895	1	1	yes	characteristic
195	C₂×C₉	C₂×Dic₁₂	1	1	yes	characteristic
196	C₂×C₉	#909	1	1	yes	characteristic
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	3	2	no	—
199	Q₁₆	—	6	4	no	—
200	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
201	C₄⋊C₄	—	3	1	no	—
202	C₄⋊C₄	—	6	1	no	—
203	C₄⋊C₄	—	6	1	no	—
204	C₄⋊C₄	—	6	1	no	—
205	C₄²	—	6	1	no	—
206	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
209	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	—	6	2	no	—
217	C₃×C₄	—	12	2	no	—
218	C₃×C₄	—	12	1	no	—
219	C₃×C₄	C₉×Q₁₆	1	2	yes	complemented
220	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
221	C₃×C₄	C₉×#41	1	1	yes	characteristic
222	C₃×C₄	C₃×Dic₁₂	1	2	yes	—
223	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
224	C₃×C₄	C₃×#239	1	1	yes	characteristic
225	Dic₃	—	2	2	no	—
226	Dic₃	—	4	2	no	—
227	Dic₃	—	4	1	no	—
228	C₃²	C₃×#462	1	1	yes	characteristic
229	C₉	—	2	1	no	—
230	C₉	#4849	1	1	yes	characteristic, direct product factor
231	Q₈	—	3	2	no	—
232	Q₈	—	6	1	no	—
233	Q₈	—	6	2	no	—
234	Q₈	—	6	2	no	—
235	C₂×C₄	—	6	1	no	—
236	C₂×C₄	—	6	1	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
239	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	C₉×#121	1	1	yes	characteristic
246	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
247	C₂×C₃	C₉×#137	1	1	yes	characteristic
248	C₂×C₃	C₃×#895	1	1	yes	characteristic
249	C₂×C₃	C₂×C₃×Dic₁₂	1	1	yes	characteristic
250	C₂×C₃	C₃×#909	1	1	yes	characteristic
251	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	6	2	no	—
254	C₄	—	12	1	no	—
255	C₄	—	12	2	no	—
256	C₄	C₉×Dic₁₂	1	2	yes	complemented
257	C₄	D₃×D₄×C₉	1	1	yes	characteristic
258	C₄	C₉×#239	1	1	yes	characteristic
259	C₃	—	2	1	no	—
260	C₃	C₉×#462	1	1	yes	characteristic, complemented
261	C₃	C₃×#4849	1	1	yes	characteristic
262	C₂	C₉×#895	1	1	yes	characteristic
263	C₂	C₂×C₉×Dic₁₂	1	1	yes	characteristic
264	C₂	C₉×#909	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×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₄⋊C₈	D₃
hypocenter	C₃	C₉×#462
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
41. #4849	C₉	direct (N × H)
66. C₃×C₄×C₉	Q₁₆	semidirect
153. C₄×C₉	Dic₁₂	semidirect
161. C₃×C₉	#462	semidirect
219. C₃×C₄	C₉×Q₁₆	semidirect
230. C₉	#4849	direct (N × H)
256. C₄	C₉×Dic₁₂	semidirect
260. C₃	C₉×#462	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×#869	2
C₄×C₉×Dic₆	2
C₃×C₉×C₄⋊C₈	2
C₉×#816	2
C₃×#4849	3
C₉×#462	3

Subgroups — C9×#4849 #13142043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4849 #13142043