Subgroups — C₉×#4887 #13142207

Order 1728.

378 conjugacy classes of subgroups.

The subgroup lattice has 268 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×D₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#897	C₂	1	2	yes	maximal
4	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#810	C₂	1	2	yes	maximal, complemented
6	C₃×#4887	C₃	1	1	yes	maximal, characteristic
7	C₉×#467	—	3	1	no	maximal
8	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₉×(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₉×C₄⋊Dic₃	—	2	2	no	—
14	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₉×Dic₁₂	—	2	2	no	—
16	D₃×C₈×C₉	—	2	2	no	—
17	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×D₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×#897	C₂×C₃	1	2	yes	—
20	C₃²×#107	C₂×C₃	1	1	yes	characteristic
21	C₃×#810	C₂×C₃	1	2	yes	—
22	C₂×C₉×Q₁₆	—	3	1	no	—
23	C₂²×C₈×C₉	—	3	1	no	—
24	C₉×C₂²⋊Q₈	—	3	2	no	—
25	C₉×#107	—	2	1	no	—
26	C₉×#107	D₃	1	1	yes	characteristic, complemented
27	C₉×#103	—	3	2	no	—
28	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₉×Dic₃	—	2	2	no	—
32	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
33	C₄×D₃×C₉	—	2	1	no	—
34	C₄×D₃×C₉	—	2	1	no	—
35	C₉×Dic₆	—	2	2	no	—
36	C₉×Dic₆	—	2	2	no	—
37	C₃×C₈×C₉	D₄	1	2	yes	—
38	C₉×C₃⋊C₈	—	2	1	no	—
39	C₃×#467	—	3	1	no	—
40	#4887	C₉	1	1	yes	characteristic, direct product factor
41	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
45	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
46	C₃×C₄⋊Dic₃	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
48	C₃×Dic₁₂	—	2	2	no	—
49	C₃×D₃×C₈	—	2	2	no	—
50	C₂×Q₈×C₉	—	3	2	no	—
51	C₂²×C₄×C₉	—	3	1	no	—
52	C₉×Q₁₆	—	6	2	no	—
53	C₂×C₈×C₉	—	2	1	no	—
54	C₂×C₈×C₉	—	3	1	no	—
55	C₂×C₈×C₉	—	6	2	no	—
56	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	6	2	no	—
59	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
60	C₉×C₂²⋊C₄	—	6	2	no	—
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₂×D₃×C₉	—	2	1	no	—
63	C₂×D₃×C₉	—	2	1	no	—
64	C₂×D₃×C₉	Q₁₆	1	2	yes	complemented
65	C₃×C₄×C₉	—	4	2	no	—
66	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₃×C₄×C₉	#41	1	1	yes	characteristic
68	C₉×Dic₃	—	2	1	no	—
69	C₉×Dic₃	—	4	2	no	—
70	C₂×C₃×Q₁₆	—	3	1	no	—
71	C₂²×C₃×C₈	—	3	1	no	—
72	C₃×C₂²⋊Q₈	—	3	2	no	—
73	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
74	C₂×D₃×C₈	C₂×C₉	1	1	yes	characteristic
75	#897	C₂×C₉	1	2	yes	—
76	C₃×#107	—	2	1	no	—
77	C₃×#107	C₂×C₉	1	1	yes	characteristic, complemented
78	C₃×#107	C₃×D₃	1	1	yes	characteristic
79	C₃×#103	—	3	2	no	—
80	#810	C₂×C₉	1	2	yes	complemented
81	C₂²×C₃×D₃	C₃×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₃×Dic₃	—	2	2	no	—
85	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
86	C₃×C₄×D₃	—	2	1	no	—
87	C₃×C₄×D₃	—	2	1	no	—
88	C₃×Dic₆	—	2	2	no	—
89	C₃×Dic₆	—	2	2	no	—
90	C₂³×C₉	—	3	1	no	—
91	C₃²×C₈	C₃×D₄	1	2	yes	—
92	C₃×C₃⋊C₈	—	2	1	no	—
93	Q₈×C₉	—	6	2	no	—
94	Q₈×C₉	—	6	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	3	1	no	—
98	C₂×C₄×C₉	—	4	2	no	—
99	C₂×C₄×C₉	—	6	1	no	—
100	C₂×C₄×C₉	—	6	1	no	—
101	C₂×C₄×C₉	—	6	2	no	—
102	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
103	C₈×C₉	—	2	2	no	—
104	C₈×C₉	—	6	1	no	—
105	C₈×C₉	D₁₂	1	2	yes	—
106	#467	—	3	1	no	Sylow
107	C₂×C₃×C₉	#121	1	1	yes	characteristic
108	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₃×C₉	#135	1	1	yes	characteristic
110	D₃×C₉	—	2	2	no	—
111	C₂×C₃×Q₈	—	3	2	no	—
112	C₂²×C₃×C₄	—	3	1	no	—
113	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
114	C₂×Dic₆	C₂²×C₉	1	2	yes	—
115	C₃×Q₁₆	—	6	2	no	—
116	C₂×C₃×C₈	—	2	1	no	—
117	C₂×C₃×C₈	—	3	1	no	—
118	C₂×C₃×C₈	—	6	2	no	—
119	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	6	2	no	—
123	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
124	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
125	C₃×C₂²⋊C₄	—	6	2	no	—
126	(C₂×D₃)⋊C₄	—	2	2	no	—
127	C₄⋊Dic₃	—	2	2	no	—
128	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
129	Dic₁₂	—	2	2	no	—
130	D₃×C₈	—	2	2	no	—
131	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
132	C₂×C₃×D₃	—	2	1	no	—
133	C₂×C₃×D₃	—	2	1	no	—
134	C₂×C₃×D₃	C₃×Q₁₆	1	2	yes	—
135	C₃²×C₄	—	4	2	no	—
136	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
137	C₃²×C₄	C₃×#41	1	1	yes	characteristic
138	C₃×Dic₃	—	2	1	no	—
139	C₃×Dic₃	—	4	2	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	—	3	2	no	—
142	C₂²×C₉	—	6	1	no	—
143	C₂²×C₉	—	6	1	no	—
144	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	2	1	no	—
147	C₄×C₉	—	4	2	no	—
148	C₄×C₉	—	6	1	no	—
149	C₄×C₉	—	8	2	no	—
150	C₄×C₉	—	12	2	no	—
151	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
152	C₄×C₉	#239	1	1	yes	characteristic
153	C₂×Q₁₆	—	3	1	no	—
154	C₂²×C₈	—	3	1	no	—
155	C₂²⋊Q₈	—	3	2	no	—
156	#107	D₃×C₉	1	1	yes	characteristic, complemented
157	#103	—	3	2	no	—
158	C₃×C₉	#467	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₃	—	3	1	no	—
160	C₂²×D₃	D₄×C₉	1	1	yes	characteristic
161	C₃×Q₈	—	6	2	no	—
162	C₃×Q₈	—	6	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	3	1	no	—
167	C₂×C₃×C₄	—	4	2	no	—
168	C₂×C₃×C₄	—	6	1	no	—
169	C₂×C₃×C₄	—	6	1	no	—
170	C₂×C₃×C₄	—	6	2	no	—
171	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
173	C₂×Dic₃	—	2	2	no	—
174	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
175	C₄×D₃	—	2	1	no	—
176	C₄×D₃	—	2	1	no	—
177	Dic₆	—	2	2	no	—
178	Dic₆	—	2	2	no	—
179	C₃×C₈	—	2	2	no	—
180	C₃×C₈	—	6	1	no	—
181	C₃×C₈	D₄×C₉	1	2	yes	—
182	C₃×C₈	C₃×D₁₂	1	2	yes	—
183	C₃⋊C₈	—	2	1	no	—
184	C₂×C₃²	C₃×#121	1	1	yes	characteristic
185	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
186	C₂×C₃²	C₃×#135	1	1	yes	characteristic
187	C₃×D₃	—	2	2	no	—
188	C₂×C₉	—	2	1	no	—
189	C₂×C₉	—	2	1	no	—
190	C₂×C₉	—	2	1	no	—
191	C₂×C₉	—	6	2	no	—
192	C₂×C₉	#895	1	1	yes	characteristic
193	C₂×C₉	#912	1	1	yes	characteristic
194	C₂×C₉	D₃×Q₁₆	1	1	yes	characteristic
195	C₂×Q₈	—	3	2	no	—
196	C₂²×C₄	—	3	1	no	—
197	Q₁₆	—	6	2	no	—
198	C₂×C₈	—	3	1	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
201	C₄⋊C₄	—	6	2	no	—
202	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
203	C₂²⋊C₄	—	6	2	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	3	2	no	—
206	C₂²×C₃	—	6	1	no	—
207	C₂²×C₃	—	6	1	no	—
208	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
209	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
210	C₂×D₃	—	2	1	no	—
211	C₂×D₃	—	2	1	no	—
212	C₂×D₃	C₉×Q₁₆	1	2	yes	complemented
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	4	2	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	—	6	1	no	—
218	C₃×C₄	—	8	2	no	—
219	C₃×C₄	—	12	2	no	—
220	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
221	C₃×C₄	C₉×#41	1	1	yes	characteristic
222	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
223	C₃×C₄	C₃×#239	1	1	yes	characteristic
224	Dic₃	—	2	1	no	—
225	Dic₃	—	4	2	no	—
226	C₃²	C₃×#467	1	1	yes	characteristic
227	C₉	—	2	1	no	—
228	C₉	#4887	1	1	yes	characteristic, direct product factor
229	C₂³	—	3	1	no	—
230	Q₈	—	6	2	no	—
231	Q₈	—	6	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	3	1	no	—
234	C₂×C₄	—	6	1	no	—
235	C₂×C₄	—	6	1	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
238	C₈	—	6	1	no	—
239	C₈	C₉×D₁₂	1	2	yes	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	6	2	no	—
244	C₂×C₃	C₉×#121	1	1	yes	characteristic
245	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
246	C₂×C₃	C₉×#135	1	1	yes	characteristic
247	C₂×C₃	C₃×#895	1	1	yes	characteristic
248	C₂×C₃	C₃×#912	1	1	yes	characteristic
249	C₂×C₃	C₃×D₃×Q₁₆	1	1	yes	characteristic
250	D₃	—	2	2	no	—
251	C₂²	—	3	2	no	—
252	C₂²	—	6	1	no	—
253	C₂²	—	6	1	no	—
254	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
255	C₄	—	4	2	no	—
256	C₄	—	6	1	no	—
257	C₄	—	12	2	no	—
258	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
259	C₄	C₉×#239	1	1	yes	characteristic
260	C₃	—	2	1	no	—
261	C₃	C₉×#467	1	1	yes	characteristic, complemented
262	C₃	C₃×#4887	1	1	yes	characteristic
263	C₂	—	6	2	no	—
264	C₂	C₉×#895	1	1	yes	characteristic
265	C₂	C₉×#912	1	1	yes	characteristic
266	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	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂²×C₃×D₃	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₉×#467
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
1. C₂×C₉×Dic₁₂	C₂	semidirect
4. C₃×C₉×#107	C₂	semidirect
5. C₉×#810	C₂	semidirect
26. C₉×#107	D₃	semidirect
40. #4887	C₉	direct (N × H)
64. C₂×D₃×C₉	Q₁₆	semidirect
73. C₂×Dic₁₂	C₂×C₉	semidirect
77. C₃×#107	C₂×C₉	semidirect
80. #810	C₂×C₉	semidirect
156. #107	D₃×C₉	semidirect
158. C₃×C₉	#467	semidirect
212. C₂×D₃	C₉×Q₁₆	semidirect
228. C₉	#4887	direct (N × H)
261. C₃	C₉×#467	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×D₃×C₈×C₉	2
C₉×#897	2
C₃×C₉×#107	2
C₉×#810	2
C₃×#4887	3
C₉×#467	3

Subgroups — C9×#4887 #13142207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4887 #13142207