Subgroups — C₉×#5135 #13143192

Order 1728.

458 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#930	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	2	yes	maximal, complemented
5	C₉×#817	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5135	C₃	1	1	yes	maximal, characteristic
7	C₉×#469	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×#67	—	2	2	no	—
10	C₂×C₉×D₁₂	C₂²	1	2	yes	—
11	C₃×C₉×M₁₆	—	2	2	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₉×(C₂×D₃)⋊C₄	—	2	2	no	—
15	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₉×D₂₄	—	2	2	no	—
17	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
18	C₃×#930	C₂×C₃	1	2	yes	—
19	C₂×C₃×D₂₄	C₂×C₃	1	1	yes	characteristic
20	C₃×#821	C₂×C₃	1	2	yes	—
21	C₃×#817	C₂×C₃	1	1	yes	characteristic
22	C₂×C₉×D₈	—	3	1	no	—
23	C₂×C₉×M₁₆	—	2	1	no	—
24	C₂×C₉×M₁₆	D₃	1	1	yes	characteristic, complemented
25	C₉×#121	—	3	2	no	—
26	C₉×#106	—	3	1	no	—
27	C₉×#102	—	3	2	no	—
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂²×D₃×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₉×#67	—	4	4	no	—
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₉×D₁₂	—	2	2	no	—
37	C₉×D₁₂	—	2	2	no	—
38	C₃×C₈×C₉	—	2	1	no	—
39	C₃×C₈×C₉	D₄	1	2	yes	complemented
40	C₃×#469	—	3	1	no	—
41	#5135	C₉	1	1	yes	characteristic, direct product factor
42	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×#67	—	2	2	no	—
44	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
45	C₃²×M₁₆	—	2	2	no	—
46	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
49	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
50	C₃×D₂₄	—	2	2	no	—
51	C₂×D₄×C₉	—	3	2	no	—
52	C₂×D₄×C₉	—	6	2	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
55	C₉×D₈	—	6	2	no	—
56	C₉×M₁₆	—	2	4	no	—
57	C₉×M₁₆	—	2	2	no	—
58	C₂×C₈×C₉	—	2	1	no	—
59	C₂×C₈×C₉	—	2	1	no	—
60	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
61	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
62	C₉×C₄⋊C₄	—	3	2	no	—
63	C₉×C₂²⋊C₄	—	6	2	no	—
64	C₂²×C₃×C₉	—	2	1	no	—
65	C₂²×C₃×C₉	—	2	2	no	—
66	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×D₃×C₉	—	4	2	no	—
68	C₂×D₃×C₉	—	4	2	no	—
69	C₂×D₃×C₉	—	4	2	no	—
70	C₃×C₄×C₉	—	2	1	no	—
71	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	#41	1	1	yes	characteristic
73	C₉×Dic₃	—	4	2	no	—
74	C₂×C₃×D₈	—	3	1	no	—
75	C₂×C₃×M₁₆	—	2	1	no	—
76	C₂×C₃×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
77	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
78	C₃×#121	—	3	2	no	—
79	#930	C₂×C₉	1	2	yes	complemented
80	C₂×D₂₄	C₂×C₉	1	1	yes	characteristic, complemented
81	C₃×#106	—	3	1	no	—
82	C₃×#102	—	3	2	no	—
83	#821	C₂×C₉	1	2	yes	complemented
84	#817	C₂×C₉	1	1	yes	characteristic, complemented
85	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
86	C₂²×C₃×D₃	—	2	2	no	—
87	C₂×C₃²×C₄	—	2	1	no	—
88	C₂×C₃²×C₄	—	2	1	no	—
89	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₃×#67	—	4	4	no	—
92	C₂×C₃×Dic₃	—	2	2	no	—
93	C₃×D₁₂	—	2	2	no	—
94	C₃×D₁₂	—	2	2	no	—
95	C₂³×C₉	—	2	1	no	—
96	C₂³×C₉	—	6	2	no	—
97	C₂³×C₉	D₁₂	1	1	yes	characteristic
98	C₃²×C₈	—	2	1	no	—
99	C₃²×C₈	C₃×D₄	1	2	yes	—
100	D₄×C₉	—	6	2	no	—
101	D₄×C₉	—	6	2	no	—
102	D₄×C₉	—	12	4	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	1	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	6	2	no	—
110	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
112	C₈×C₉	—	2	2	no	—
113	C₈×C₉	—	2	1	no	—
114	C₈×C₉	—	2	2	no	—
115	C₈×C₉	#67	1	2	yes	complemented
116	#469	—	3	1	no	Sylow
117	C₂×C₃×C₉	—	4	1	no	—
118	C₂×C₃×C₉	#121	1	1	yes	characteristic
119	C₂×C₃×C₉	AGL(1,ℤ/8)	1	2	yes	—
120	D₃×C₉	—	8	2	no	—
121	C₂×C₃×D₄	—	3	2	no	—
122	C₂×C₃×D₄	—	6	2	no	—
123	C₂²×C₃×C₄	—	2	1	no	—
124	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
126	C₂×#67	—	2	2	no	—
127	C₂×D₁₂	C₂²×C₉	1	2	yes	—
128	C₃×D₈	—	6	2	no	—
129	C₃×M₁₆	—	2	4	no	—
130	C₃×M₁₆	—	2	2	no	—
131	C₃×M₁₆	—	2	2	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	3	2	no	—
139	C₃×C₂²⋊C₄	—	6	2	no	—
140	(C₂×D₃)⋊C₄	—	2	2	no	—
141	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
142	D₂₄	—	2	2	no	—
143	C₂²×C₃²	—	2	1	no	—
144	C₂²×C₃²	—	2	2	no	—
145	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂×C₃×D₃	—	4	2	no	—
147	C₂×C₃×D₃	—	4	2	no	—
148	C₂×C₃×D₃	—	4	2	no	—
149	C₃²×C₄	—	2	1	no	—
150	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
151	C₃²×C₄	C₃×#41	1	1	yes	characteristic
152	C₃×Dic₃	—	4	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	2	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	4	2	no	—
158	C₂²×C₉	—	12	2	no	—
159	C₂²×C₉	—	12	2	no	—
160	C₂²×C₉	—	12	2	no	—
161	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	12	2	no	—
167	C₄×C₉	#235	1	1	yes	characteristic
168	C₄×C₉	C₂×#67	1	1	yes	characteristic
169	C₂×D₈	—	3	1	no	—
170	C₂×M₁₆	D₃×C₉	1	1	yes	characteristic, complemented
171	#121	—	3	2	no	—
172	#106	—	3	1	no	—
173	#102	—	3	2	no	—
174	C₃×C₉	#469	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	2	1	no	—
176	C₂³×C₃	—	6	2	no	—
177	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
178	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
179	C₂²×D₃	—	2	2	no	—
180	C₃×D₄	—	6	2	no	—
181	C₃×D₄	—	6	2	no	—
182	C₃×D₄	—	12	4	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	6	2	no	—
192	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
196	#67	—	4	4	no	—
197	C₂×Dic₃	—	2	2	no	—
198	D₁₂	—	2	2	no	—
199	D₁₂	—	2	2	no	—
200	C₃×C₈	—	2	2	no	—
201	C₃×C₈	—	2	1	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	D₄×C₉	1	2	yes	complemented
205	C₃×C₈	C₃×#67	1	2	yes	—
206	C₂×C₃²	—	4	1	no	—
207	C₂×C₃²	C₃×#121	1	1	yes	characteristic
208	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	2	yes	—
209	C₃×D₃	—	8	2	no	—
210	C₂×C₉	—	2	2	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	4	2	no	—
213	C₂×C₉	—	4	1	no	—
214	C₂×C₉	—	24	2	no	—
215	C₂×C₉	#908	1	2	yes	—
216	C₂×C₉	#930	1	1	yes	characteristic
217	C₂×D₄	—	3	2	no	—
218	C₂×D₄	—	6	2	no	—
219	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
220	D₈	—	6	2	no	—
221	M₁₆	—	2	2	no	—
222	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
223	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
224	C₄⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	—	6	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	4	2	no	—
233	C₂²×C₃	—	12	2	no	—
234	C₂²×C₃	—	12	2	no	—
235	C₂²×C₃	—	12	2	no	—
236	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
238	C₂×D₃	—	4	2	no	—
239	C₂×D₃	—	4	2	no	—
240	C₂×D₃	—	4	2	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	12	2	no	—
247	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
248	C₃×C₄	C₉×#41	1	1	yes	characteristic
249	C₃×C₄	C₃×#235	1	1	yes	characteristic
250	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
251	Dic₃	—	4	2	no	—
252	C₃²	C₃×#469	1	1	yes	characteristic
253	C₉	—	2	1	no	—
254	C₉	#5135	1	1	yes	characteristic, direct product factor
255	C₂³	—	6	2	no	—
256	C₂³	C₉×D₁₂	1	1	yes	characteristic
257	D₄	—	6	2	no	—
258	D₄	—	6	2	no	—
259	D₄	—	12	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	6	2	no	—
263	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	C₉×#67	1	2	yes	complemented
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	4	2	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	—	4	1	no	—
272	C₂×C₃	—	24	2	no	—
273	C₂×C₃	C₉×#121	1	1	yes	characteristic
274	C₂×C₃	C₉×AGL(1,ℤ/8)	1	2	yes	—
275	C₂×C₃	C₃×#908	1	2	yes	—
276	C₂×C₃	C₃×#930	1	1	yes	characteristic
277	D₃	—	8	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	12	2	no	—
281	C₂²	—	12	2	no	—
282	C₂²	—	12	2	no	—
283	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	12	2	no	—
286	C₄	C₉×#235	1	1	yes	characteristic
287	C₄	C₂×C₉×#67	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	C₉×#469	1	1	yes	characteristic, complemented
290	C₃	C₃×#5135	1	1	yes	characteristic
291	C₂	—	4	1	no	—
292	C₂	—	24	2	no	—
293	C₂	C₉×#908	1	2	yes	—
294	C₂	C₉×#930	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	C₂×C₃×C₉×M₁₆	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#5135	C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₉×M₁₆	D₃
hypocenter	C₃	C₉×#469
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂×C₉×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×M₁₆	C₂	semidirect
2. C₉×#930	C₂	semidirect
3. C₂×C₉×D₂₄	C₂	semidirect
4. C₉×#821	C₂	semidirect
5. C₉×#817	C₂	semidirect
24. C₂×C₉×M₁₆	D₃	semidirect
39. C₃×C₈×C₉	D₄	semidirect
41. #5135	C₉	direct (N × H)
76. C₂×C₃×M₁₆	C₂×C₉	semidirect
79. #930	C₂×C₉	semidirect
80. C₂×D₂₄	C₂×C₉	semidirect
83. #821	C₂×C₉	semidirect
84. #817	C₂×C₉	semidirect
115. C₈×C₉	#67	semidirect
170. C₂×M₁₆	D₃×C₉	semidirect
174. C₃×C₉	#469	semidirect
204. C₃×C₈	D₄×C₉	semidirect
254. C₉	#5135	direct (N × H)
266. C₈	C₉×#67	semidirect
289. C₃	C₉×#469	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×M₁₆	2
C₉×#930	2
C₂×C₉×D₂₄	2
C₉×#821	2
C₉×#817	2
C₃×#5135	3
C₉×#469	3

Subgroups — C9×#5135 #13143192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5135 #13143192