Subgroups — C₉×#5131 #13143188

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#926	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	4	yes	maximal, complemented
5	C₃×#5131	C₃	1	1	yes	maximal, characteristic
6	C₉×#418	—	3	1	no	maximal
7	C₂³×D₃×C₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×D₁₂	—	2	1	no	—
10	C₂×C₉×D₁₂	—	2	2	no	—
11	C₂×C₉×D₁₂	C₄	1	4	yes	complemented
12	C₂×C₉×D₁₂	C₂²	1	2	yes	—
13	C₃×C₉×M₁₆	—	2	2	no	—
14	C₂×C₃×C₈×C₉	C₂²	1	2	yes	complemented
15	C₉×#217	—	2	1	no	—
16	C₉×C₄⋊Dic₃	C₂²	1	2	yes	complemented
17	C₄×C₉×Dic₃	—	2	1	no	—
18	C₂²×C₃×D₁₂	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
20	C₃×#926	C₂×C₃	1	1	yes	characteristic
21	C₂²×D₄×C₉	—	3	1	no	—
22	C₃×#821	C₂×C₃	1	4	yes	—
23	C₂×C₉×M₁₆	—	2	1	no	—
24	C₂×C₉×M₁₆	D₃	1	1	yes	characteristic, complemented
25	C₉×#117	—	3	1	no	—
26	C₉×#102	—	3	4	no	—
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂²×D₃×C₉	—	2	4	no	—
29	C₂²×D₃×C₉	—	2	2	no	—
30	C₂²×D₃×C₉	—	4	4	no	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₉×D₁₂	—	2	4	no	—
37	C₉×D₁₂	—	2	2	no	—
38	C₉×D₁₂	C₂×C₄	1	4	yes	—
39	C₃×C₈×C₉	—	2	2	no	—
40	C₃×#418	—	3	1	no	—
41	#5131	C₉	1	1	yes	characteristic, direct product factor
42	C₂³×C₃×D₃	—	2	1	no	—
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×D₁₂	—	2	1	no	—
45	C₂×C₃×D₁₂	—	2	2	no	—
46	C₂×C₃×D₁₂	C₃×C₄	1	4	yes	—
47	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
48	C₂⁴×C₉	—	6	1	no	—
49	C₃²×M₁₆	—	2	2	no	—
50	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
51	C₃×#217	—	2	1	no	—
52	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
53	C₃×C₄×Dic₃	—	2	1	no	—
54	C₂×D₄×C₉	—	3	4	no	—
55	C₂×D₄×C₉	—	3	2	no	—
56	C₂×D₄×C₉	—	6	1	no	—
57	C₂×D₄×C₉	—	6	2	no	—
58	C₂²×C₄×C₉	—	2	1	no	—
59	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
60	C₉×M₁₆	—	2	4	no	—
61	C₉×M₁₆	—	2	2	no	—
62	C₂×C₈×C₉	—	2	2	no	—
63	C₂×C₈×C₉	C₂×D₃	1	2	yes	complemented
64	C₉×C₄⋊C₄	—	3	2	no	—
65	C₉×C₂²⋊C₄	—	6	1	no	—
66	C₄²×C₉	—	6	1	no	—
67	C₂²×C₃×C₉	—	2	2	no	—
68	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×D₃×C₉	—	2	4	no	—
71	C₂×D₃×C₉	—	4	8	no	—
72	C₂×D₃×C₉	—	4	4	no	—
73	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
74	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₉×Dic₃	—	4	2	no	—
77	C₂²×C₃×D₄	—	3	1	no	—
78	C₂²×D₁₂	C₂×C₉	1	1	yes	characteristic
79	C₂×C₃×M₁₆	—	2	1	no	—
80	C₂×C₃×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
81	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
82	C₃×#117	—	3	1	no	—
83	#926	C₂×C₉	1	1	yes	characteristic, complemented
84	C₃×#102	—	3	4	no	—
85	#821	C₂×C₉	1	4	yes	complemented
86	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
87	C₂²×C₃×D₃	—	2	4	no	—
88	C₂²×C₃×D₃	—	2	2	no	—
89	C₂²×C₃×D₃	—	4	4	no	—
90	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
93	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₃×Dic₃	—	2	2	no	—
95	C₃×D₁₂	—	2	4	no	—
96	C₃×D₁₂	—	2	2	no	—
97	C₃×D₁₂	C₂×C₃×C₄	1	4	yes	—
98	C₂³×C₉	—	2	1	no	—
99	C₂³×C₉	—	6	4	no	—
100	C₂³×C₉	—	6	2	no	—
101	C₂³×C₉	—	12	4	no	—
102	C₂³×C₉	D₁₂	1	1	yes	characteristic
103	C₃²×C₈	—	2	2	no	—
104	D₄×C₉	—	3	4	no	—
105	D₄×C₉	—	6	4	no	—
106	D₄×C₉	—	6	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	—	6	2	no	—
112	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
113	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
114	C₂×C₄×C₉	#67	1	2	yes	—
115	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
116	C₈×C₉	—	2	4	no	—
117	C₈×C₉	—	2	2	no	—
118	#418	—	3	1	no	Sylow
119	C₂×C₃×C₉	—	2	2	no	—
120	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₃×C₉	AGL(1,ℤ/8)	1	2	yes	—
122	D₃×C₉	—	4	4	no	—
123	C₂⁴×C₃	—	6	1	no	—
124	C₂³×D₃	—	2	1	no	—
125	C₂×C₃×D₄	—	3	4	no	—
126	C₂×C₃×D₄	—	3	2	no	—
127	C₂×C₃×D₄	—	6	1	no	—
128	C₂×C₃×D₄	—	6	2	no	—
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
132	C₂×D₁₂	—	2	1	no	—
133	C₂×D₁₂	—	2	2	no	—
134	C₂×D₁₂	C₄×C₉	1	4	yes	complemented
135	C₂×D₁₂	C₂²×C₉	1	2	yes	—
136	C₃×M₁₆	—	2	4	no	—
137	C₃×M₁₆	—	2	2	no	—
138	C₃×M₁₆	—	2	2	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	C₂²×C₉	1	2	yes	complemented
141	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
142	C₃×C₄⋊C₄	—	3	2	no	—
143	C₃×C₂²⋊C₄	—	6	1	no	—
144	C₃×C₄²	—	6	1	no	—
145	#217	—	2	1	no	—
146	C₄⋊Dic₃	C₂²×C₉	1	2	yes	complemented
147	C₄×Dic₃	—	2	1	no	—
148	C₂²×C₃²	—	2	2	no	—
149	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
150	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×D₃	—	2	4	no	—
152	C₂×C₃×D₃	—	4	8	no	—
153	C₂×C₃×D₃	—	4	4	no	—
154	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
155	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
157	C₃×Dic₃	—	4	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	4	2	no	—
162	C₂²×C₉	—	6	4	no	—
163	C₂²×C₉	—	12	8	no	—
164	C₂²×C₉	—	12	4	no	—
165	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
166	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	2	1	no	—
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	—	12	2	no	—
171	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
172	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
173	C₄×C₉	C₂×#67	1	1	yes	characteristic
174	C₂²×D₄	—	3	1	no	—
175	C₂×M₁₆	D₃×C₉	1	1	yes	characteristic, complemented
176	#117	—	3	1	no	—
177	#102	—	3	4	no	—
178	C₃×C₉	#418	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	2	1	no	—
180	C₂³×C₃	—	6	4	no	—
181	C₂³×C₃	—	6	2	no	—
182	C₂³×C₃	—	12	4	no	—
183	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
184	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
185	C₂²×D₃	—	2	4	no	—
186	C₂²×D₃	—	2	2	no	—
187	C₂²×D₃	—	4	4	no	—
188	C₃×D₄	—	3	4	no	—
189	C₃×D₄	—	6	4	no	—
190	C₃×D₄	—	6	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	6	2	no	—
196	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
197	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
198	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
199	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
200	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
201	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
202	C₂×C₃×C₄	C₃×#67	1	2	yes	—
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
204	C₂×Dic₃	—	2	2	no	—
205	D₁₂	—	2	4	no	—
206	D₁₂	—	2	2	no	—
207	D₁₂	C₂×C₄×C₉	1	4	yes	—
208	C₃×C₈	—	2	4	no	—
209	C₃×C₈	—	2	2	no	—
210	C₃×C₈	—	2	2	no	—
211	C₂×C₃²	—	2	2	no	—
212	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	2	yes	—
214	C₃×D₃	—	4	4	no	—
215	C₂×C₉	—	2	2	no	—
216	C₂×C₉	—	2	2	no	—
217	C₂×C₉	—	2	1	no	—
218	C₂×C₉	—	4	2	no	—
219	C₂×C₉	—	12	4	no	—
220	C₂×C₉	#908	1	2	yes	—
221	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
222	C₂⁴	—	6	1	no	—
223	C₂×D₄	—	3	4	no	—
224	C₂×D₄	—	3	2	no	—
225	C₂×D₄	—	6	1	no	—
226	C₂×D₄	—	6	2	no	—
227	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	C₂×C₈	C₂×D₃×C₉	1	2	yes	complemented
230	C₄⋊C₄	—	3	2	no	—
231	C₂²⋊C₄	—	6	1	no	—
232	C₄²	—	6	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	4	2	no	—
238	C₂²×C₃	—	6	4	no	—
239	C₂²×C₃	—	12	8	no	—
240	C₂²×C₃	—	12	4	no	—
241	C₂²×C₃	C₉×C₂²⋊C₄	1	2	yes	—
242	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
243	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
244	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
245	C₂×D₃	—	2	4	no	—
246	C₂×D₃	—	4	8	no	—
247	C₂×D₃	—	4	4	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	12	2	no	—
252	C₃×C₄	C₉×C₂²⋊C₄	1	2	yes	—
253	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
254	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
255	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
256	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
257	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
258	Dic₃	—	4	2	no	—
259	C₃²	C₃×#418	1	1	yes	characteristic
260	C₉	—	2	1	no	—
261	C₉	#5131	1	1	yes	characteristic, direct product factor
262	C₂³	—	6	2	no	—
263	C₂³	—	6	4	no	—
264	C₂³	—	12	4	no	—
265	C₂³	C₉×D₁₂	1	1	yes	characteristic
266	D₄	—	3	4	no	—
267	D₄	—	6	2	no	—
268	D₄	—	6	4	no	—
269	C₂×C₄	—	6	2	no	—
270	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
271	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
272	C₂×C₄	C₉×#67	1	2	yes	—
273	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	—	2	2	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	4	2	no	—
280	C₂×C₃	—	12	4	no	—
281	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₃	C₉×AGL(1,ℤ/8)	1	2	yes	—
283	C₂×C₃	C₃×#908	1	2	yes	—
284	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
285	D₃	—	4	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	6	4	no	—
288	C₂²	—	12	8	no	—
289	C₂²	—	12	4	no	—
290	C₂²	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
291	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
292	C₄	—	12	2	no	—
293	C₄	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
294	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
295	C₄	C₂×C₉×#67	1	1	yes	characteristic
296	C₃	—	2	1	no	—
297	C₃	C₉×#418	1	1	yes	characteristic, complemented
298	C₃	C₃×#5131	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	12	4	no	—
301	C₂	C₉×#908	1	2	yes	—
302	C₂	C₂×C₉×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×M₁₆	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂²×C₃×D₁₂	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₉×M₁₆	D₃
hypocenter	C₃	C₉×#418
LCS	G ▷ 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
2. C₂×C₃×C₉×M₁₆	C₂	semidirect
3. C₉×#926	C₂	semidirect
4. C₉×#821	C₂	semidirect
11. C₂×C₉×D₁₂	C₄	semidirect
14. C₂×C₃×C₈×C₉	C₂²	semidirect
16. C₉×C₄⋊Dic₃	C₂²	semidirect
24. C₂×C₉×M₁₆	D₃	semidirect
41. #5131	C₉	direct (N × H)
63. C₂×C₈×C₉	C₂×D₃	semidirect
80. C₂×C₃×M₁₆	C₂×C₉	semidirect
83. #926	C₂×C₉	semidirect
85. #821	C₂×C₉	semidirect
134. C₂×D₁₂	C₄×C₉	semidirect
140. C₂×C₃×C₈	C₂²×C₉	semidirect
146. C₄⋊Dic₃	C₂²×C₉	semidirect
175. C₂×M₁₆	D₃×C₉	semidirect
178. C₃×C₉	#418	semidirect
229. C₂×C₈	C₂×D₃×C₉	semidirect
261. C₉	#5131	direct (N × H)
297. C₃	C₉×#418	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₁₂	2
C₂×C₃×C₉×M₁₆	2
C₉×#926	2
C₉×#821	2
C₃×#5131	3
C₉×#418	3

Subgroups — C9×#5131 #13143188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5131 #13143188