Subgroups — C₉×#4843 #13142037

Order 1728.

458 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×D₂₄	C₂	1	2	yes	maximal, complemented
2	C₉×#874	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#821	C₂	1	2	yes	maximal, complemented
6	C₃×#4843	C₃	1	1	yes	maximal, characteristic
7	C₉×#459	—	3	1	no	maximal
8	C₂×C₉×D₁₂	—	2	1	no	—
9	C₂×C₉×D₁₂	—	2	1	no	—
10	C₂×C₉×D₁₂	C₂²	1	2	yes	—
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃×C₉	—	2	1	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₉×D₂₄	—	2	4	no	—
18	C₂×C₃×D₂₄	C₂×C₃	1	2	yes	—
19	C₃×#874	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×#821	C₂×C₃	1	2	yes	—
23	C₂×C₉×D₈	—	3	2	no	—
24	C₉×#127	—	3	1	no	—
25	C₄×D₄×C₉	—	3	1	no	—
26	C₉×C₄⋊C₈	—	2	1	no	—
27	C₉×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
28	C₉×#102	—	3	2	no	—
29	C₂²×D₃×C₉	—	2	1	no	—
30	C₂²×D₃×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₉×Dic₃	—	2	1	no	—
35	C₉×D₁₂	—	2	2	no	—
36	C₉×D₁₂	—	2	1	no	—
37	C₉×D₁₂	—	2	2	no	—
38	C₉×D₁₂	—	4	2	no	—
39	C₉×D₁₂	—	4	2	no	—
40	C₉×D₁₂	D₄	1	2	yes	complemented
41	C₄×D₃×C₉	—	4	2	no	—
42	C₃×C₈×C₉	—	2	2	no	—
43	C₃×#459	—	3	1	no	—
44	#4843	C₉	1	1	yes	characteristic, direct product factor
45	C₂×C₃×D₁₂	—	2	1	no	—
46	C₂×C₃×D₁₂	—	2	1	no	—
47	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×C₄×D₃	—	2	1	no	—
50	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
51	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
53	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
54	C₃×D₂₄	—	2	4	no	—
55	C₂×D₄×C₉	—	3	1	no	—
56	C₂×D₄×C₉	—	3	2	no	—
57	C₂×D₄×C₉	—	6	1	no	—
58	C₂×D₄×C₉	—	6	1	no	—
59	C₂²×C₄×C₉	—	6	1	no	—
60	C₉×D₈	—	6	4	no	—
61	C₂×C₈×C₉	—	2	2	no	—
62	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
63	C₉×C₄⋊C₄	—	3	1	no	—
64	C₉×C₂²⋊C₄	—	6	1	no	—
65	C₄²×C₉	—	2	1	no	—
66	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×D₃×C₉	—	2	2	no	—
69	C₂×D₃×C₉	—	4	1	no	—
70	C₂×D₃×C₉	—	4	1	no	—
71	C₂×D₃×C₉	—	4	2	no	—
72	C₂×D₃×C₉	—	4	2	no	—
73	C₂×D₃×C₉	—	4	2	no	—
74	C₃×C₄×C₉	—	2	1	no	—
75	C₃×C₄×C₉	D₈	1	2	yes	complemented
76	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄×C₉	#41	1	1	yes	characteristic
78	C₉×Dic₃	—	4	1	no	—
79	C₂×C₃×D₈	—	3	2	no	—
80	C₃×#127	—	3	1	no	—
81	C₃×C₄×D₄	—	3	1	no	—
82	C₂×D₂₄	C₂×C₉	1	2	yes	complemented
83	#874	C₂×C₉	1	1	yes	characteristic, complemented
84	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic, complemented
85	C₃×C₄⋊C₈	—	2	1	no	—
86	C₃×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
87	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
88	C₃×#102	—	3	2	no	—
89	#821	C₂×C₉	1	2	yes	complemented
90	C₂²×C₃×D₃	—	2	1	no	—
91	C₂²×C₃×D₃	—	2	2	no	—
92	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₃×Dic₃	—	2	1	no	—
96	C₃×D₁₂	—	2	2	no	—
97	C₃×D₁₂	—	2	1	no	—
98	C₃×D₁₂	—	2	2	no	—
99	C₃×D₁₂	—	4	2	no	—
100	C₃×D₁₂	—	4	2	no	—
101	C₃×D₁₂	C₃×D₄	1	2	yes	—
102	C₃×C₄×D₃	—	4	2	no	—
103	C₂³×C₉	—	6	1	no	—
104	C₂³×C₉	—	6	2	no	—
105	C₃²×C₈	—	2	2	no	—
106	D₄×C₉	—	3	2	no	—
107	D₄×C₉	—	6	2	no	—
108	D₄×C₉	—	6	1	no	—
109	D₄×C₉	—	6	2	no	—
110	D₄×C₉	—	12	2	no	—
111	D₄×C₉	—	12	2	no	—
112	C₂×C₄×C₉	—	2	1	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	2	1	no	—
115	C₂×C₄×C₉	—	6	1	no	—
116	C₂×C₄×C₉	—	12	2	no	—
117	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
118	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
119	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
120	C₈×C₉	—	2	2	no	—
121	C₈×C₉	—	4	2	no	—
122	#459	—	3	1	no	Sylow
123	C₂×C₃×C₉	#121	1	1	yes	characteristic
124	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	D₃×C₉	—	4	2	no	—
127	D₃×C₉	—	8	2	no	—
128	C₂×C₃×D₄	—	3	1	no	—
129	C₂×C₃×D₄	—	3	2	no	—
130	C₂×C₃×D₄	—	6	1	no	—
131	C₂×C₃×D₄	—	6	1	no	—
132	C₂²×C₃×C₄	—	6	1	no	—
133	C₂×D₁₂	—	2	1	no	—
134	C₂×D₁₂	—	2	1	no	—
135	C₂×D₁₂	C₂²×C₉	1	2	yes	—
136	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
137	C₂×C₄×D₃	—	2	1	no	—
138	C₃×D₈	—	6	4	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
141	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
142	C₃×C₄⋊C₄	—	3	1	no	—
143	C₃×C₂²⋊C₄	—	6	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
146	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
147	(C₂×D₃)⋊C₄	—	2	1	no	—
148	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
149	D₂₄	—	2	4	no	—
150	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×D₃	—	2	2	no	—
152	C₂×C₃×D₃	—	4	1	no	—
153	C₂×C₃×D₃	—	4	1	no	—
154	C₂×C₃×D₃	—	4	2	no	—
155	C₂×C₃×D₃	—	4	2	no	—
156	C₂×C₃×D₃	—	4	2	no	—
157	C₃²×C₄	—	2	1	no	—
158	C₃²×C₄	C₃×D₈	1	2	yes	—
159	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
160	C₃²×C₄	C₃×#41	1	1	yes	characteristic
161	C₃×Dic₃	—	4	1	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	—	6	2	no	—
164	C₂²×C₉	—	12	1	no	—
165	C₂²×C₉	—	12	1	no	—
166	C₂²×C₉	—	12	2	no	—
167	C₂²×C₉	—	12	2	no	—
168	C₂²×C₉	—	12	2	no	—
169	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	2	1	no	—
173	C₄×C₉	—	2	1	no	—
174	C₄×C₉	—	4	1	no	—
175	C₄×C₉	—	12	1	no	—
176	C₄×C₉	D₂₄	1	2	yes	complemented
177	C₄×C₉	D₃×D₄	1	1	yes	characteristic
178	C₄×C₉	#239	1	1	yes	characteristic
179	C₂×D₈	—	3	2	no	—
180	#127	—	3	1	no	—
181	C₄×D₄	—	3	1	no	—
182	C₄⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
183	#102	—	3	2	no	—
184	C₃×C₉	#459	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₃	—	6	1	no	—
186	C₂³×C₃	—	6	2	no	—
187	C₂²×D₃	—	2	1	no	—
188	C₂²×D₃	—	2	2	no	—
189	C₃×D₄	—	3	2	no	—
190	C₃×D₄	—	6	2	no	—
191	C₃×D₄	—	6	1	no	—
192	C₃×D₄	—	6	2	no	—
193	C₃×D₄	—	12	2	no	—
194	C₃×D₄	—	12	2	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	12	2	no	—
200	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
201	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
203	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
204	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
206	C₂×Dic₃	—	2	1	no	—
207	D₁₂	—	2	2	no	—
208	D₁₂	—	2	1	no	—
209	D₁₂	—	2	2	no	—
210	D₁₂	—	4	2	no	—
211	D₁₂	—	4	2	no	—
212	D₁₂	D₄×C₉	1	2	yes	complemented
213	C₄×D₃	—	4	2	no	—
214	C₃×C₈	—	2	2	no	—
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	4	2	no	—
217	C₂×C₃²	C₃×#121	1	1	yes	characteristic
218	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
219	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₃×D₃	—	4	2	no	—
221	C₃×D₃	—	8	2	no	—
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	12	2	no	—
226	C₂×C₉	—	24	2	no	—
227	C₂×C₉	#895	1	1	yes	characteristic
228	C₂×C₉	C₂×D₂₄	1	1	yes	characteristic
229	C₂×C₉	#908	1	1	yes	characteristic
230	C₂×D₄	—	3	1	no	—
231	C₂×D₄	—	3	2	no	—
232	C₂×D₄	—	6	1	no	—
233	C₂×D₄	—	6	1	no	—
234	C₂²×C₄	—	6	1	no	—
235	D₈	—	6	4	no	—
236	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
237	C₄⋊C₄	—	3	1	no	—
238	C₂²⋊C₄	—	6	1	no	—
239	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	6	2	no	—
242	C₂²×C₃	—	12	1	no	—
243	C₂²×C₃	—	12	1	no	—
244	C₂²×C₃	—	12	2	no	—
245	C₂²×C₃	—	12	2	no	—
246	C₂²×C₃	—	12	2	no	—
247	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
248	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
249	C₂×D₃	—	2	2	no	—
250	C₂×D₃	—	4	1	no	—
251	C₂×D₃	—	4	1	no	—
252	C₂×D₃	—	4	2	no	—
253	C₂×D₃	—	4	2	no	—
254	C₂×D₃	—	4	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	12	1	no	—
262	C₃×C₄	C₉×D₈	1	2	yes	complemented
263	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
264	C₃×C₄	C₉×#41	1	1	yes	characteristic
265	C₃×C₄	C₃×D₂₄	1	2	yes	—
266	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
267	C₃×C₄	C₃×#239	1	1	yes	characteristic
268	Dic₃	—	4	1	no	—
269	C₃²	C₃×#459	1	1	yes	characteristic
270	C₉	—	2	1	no	—
271	C₉	#4843	1	1	yes	characteristic, direct product factor
272	C₂³	—	6	1	no	—
273	C₂³	—	6	2	no	—
274	D₄	—	3	2	no	—
275	D₄	—	6	2	no	—
276	D₄	—	6	1	no	—
277	D₄	—	6	2	no	—
278	D₄	—	12	2	no	—
279	D₄	—	12	2	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	12	2	no	—
282	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
283	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	12	2	no	—
290	C₂×C₃	—	24	2	no	—
291	C₂×C₃	C₉×#121	1	1	yes	characteristic
292	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
293	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
294	C₂×C₃	C₃×#895	1	1	yes	characteristic
295	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
296	C₂×C₃	C₃×#908	1	1	yes	characteristic
297	D₃	—	4	2	no	—
298	D₃	—	8	2	no	—
299	C₂²	—	6	2	no	—
300	C₂²	—	12	1	no	—
301	C₂²	—	12	1	no	—
302	C₂²	—	12	2	no	—
303	C₂²	—	12	2	no	—
304	C₂²	—	12	2	no	—
305	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	12	1	no	—
308	C₄	C₉×D₂₄	1	2	yes	complemented
309	C₄	D₃×D₄×C₉	1	1	yes	characteristic
310	C₄	C₉×#239	1	1	yes	characteristic
311	C₃	—	2	1	no	—
312	C₃	C₉×#459	1	1	yes	characteristic, complemented
313	C₃	C₃×#4843	1	1	yes	characteristic
314	C₂	—	12	2	no	—
315	C₂	—	24	2	no	—
316	C₂	C₉×#895	1	1	yes	characteristic
317	C₂	C₂×C₉×D₂₄	1	1	yes	characteristic
318	C₂	C₉×#908	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	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#4843	C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₄⋊C₈	D₃
hypocenter	C₃	C₉×#459
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
1. C₂×C₉×D₂₄	C₂	semidirect
2. C₉×#874	C₂	semidirect
3. C₄×C₉×D₁₂	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₉×#821	C₂	semidirect
27. C₉×C₄⋊C₈	D₃	semidirect
40. C₉×D₁₂	D₄	semidirect
44. #4843	C₉	direct (N × H)
75. C₃×C₄×C₉	D₈	semidirect
82. C₂×D₂₄	C₂×C₉	semidirect
83. #874	C₂×C₉	semidirect
84. C₄×D₁₂	C₂×C₉	semidirect
86. C₃×C₄⋊C₈	C₂×C₉	semidirect
89. #821	C₂×C₉	semidirect
176. C₄×C₉	D₂₄	semidirect
182. C₄⋊C₈	D₃×C₉	semidirect
184. C₃×C₉	#459	semidirect
212. D₁₂	D₄×C₉	semidirect
262. C₃×C₄	C₉×D₈	semidirect
271. C₉	#4843	direct (N × H)
308. C₄	C₉×D₂₄	semidirect
312. C₃	C₉×#459	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×D₂₄	2
C₉×#874	2
C₄×C₉×D₁₂	2
C₃×C₉×C₄⋊C₈	2
C₉×#821	2
C₃×#4843	3
C₉×#459	3

Subgroups — C9×#4843 #13142037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4843 #13142037