Subgroups — C₉×#5027 #13142783

Order 1728.

454 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#215	C₂	1	2	yes	maximal, complemented
3	C₉×#874	C₂	1	1	yes	maximal, characteristic
4	C₉×#809	C₂	1	2	yes	maximal, complemented
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5027	C₃	1	1	yes	maximal, characteristic
7	C₉×#460	—	3	1	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×D₁₂	—	2	1	no	—
10	C₂×C₉×D₁₂	—	2	1	no	—
11	C₂×C₉×D₁₂	C₂²	1	2	yes	—
12	C₃×C₉×C₄⋊C₄	—	2	1	no	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₄²×C₉	—	2	1	no	—
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×#215	—	2	4	no	—
17	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#215	C₂×C₃	1	2	yes	—
20	C₃×#874	C₂×C₃	1	1	yes	characteristic
21	C₃×#809	C₂×C₃	1	2	yes	—
22	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	C₂×C₉×SD₁₆	—	3	2	no	—
24	C₉×#127	—	3	1	no	—
25	C₄×Q₈×C₉	—	2	1	no	—
26	C₄×Q₈×C₉	D₃	1	1	yes	characteristic, complemented
27	C₉×C₄⋊C₈	—	3	1	no	—
28	C₉×#102	—	3	2	no	—
29	C₂²×D₃×C₉	—	2	2	no	—
30	C₃×Q₈×C₉	—	2	1	no	—
31	C₃×Q₈×C₉	D₄	1	2	yes	complemented
32	C₂×C₃×C₄×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	—	2	1	no	—
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₉×D₁₂	—	2	2	no	—
38	C₉×D₁₂	—	2	2	no	—
39	C₉×D₁₂	—	4	2	no	—
40	C₉×D₁₂	—	4	2	no	—
41	C₉×C₃⋊C₈	—	2	2	no	—
42	C₃×#460	—	3	1	no	—
43	#5027	C₉	1	1	yes	characteristic, direct product factor
44	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×D₁₂	—	2	1	no	—
46	C₂×C₃×D₁₂	—	2	1	no	—
47	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
48	C₃²×C₄⋊C₄	—	2	1	no	—
49	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
50	C₃²×C₄²	—	2	1	no	—
51	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₃×#215	—	2	4	no	—
53	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
54	C₂×Q₈×C₉	—	2	1	no	—
55	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂×D₄×C₉	—	3	2	no	—
57	C₂×D₄×C₉	—	6	1	no	—
58	C₂×D₄×C₉	—	6	1	no	—
59	C₉×SD₁₆	—	6	4	no	—
60	C₂×C₈×C₉	—	3	2	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	1	no	—
64	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
65	C₄²×C₉	—	2	2	no	—
66	C₄²×C₉	—	2	1	no	—
67	C₄²×C₉	—	2	1	no	—
68	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×D₃×C₉	—	4	2	no	—
71	C₂×D₃×C₉	—	4	2	no	—
72	C₂×D₃×C₉	—	4	2	no	—
73	C₃×C₄×C₉	—	2	2	no	—
74	C₃×C₄×C₉	—	2	1	no	—
75	C₃×C₄×C₉	—	4	1	no	—
76	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
77	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄×C₉	#41	1	1	yes	characteristic
79	C₂×C₃×SD₁₆	—	3	2	no	—
80	C₃×#127	—	3	1	no	—
81	C₃×C₄×Q₈	—	2	1	no	—
82	C₃×C₄×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
83	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
84	C₂×#215	C₂×C₉	1	2	yes	complemented
85	#874	C₂×C₉	1	1	yes	characteristic
86	C₃×C₄⋊C₈	—	3	1	no	—
87	C₃×#102	—	3	2	no	—
88	#809	C₂×C₉	1	2	yes	complemented
89	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic, complemented
90	C₂²×C₃×D₃	—	2	2	no	—
91	C₃²×Q₈	—	2	1	no	—
92	C₃²×Q₈	C₃×D₄	1	2	yes	—
93	C₂×C₃²×C₄	—	2	1	no	—
94	C₂×C₃²×C₄	—	2	1	no	—
95	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₃×D₁₂	—	2	2	no	—
99	C₃×D₁₂	—	2	2	no	—
100	C₃×D₁₂	—	4	2	no	—
101	C₃×D₁₂	—	4	2	no	—
102	C₂³×C₉	—	6	2	no	—
103	C₃×C₃⋊C₈	—	2	2	no	—
104	Q₈×C₉	—	2	2	no	—
105	Q₈×C₉	—	2	1	no	—
106	Q₈×C₉	—	2	2	no	—
107	Q₈×C₉	D₁₂	1	2	yes	complemented
108	D₄×C₉	—	6	2	no	—
109	D₄×C₉	—	6	2	no	—
110	D₄×C₉	—	12	2	no	—
111	D₄×C₉	—	12	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	1	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	#67	1	1	yes	characteristic
120	C₂×C₄×C₉	#67	1	1	yes	characteristic
121	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
122	C₈×C₉	—	6	2	no	—
123	#460	—	3	1	no	Sylow
124	C₂×C₃×C₉	#121	1	1	yes	characteristic
125	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	D₃×C₉	—	8	2	no	—
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
130	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
131	C₂×C₃×D₄	—	3	2	no	—
132	C₂×C₃×D₄	—	6	1	no	—
133	C₂×C₃×D₄	—	6	1	no	—
134	C₂×D₁₂	—	2	1	no	—
135	C₂×D₁₂	—	2	1	no	—
136	C₂×D₁₂	C₂²×C₉	1	2	yes	—
137	C₃×SD₁₆	—	6	4	no	—
138	C₂×C₃×C₈	—	3	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	2	1	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
144	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	—	2	1	no	—
149	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
150	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
151	#215	—	2	4	no	—
152	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
153	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
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	2	no	—
158	C₃²×C₄	—	2	1	no	—
159	C₃²×C₄	—	4	1	no	—
160	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
161	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
162	C₃²×C₄	C₃×#41	1	1	yes	characteristic
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	12	2	no	—
165	C₂²×C₉	—	12	2	no	—
166	C₂²×C₉	—	12	2	no	—
167	C₂²×C₉	C₂×#67	1	1	yes	characteristic
168	C₄×C₉	—	2	4	no	—
169	C₄×C₉	—	2	2	no	—
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	—	4	2	no	—
176	C₄×C₉	—	4	1	no	—
177	C₄×C₉	#215	1	2	yes	complemented
178	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
179	C₄×C₉	#235	1	1	yes	characteristic
180	C₂×SD₁₆	—	3	2	no	—
181	#127	—	3	1	no	—
182	C₄×Q₈	D₃×C₉	1	1	yes	characteristic, complemented
183	C₄⋊C₈	—	3	1	no	—
184	#102	—	3	2	no	—
185	C₃×C₉	#460	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₃	—	6	2	no	—
187	C₂²×D₃	—	2	2	no	—
188	C₃×Q₈	—	2	2	no	—
189	C₃×Q₈	—	2	1	no	—
190	C₃×Q₈	—	2	1	no	—
191	C₃×Q₈	—	2	2	no	—
192	C₃×Q₈	D₄×C₉	1	2	yes	complemented
193	C₃×Q₈	C₃×D₁₂	1	2	yes	—
194	C₃×D₄	—	6	2	no	—
195	C₃×D₄	—	6	2	no	—
196	C₃×D₄	—	12	2	no	—
197	C₃×D₄	—	12	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
208	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
210	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
211	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
213	D₁₂	—	2	2	no	—
214	D₁₂	—	2	2	no	—
215	D₁₂	—	4	2	no	—
216	D₁₂	—	4	2	no	—
217	C₃×C₈	—	6	2	no	—
218	C₃⋊C₈	—	2	2	no	—
219	C₂×C₃²	C₃×#121	1	1	yes	characteristic
220	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
221	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₃×D₃	—	8	2	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	24	2	no	—
227	C₂×C₉	#930	1	1	yes	characteristic
228	C₂×C₉	C₂×#215	1	1	yes	characteristic
229	C₂×C₉	#949	1	1	yes	characteristic
230	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
231	C₂×D₄	—	3	2	no	—
232	C₂×D₄	—	6	1	no	—
233	C₂×D₄	—	6	1	no	—
234	SD₁₆	—	6	4	no	—
235	C₂×C₈	—	3	2	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
238	C₄²	—	2	1	no	—
239	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	12	2	no	—
242	C₂²×C₃	—	12	2	no	—
243	C₂²×C₃	—	12	2	no	—
244	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
245	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
246	C₂×D₃	—	4	2	no	—
247	C₂×D₃	—	4	2	no	—
248	C₂×D₃	—	4	2	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	C₉×SD₁₆	1	2	yes	complemented
262	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
263	C₃×C₄	C₉×#41	1	1	yes	characteristic
264	C₃×C₄	C₃×#215	1	2	yes	—
265	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
266	C₃×C₄	C₃×#235	1	1	yes	characteristic
267	C₃²	C₃×#460	1	1	yes	characteristic
268	C₉	—	2	1	no	—
269	C₉	#5027	1	1	yes	characteristic, direct product factor
270	C₂³	—	6	2	no	—
271	Q₈	—	2	1	no	—
272	Q₈	C₉×D₁₂	1	2	yes	complemented
273	D₄	—	6	2	no	—
274	D₄	—	6	2	no	—
275	D₄	—	12	2	no	—
276	D₄	—	12	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	C₉×#67	1	1	yes	characteristic
280	C₂×C₄	C₉×#67	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
282	C₈	—	6	2	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	24	2	no	—
287	C₂×C₃	C₉×#121	1	1	yes	characteristic
288	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
289	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₃	C₃×#930	1	1	yes	characteristic
291	C₂×C₃	C₂×C₃×#215	1	1	yes	characteristic
292	C₂×C₃	C₃×#949	1	1	yes	characteristic
293	D₃	—	8	2	no	—
294	C₂²	—	12	2	no	—
295	C₂²	—	12	2	no	—
296	C₂²	—	12	2	no	—
297	C₂²	C₂×C₉×#67	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₉×#215	1	2	yes	complemented
302	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
303	C₄	C₉×#235	1	1	yes	characteristic
304	C₃	—	2	1	no	—
305	C₃	C₉×#460	1	1	yes	characteristic, complemented
306	C₃	C₃×#5027	1	1	yes	characteristic
307	C₂	—	24	2	no	—
308	C₂	C₉×#930	1	1	yes	characteristic
309	C₂	C₂×C₉×#215	1	1	yes	characteristic
310	C₂	C₉×#949	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#874	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×Q₈×C₉	D₃
hypocenter	C₃	C₉×#460
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×Q₈×C₉	C₂	semidirect
2. C₂×C₉×#215	C₂	semidirect
4. C₉×#809	C₂	semidirect
5. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
26. C₄×Q₈×C₉	D₃	semidirect
31. C₃×Q₈×C₉	D₄	semidirect
43. #5027	C₉	direct (N × H)
76. C₃×C₄×C₉	SD₁₆	semidirect
82. C₃×C₄×Q₈	C₂×C₉	semidirect
84. C₂×#215	C₂×C₉	semidirect
88. #809	C₂×C₉	semidirect
89. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
107. Q₈×C₉	D₁₂	semidirect
177. C₄×C₉	#215	semidirect
182. C₄×Q₈	D₃×C₉	semidirect
185. C₃×C₉	#460	semidirect
192. C₃×Q₈	D₄×C₉	semidirect
261. C₃×C₄	C₉×SD₁₆	semidirect
269. C₉	#5027	direct (N × H)
272. Q₈	C₉×D₁₂	semidirect
301. C₄	C₉×#215	semidirect
305. C₃	C₉×#460	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₂×C₉×#215	2
C₉×#874	2
C₉×#809	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#5027	3
C₉×#460	3

Subgroups — C9×#5027 #13142783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5027 #13142783