Subgroups — C₂×C₉×#459 #11767750

Order 1152.

972 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#459	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#459	C₃	1	1	yes	maximal, characteristic
8	C₂²×D₄×C₉	—	2	1	no	—
9	C₂²×D₄×C₉	—	2	1	no	—
10	C₂²×D₄×C₉	C₂²	1	2	yes	—
11	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₉	—	2	1	no	—
13	C₂×C₉×D₈	—	2	8	no	—
14	C₂×C₉×D₈	C₂²	1	8	yes	complemented
15	C₂²×C₈×C₉	C₂²	1	2	yes	—
16	C₉×#127	—	2	2	no	—
17	C₉×#127	C₂²	1	4	yes	complemented
18	C₄×D₄×C₉	—	2	2	no	—
19	C₄×D₄×C₉	C₂²	1	4	yes	complemented
20	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
24	C₉×#102	C₂²	1	8	yes	complemented
25	C₂²×C₃×D₈	C₂×C₃	1	2	yes	—
26	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
28	C₃×#459	C₂×C₃	1	8	yes	—
29	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#102	C₂×C₃	1	2	yes	—
31	C₂⁴×C₉	—	2	1	no	—
32	C₂⁴×C₉	—	2	2	no	—
33	C₂×D₄×C₉	—	2	1	no	—
34	C₂×D₄×C₉	—	2	4	no	—
35	C₂×D₄×C₉	—	2	2	no	—
36	C₂×D₄×C₉	—	2	4	no	—
37	C₂×D₄×C₉	—	2	4	no	—
38	C₂×D₄×C₉	—	2	2	no	—
39	C₂×D₄×C₉	—	2	4	no	—
40	C₂×D₄×C₉	—	4	4	no	—
41	C₂×D₄×C₉	—	4	4	no	—
42	C₂×D₄×C₉	D₄	1	4	yes	complemented
43	C₂×D₄×C₉	C₂³	1	4	yes	—
44	C₂×D₄×C₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	—	2	4	no	—
47	C₂²×C₄×C₉	—	4	4	no	—
48	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×D₈	—	2	16	no	—
52	C₂×C₈×C₉	—	2	4	no	—
53	C₂×C₈×C₉	C₂³	1	4	yes	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	C₂³	1	2	yes	—
56	C₉×C₂²⋊C₄	—	2	4	no	—
57	C₄²×C₉	D₄	1	2	yes	—
58	C₄²×C₉	C₂³	1	2	yes	—
59	C₂×#459	C₉	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²×C₃×D₄	—	2	1	no	—
61	C₂²×C₃×D₄	—	2	1	no	—
62	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
63	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
64	C₂³×C₃×C₄	—	2	1	no	—
65	C₂×C₃×D₈	—	2	8	no	—
66	C₂×C₃×D₈	C₂²×C₃	1	8	yes	—
67	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
68	C₃×#127	—	2	2	no	—
69	C₃×#127	C₂²×C₃	1	4	yes	—
70	C₃×C₄×D₄	—	2	2	no	—
71	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
72	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
74	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
76	C₃×#102	C₂²×C₃	1	8	yes	—
77	C₂³×C₉	—	2	4	no	—
78	C₂³×C₉	—	2	2	no	—
79	C₂³×C₉	—	2	4	no	—
80	C₂³×C₉	—	4	2	no	—
81	C₂³×C₉	—	4	2	no	—
82	C₂³×C₉	—	4	4	no	—
83	C₂³×C₉	—	4	4	no	—
84	C₂³×C₉	—	4	4	no	—
85	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
86	D₄×C₉	—	2	4	no	—
87	D₄×C₉	—	2	4	no	—
88	D₄×C₉	—	2	2	no	—
89	D₄×C₉	—	2	4	no	—
90	D₄×C₉	—	4	8	no	—
91	D₄×C₉	—	4	4	no	—
92	D₄×C₉	—	4	8	no	—
93	D₄×C₉	C₂×D₄	1	4	yes	complemented
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	4	2	no	—
97	C₂×C₄×C₉	—	4	8	no	—
98	C₂×C₄×C₉	D₈	1	4	yes	complemented
99	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₉	#41	1	2	yes	—
104	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
105	C₈×C₉	—	2	4	no	—
106	C₂²×D₈	C₂×C₉	1	2	yes	complemented
107	C₂×#127	C₂×C₉	1	1	yes	characteristic, complemented
108	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic, complemented
109	#459	C₂×C₉	1	8	yes	direct product factor
110	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
111	C₂×#102	C₂×C₉	1	2	yes	complemented
112	C₂⁴×C₃	—	2	1	no	—
113	C₂⁴×C₃	—	2	2	no	—
114	C₂×C₃×D₄	—	2	1	no	—
115	C₂×C₃×D₄	—	2	4	no	—
116	C₂×C₃×D₄	—	2	2	no	—
117	C₂×C₃×D₄	—	2	4	no	—
118	C₂×C₃×D₄	—	2	4	no	—
119	C₂×C₃×D₄	—	2	2	no	—
120	C₂×C₃×D₄	—	2	4	no	—
121	C₂×C₃×D₄	—	4	4	no	—
122	C₂×C₃×D₄	—	4	4	no	—
123	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
124	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
125	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	4	no	—
128	C₂²×C₃×C₄	—	4	4	no	—
129	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₃×D₈	—	2	16	no	—
133	C₂×C₃×C₈	—	2	4	no	—
134	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
135	C₃×C₄⋊C₄	—	2	1	no	—
136	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
137	C₃×C₂²⋊C₄	—	2	4	no	—
138	C₃×C₄²	C₃×D₄	1	2	yes	—
139	C₃×C₄²	C₂³×C₃	1	2	yes	—
140	C₂²×C₉	—	2	4	no	—
141	C₂²×C₉	—	4	2	no	—
142	C₂²×C₉	—	4	2	no	—
143	C₂²×C₉	—	4	8	no	—
144	C₂²×C₉	—	4	4	no	—
145	C₂²×C₉	—	4	4	no	—
146	C₂²×C₉	—	4	4	no	—
147	C₂²×C₉	—	8	8	no	—
148	C₂²×C₉	#121	1	2	yes	—
149	C₂²×C₉	C₂×D₈	1	2	yes	—
150	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
151	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₉	—	2	2	no	—
153	C₄×C₉	—	4	2	no	—
154	C₄×C₉	#121	1	2	yes	—
155	C₄×C₉	C₂×D₈	1	4	yes	complemented
156	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
157	C₄×C₉	C₂×#41	1	1	yes	characteristic
158	C₂²×D₄	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂²×D₄	C₂²×C₉	1	2	yes	—
161	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
162	C₂³×C₄	—	2	1	no	—
163	C₂×D₈	—	2	8	no	—
164	C₂×D₈	C₂²×C₉	1	8	yes	complemented
165	C₂²×C₈	C₂²×C₉	1	2	yes	—
166	#127	—	2	2	no	—
167	#127	C₂²×C₉	1	4	yes	complemented
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	C₂²×C₉	1	4	yes	complemented
170	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	2	1	no	—
172	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
173	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
174	#102	C₂²×C₉	1	8	yes	complemented
175	C₂³×C₃	—	2	4	no	—
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	2	4	no	—
178	C₂³×C₃	—	4	2	no	—
179	C₂³×C₃	—	4	2	no	—
180	C₂³×C₃	—	4	4	no	—
181	C₂³×C₃	—	4	4	no	—
182	C₂³×C₃	—	4	4	no	—
183	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
184	C₃×D₄	—	2	4	no	—
185	C₃×D₄	—	2	4	no	—
186	C₃×D₄	—	2	2	no	—
187	C₃×D₄	—	2	4	no	—
188	C₃×D₄	—	4	8	no	—
189	C₃×D₄	—	4	4	no	—
190	C₃×D₄	—	4	8	no	—
191	C₃×D₄	C₂×C₃×D₄	1	4	yes	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	4	2	no	—
195	C₂×C₃×C₄	—	4	8	no	—
196	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
197	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
200	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
201	C₂×C₃×C₄	C₃×#41	1	2	yes	—
202	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
203	C₃×C₈	—	2	4	no	—
204	C₂×C₉	—	4	4	no	—
205	C₂×C₉	—	8	4	no	—
206	C₂×C₉	#459	1	4	yes	direct product factor
207	C₂×C₉	C₂×#121	1	1	yes	characteristic
208	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
209	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂⁴	—	2	1	no	—
211	C₂⁴	—	2	2	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	D₄×C₉	1	4	yes	complemented
222	C₂×D₄	C₂³×C₉	1	4	yes	—
223	C₂×D₄	C₂³×C₉	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
228	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
230	D₈	—	2	16	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂³×C₉	1	4	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	C₂³×C₉	1	2	yes	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₄²	D₄×C₉	1	2	yes	—
237	C₄²	C₂³×C₉	1	2	yes	—
238	C₂²×C₃	—	2	4	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	—	4	2	no	—
241	C₂²×C₃	—	4	8	no	—
242	C₂²×C₃	—	4	4	no	—
243	C₂²×C₃	—	4	4	no	—
244	C₂²×C₃	—	4	4	no	—
245	C₂²×C₃	—	8	8	no	—
246	C₂²×C₃	C₃×#121	1	2	yes	—
247	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
248	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
249	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	4	2	no	—
252	C₃×C₄	C₃×#121	1	2	yes	—
253	C₃×C₄	C₂×C₃×D₈	1	4	yes	—
254	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
255	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
256	C₉	C₂×#459	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	2	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	4	no	—
265	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	4	8	no	—
271	D₄	—	4	4	no	—
272	D₄	—	4	8	no	—
273	D₄	C₂×D₄×C₉	1	4	yes	complemented
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	C₉×D₈	1	4	yes	complemented
279	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
280	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
282	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
283	C₂×C₄	C₉×#41	1	2	yes	—
284	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₂×C₃	—	4	4	no	—
287	C₂×C₃	—	8	4	no	—
288	C₂×C₃	C₃×#459	1	4	yes	—
289	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
290	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
291	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
292	C₂²	—	2	4	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	8	no	—
300	C₂²	C₉×#121	1	2	yes	—
301	C₂²	C₂×C₉×D₈	1	2	yes	—
302	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
303	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	C₉×#121	1	2	yes	—
307	C₄	C₂×C₉×D₈	1	4	yes	complemented
308	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
309	C₄	C₂×C₉×#41	1	1	yes	characteristic
310	C₃	C₂×C₃×#459	1	1	yes	characteristic
311	C₂	—	4	4	no	—
312	C₂	—	8	4	no	—
313	C₂	C₉×#459	1	4	yes	direct product factor
314	C₂	C₂×C₉×#121	1	1	yes	characteristic
315	C₂	C₂²×C₉×D₈	1	1	yes	characteristic
316	C₂	C₂×C₉×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#459	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×D₈	C₂	semidirect
2. C₂×C₉×#127	C₂	semidirect
3. C₂×C₄×D₄×C₉	C₂	semidirect
4. C₉×#459	C₂	direct (N × H)
5. C₂×C₉×C₄⋊C₈	C₂	semidirect
6. C₂×C₉×#102	C₂	semidirect
14. C₂×C₉×D₈	C₂²	semidirect
17. C₉×#127	C₂²	semidirect
19. C₄×D₄×C₉	C₂²	semidirect
23. C₉×C₄⋊C₈	C₂²	semidirect
24. C₉×#102	C₂²	semidirect
42. C₂×D₄×C₉	D₄	semidirect
59. C₂×#459	C₉	direct (N × H)
93. D₄×C₉	C₂×D₄	semidirect
98. C₂×C₄×C₉	D₈	semidirect
106. C₂²×D₈	C₂×C₉	semidirect
107. C₂×#127	C₂×C₉	semidirect
108. C₂×C₄×D₄	C₂×C₉	semidirect
109. #459	C₂×C₉	direct (N × H)
110. C₂×C₄⋊C₈	C₂×C₉	semidirect
111. C₂×#102	C₂×C₉	semidirect
155. C₄×C₉	C₂×D₈	semidirect
164. C₂×D₈	C₂²×C₉	semidirect
167. #127	C₂²×C₉	semidirect
169. C₄×D₄	C₂²×C₉	semidirect
173. C₄⋊C₈	C₂²×C₉	semidirect
174. #102	C₂²×C₉	semidirect
206. C₂×C₉	#459	direct (N × H)
221. C₂×D₄	D₄×C₉	semidirect
256. C₉	C₂×#459	direct (N × H)
273. D₄	C₂×D₄×C₉	semidirect
278. C₂×C₄	C₉×D₈	semidirect
307. C₄	C₂×C₉×D₈	semidirect
313. C₂	C₉×#459	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₈	2
C₂×C₉×#127	2
C₂×C₄×D₄×C₉	2
C₉×#459	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#102	2
C₂×C₃×#459	3

Subgroups — C2×C9×#459 #11767750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#459 #11767750