Subgroups — C₉×#2048 #11766769

Order 1152.

411 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₉×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#421	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#413	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×#100	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2048	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₉×AGL(1,ℤ/8)	—	2	2	no	—
12	C₉×AGL(1,ℤ/8)	—	2	2	no	—
13	C₂×C₉×SD₁₆	—	2	1	no	—
14	C₂×C₉×D₈	—	2	1	no	—
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
19	C₉×#117	C₂²	1	1	yes	characteristic
20	C₉×#108	—	2	2	no	—
21	C₉×#104	—	2	2	no	—
22	C₉×#102	—	2	2	no	—
23	C₉×#101	—	2	1	no	—
24	C₉×#100	—	2	1	no	—
25	C₉×#100	—	2	2	no	—
26	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
27	C₃×#430	C₂×C₃	1	1	yes	characteristic
28	C₃×#421	C₂×C₃	1	1	yes	characteristic
29	C₃×#418	C₂×C₃	1	1	yes	characteristic
30	C₃×#413	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×#100	C₂×C₃	1	1	yes	characteristic
32	C₃×#344	C₂×C₃	1	1	yes	characteristic
33	C₂⁴×C₉	—	2	1	no	—
34	C₉×#41	—	2	1	no	—
35	C₉×#41	—	2	1	no	—
36	C₉×#41	—	2	1	no	—
37	C₉×#41	—	2	1	no	—
38	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
39	C₂×D₄×C₉	—	2	1	no	—
40	C₂×D₄×C₉	—	2	1	no	—
41	C₂×D₄×C₉	—	2	1	no	—
42	C₂×D₄×C₉	—	2	1	no	—
43	C₂×D₄×C₉	—	2	1	no	—
44	C₂×D₄×C₉	—	2	1	no	—
45	C₂×D₄×C₉	D₄	1	1	yes	characteristic
46	C₂×D₄×C₉	D₄	1	2	yes	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₉×SD₁₆	—	2	4	no	—
50	C₉×D₈	—	2	4	no	—
51	C₉×M₁₆	—	2	1	no	—
52	C₉×M₁₆	—	2	1	no	—
53	C₉×M₁₆	—	2	1	no	—
54	C₉×M₁₆	—	2	1	no	—
55	C₉×M₁₆	—	2	1	no	—
56	C₉×M₁₆	—	2	1	no	—
57	C₉×M₁₆	D₄	1	2	yes	—
58	C₉×M₁₆	D₄	1	2	yes	—
59	C₂×C₈×C₉	—	2	1	no	—
60	C₂×C₈×C₉	—	2	1	no	—
61	C₂×C₈×C₉	—	2	1	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	1	no	—
64	C₉×C₂²⋊C₄	—	2	1	no	—
65	C₄²×C₉	—	2	1	no	—
66	#2048	C₉	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
68	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₃×AGL(1,ℤ/8)	—	2	2	no	—
70	C₃×AGL(1,ℤ/8)	—	2	2	no	—
71	C₂×C₃×SD₁₆	—	2	1	no	—
72	C₂×C₃×D₈	—	2	1	no	—
73	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
77	C₃×#117	C₂²×C₃	1	1	yes	characteristic
78	C₃×#108	—	2	2	no	—
79	C₃×#104	—	2	2	no	—
80	C₃×#102	—	2	2	no	—
81	C₃×#101	—	2	1	no	—
82	C₃×#100	—	2	1	no	—
83	C₃×#100	—	2	2	no	—
84	C₂³×C₉	—	2	2	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	—	4	1	no	—
87	C₂³×C₉	—	4	1	no	—
88	C₂³×C₉	—	8	2	no	—
89	C₂³×C₉	#41	1	1	yes	characteristic
90	Q₈×C₉	—	2	1	no	—
91	Q₈×C₉	—	2	1	no	—
92	D₄×C₉	—	2	1	no	—
93	D₄×C₉	—	2	2	no	—
94	D₄×C₉	—	2	1	no	—
95	D₄×C₉	—	2	2	no	—
96	D₄×C₉	—	4	1	no	—
97	D₄×C₉	—	4	1	no	—
98	D₄×C₉	—	4	1	no	—
99	D₄×C₉	—	4	1	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	4	1	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	—	4	1	no	—
104	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	#41	1	1	yes	characteristic
109	C₂×C₄×C₉	#41	1	1	yes	characteristic
110	C₈×C₉	—	2	2	no	—
111	C₈×C₉	—	2	2	no	—
112	C₈×C₉	—	4	1	no	—
113	C₈×C₉	—	4	1	no	—
114	C₂×AGL(1,ℤ/8)	C₂×C₉	1	1	yes	characteristic
115	#430	C₂×C₉	1	1	yes	characteristic, complemented
116	#421	C₂×C₉	1	1	yes	characteristic, complemented
117	#418	C₂×C₉	1	1	yes	characteristic, complemented
118	#413	C₂×C₉	1	1	yes	characteristic, complemented
119	C₂×#100	C₂×C₉	1	1	yes	characteristic, complemented
120	#344	C₂×C₉	1	1	yes	characteristic, complemented
121	C₂⁴×C₃	—	2	1	no	—
122	C₃×#41	—	2	1	no	—
123	C₃×#41	—	2	1	no	—
124	C₃×#41	—	2	1	no	—
125	C₃×#41	—	2	1	no	—
126	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
127	C₂×C₃×D₄	—	2	1	no	—
128	C₂×C₃×D₄	—	2	1	no	—
129	C₂×C₃×D₄	—	2	1	no	—
130	C₂×C₃×D₄	—	2	1	no	—
131	C₂×C₃×D₄	—	2	1	no	—
132	C₂×C₃×D₄	—	2	1	no	—
133	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
134	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₃×SD₁₆	—	2	4	no	—
138	C₃×D₈	—	2	4	no	—
139	C₃×M₁₆	—	2	1	no	—
140	C₃×M₁₆	—	2	1	no	—
141	C₃×M₁₆	—	2	1	no	—
142	C₃×M₁₆	—	2	1	no	—
143	C₃×M₁₆	—	2	1	no	—
144	C₃×M₁₆	—	2	1	no	—
145	C₃×M₁₆	C₃×D₄	1	2	yes	—
146	C₃×M₁₆	C₃×D₄	1	2	yes	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₂²⋊C₄	—	2	1	no	—
153	C₃×C₄²	—	2	1	no	—
154	C₂²×C₉	—	4	1	no	—
155	C₂²×C₉	—	4	1	no	—
156	C₂²×C₉	—	4	2	no	—
157	C₂²×C₉	—	8	1	no	—
158	C₂²×C₉	—	8	2	no	—
159	C₂²×C₉	—	8	2	no	—
160	C₂²×C₉	—	8	2	no	—
161	C₂²×C₉	#121	1	1	yes	characteristic
162	C₂²×C₉	#123	1	1	yes	characteristic
163	C₂²×C₉	#124	1	1	yes	characteristic
164	C₄×C₉	—	4	1	no	—
165	C₄×C₉	—	8	1	no	—
166	C₄×C₉	#120	1	1	yes	characteristic
167	C₄×C₉	#121	1	1	yes	characteristic
168	C₄×C₉	#121	1	1	yes	characteristic
169	C₄×C₉	#123	1	1	yes	characteristic
170	C₂×#41	C₂²×C₉	1	1	yes	characteristic
171	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
172	AGL(1,ℤ/8)	—	2	2	no	—
173	AGL(1,ℤ/8)	—	2	2	no	—
174	C₂×SD₁₆	—	2	1	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
177	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
178	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
179	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
180	#117	C₂²×C₉	1	1	yes	characteristic
181	#108	—	2	2	no	—
182	#104	—	2	2	no	—
183	#102	—	2	2	no	—
184	#101	—	2	1	no	—
185	#100	—	2	1	no	—
186	#100	—	2	2	no	—
187	C₂³×C₃	—	2	2	no	—
188	C₂³×C₃	—	2	1	no	—
189	C₂³×C₃	—	4	1	no	—
190	C₂³×C₃	—	4	1	no	—
191	C₂³×C₃	—	8	2	no	—
192	C₂³×C₃	C₃×#41	1	1	yes	characteristic
193	C₃×Q₈	—	2	1	no	—
194	C₃×Q₈	—	2	1	no	—
195	C₃×D₄	—	2	1	no	—
196	C₃×D₄	—	2	2	no	—
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	—	2	2	no	—
199	C₃×D₄	—	4	1	no	—
200	C₃×D₄	—	4	1	no	—
201	C₃×D₄	—	4	1	no	—
202	C₃×D₄	—	4	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
212	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	—	2	2	no	—
215	C₃×C₈	—	4	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₂×C₉	—	2	1	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	8	1	no	—
221	C₂×C₉	—	8	2	no	—
222	C₂×C₉	#394	1	1	yes	characteristic
223	C₂⁴	—	2	1	no	—
224	#41	—	2	1	no	—
225	#41	—	2	1	no	—
226	#41	—	2	1	no	—
227	#41	—	2	1	no	—
228	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	D₄×C₉	1	1	yes	characteristic
236	C₂×D₄	D₄×C₉	1	2	yes	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
239	SD₁₆	—	2	4	no	—
240	D₈	—	2	4	no	—
241	M₁₆	—	2	1	no	—
242	M₁₆	—	2	1	no	—
243	M₁₆	—	2	1	no	—
244	M₁₆	—	2	1	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	2	1	no	—
247	M₁₆	D₄×C₉	1	2	yes	—
248	M₁₆	D₄×C₉	1	2	yes	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₂²×C₃	—	4	1	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	8	2	no	—
260	C₂²×C₃	—	8	1	no	—
261	C₂²×C₃	—	8	2	no	—
262	C₂²×C₃	—	8	2	no	—
263	C₂²×C₃	C₃×#121	1	1	yes	characteristic
264	C₂²×C₃	C₃×#123	1	1	yes	characteristic
265	C₂²×C₃	C₃×#124	1	1	yes	characteristic
266	C₃×C₄	—	4	1	no	—
267	C₃×C₄	—	8	1	no	—
268	C₃×C₄	C₃×#120	1	1	yes	characteristic
269	C₃×C₄	C₃×#121	1	1	yes	characteristic
270	C₃×C₄	C₃×#121	1	1	yes	characteristic
271	C₃×C₄	C₃×#123	1	1	yes	characteristic
272	C₉	#2048	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	—	2	2	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	8	2	no	—
278	C₂³	C₉×#41	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	D₄	—	2	1	no	—
282	D₄	—	2	2	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	2	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
297	C₂×C₄	C₉×#41	1	1	yes	characteristic
298	C₂×C₄	C₉×#41	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	2	2	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	8	2	no	—
307	C₂×C₃	—	8	1	no	—
308	C₂×C₃	C₃×#394	1	1	yes	characteristic
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₉×#121	1	1	yes	characteristic
317	C₂²	C₉×#123	1	1	yes	characteristic
318	C₂²	C₉×#124	1	1	yes	characteristic
319	C₄	—	4	1	no	—
320	C₄	—	8	1	no	—
321	C₄	C₉×#120	1	1	yes	characteristic
322	C₄	C₉×#121	1	1	yes	characteristic
323	C₄	C₉×#121	1	1	yes	characteristic
324	C₄	C₉×#123	1	1	yes	characteristic
325	C₃	C₃×#2048	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	8	2	no	—
330	C₂	—	8	1	no	—
331	C₂	C₉×#394	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#394
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#394
prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#430	C₂	semidirect
3. C₉×#421	C₂	semidirect
4. C₉×#418	C₂	semidirect
5. C₉×#413	C₂	semidirect
6. C₂×C₉×#100	C₂	semidirect
7. C₉×#344	C₂	semidirect
66. #2048	C₉	direct (N × H)
115. #430	C₂×C₉	semidirect
116. #421	C₂×C₉	semidirect
117. #418	C₂×C₉	semidirect
118. #413	C₂×C₉	semidirect
119. C₂×#100	C₂×C₉	semidirect
120. #344	C₂×C₉	semidirect
272. C₉	#2048	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×AGL(1,ℤ/8)	2
C₉×#430	2
C₉×#421	2
C₉×#418	2
C₉×#413	2
C₂×C₉×#100	2
C₉×#344	2
C₃×#2048	3

Subgroups — C9×#2048 #11766769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2048 #11766769