Subgroups — C₉×#1903 #11766624

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 339 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×#568	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	C₂×C₉×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#338	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1903	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	C₄	1	2	yes	complemented
14	C₂×C₉×D₈	C₄	1	2	yes	complemented
15	C₉×#131	—	2	2	no	—
16	C₉×#131	—	2	2	no	—
17	C₂×C₉×M₁₆	—	2	2	no	—
18	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
22	C₉×#117	C₂²	1	1	yes	characteristic
23	C₉×#104	—	2	2	no	—
24	C₉×#102	—	2	2	no	—
25	C₉×#100	—	2	2	no	—
26	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
27	C₃×#568	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	C₂×C₃×#100	C₂×C₃	1	1	yes	characteristic
31	C₃×#344	C₂×C₃	1	1	yes	characteristic
32	C₃×#338	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	D₄	1	2	yes	—
37	C₉×#41	D₄	1	2	yes	—
38	C₂×Q₈×C₉	C₂×C₄	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	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
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	4	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	—	2	1	no	—
58	C₉×M₁₆	—	2	1	no	—
59	C₉×M₁₆	D₄	1	2	yes	—
60	C₂×C₈×C₉	—	2	4	no	—
61	C₂×C₈×C₉	—	2	1	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
64	C₂×C₈×C₉	D₄	1	2	yes	—
65	C₉×C₄⋊C₄	—	2	1	no	—
66	C₉×C₂²⋊C₄	—	2	1	no	—
67	C₄²×C₉	—	2	1	no	—
68	#1903	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₃×AGL(1,ℤ/8)	—	2	2	no	—
72	C₃×AGL(1,ℤ/8)	—	2	2	no	—
73	C₂×C₃×SD₁₆	C₃×C₄	1	2	yes	—
74	C₂×C₃×D₈	C₃×C₄	1	2	yes	—
75	C₃×#131	—	2	2	no	—
76	C₃×#131	—	2	2	no	—
77	C₂×C₃×M₁₆	—	2	2	no	—
78	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₃×#117	C₂²×C₃	1	1	yes	characteristic
83	C₃×#104	—	2	2	no	—
84	C₃×#102	—	2	2	no	—
85	C₃×#100	—	2	2	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	—	2	2	no	—
88	C₂³×C₉	—	4	1	no	—
89	C₂³×C₉	—	4	1	no	—
90	C₂³×C₉	—	8	2	no	—
91	C₂³×C₉	#41	1	1	yes	characteristic
92	Q₈×C₉	—	2	1	no	—
93	Q₈×C₉	C₂²⋊C₄	1	2	yes	—
94	D₄×C₉	—	2	2	no	—
95	D₄×C₉	—	2	2	no	—
96	D₄×C₉	—	2	2	no	—
97	D₄×C₉	—	2	1	no	—
98	D₄×C₉	—	2	2	no	—
99	D₄×C₉	—	4	1	no	—
100	D₄×C₉	—	4	1	no	—
101	D₄×C₉	C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	4	1	no	—
106	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	#41	1	1	yes	characteristic
112	C₈×C₉	—	2	2	no	—
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	—	2	2	no	—
115	C₈×C₉	—	4	1	no	—
116	C₂×AGL(1,ℤ/8)	C₂×C₉	1	1	yes	characteristic
117	#568	C₂×C₉	1	1	yes	characteristic, complemented
118	#421	C₂×C₉	1	1	yes	characteristic, complemented
119	#418	C₂×C₉	1	1	yes	characteristic, complemented
120	C₂×#100	C₂×C₉	1	1	yes	characteristic, complemented
121	#344	C₂×C₉	1	1	yes	characteristic, complemented
122	#338	C₂×C₉	1	1	yes	characteristic, complemented
123	C₂⁴×C₃	—	2	1	no	—
124	C₃×#41	—	2	1	no	—
125	C₃×#41	—	2	1	no	—
126	C₃×#41	C₃×D₄	1	2	yes	—
127	C₃×#41	C₃×D₄	1	2	yes	—
128	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
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	—	2	1	no	—
134	C₂×C₃×D₄	—	2	1	no	—
135	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
136	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
137	C₂²×C₃×C₄	—	2	1	no	—
138	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₃×SD₁₆	—	2	4	no	—
140	C₃×D₈	—	2	4	no	—
141	C₃×M₁₆	—	2	4	no	—
142	C₃×M₁₆	—	2	1	no	—
143	C₃×M₁₆	—	2	1	no	—
144	C₃×M₁₆	—	2	1	no	—
145	C₃×M₁₆	—	2	1	no	—
146	C₃×M₁₆	—	2	1	no	—
147	C₃×M₁₆	—	2	1	no	—
148	C₃×M₁₆	—	2	1	no	—
149	C₃×M₁₆	C₃×D₄	1	2	yes	—
150	C₂×C₃×C₈	—	2	4	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
154	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₂²⋊C₄	—	2	1	no	—
157	C₃×C₄²	—	2	1	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	4	1	no	—
160	C₂²×C₉	—	4	2	no	—
161	C₂²×C₉	—	8	1	no	—
162	C₂²×C₉	—	8	2	no	—
163	C₂²×C₉	—	8	2	no	—
164	C₂²×C₉	—	8	2	no	—
165	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
166	C₂²×C₉	#121	1	1	yes	characteristic
167	C₂²×C₉	#123	1	1	yes	characteristic
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	—	8	1	no	—
170	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₄×C₉	C₄×D₄	1	1	yes	characteristic
172	C₄×C₉	#120	1	1	yes	characteristic
173	C₄×C₉	#121	1	1	yes	characteristic
174	C₂×#41	C₂²×C₉	1	1	yes	characteristic
175	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
176	AGL(1,ℤ/8)	—	2	2	no	—
177	AGL(1,ℤ/8)	—	2	2	no	—
178	C₂×SD₁₆	C₄×C₉	1	2	yes	complemented
179	C₂×D₈	C₄×C₉	1	2	yes	complemented
180	#131	—	2	2	no	—
181	#131	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
184	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
185	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
186	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
187	#117	C₂²×C₉	1	1	yes	characteristic
188	#104	—	2	2	no	—
189	#102	—	2	2	no	—
190	#100	—	2	2	no	—
191	C₂³×C₃	—	2	1	no	—
192	C₂³×C₃	—	2	2	no	—
193	C₂³×C₃	—	4	1	no	—
194	C₂³×C₃	—	4	1	no	—
195	C₂³×C₃	—	8	2	no	—
196	C₂³×C₃	C₃×#41	1	1	yes	characteristic
197	C₃×Q₈	—	2	1	no	—
198	C₃×Q₈	C₃×C₂²⋊C₄	1	2	yes	—
199	C₃×D₄	—	2	2	no	—
200	C₃×D₄	—	2	2	no	—
201	C₃×D₄	—	2	2	no	—
202	C₃×D₄	—	2	1	no	—
203	C₃×D₄	—	2	2	no	—
204	C₃×D₄	—	4	1	no	—
205	C₃×D₄	—	4	1	no	—
206	C₃×D₄	C₃×C₂²⋊C₄	1	2	yes	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
217	C₃×C₈	—	2	2	no	—
218	C₃×C₈	—	2	2	no	—
219	C₃×C₈	—	2	2	no	—
220	C₃×C₈	—	4	1	no	—
221	C₂×C₉	—	2	1	no	—
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	4	2	no	—
225	C₂×C₉	—	8	2	no	—
226	C₂×C₉	#386	1	1	yes	characteristic
227	C₂⁴	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	D₄×C₉	1	2	yes	—
231	#41	D₄×C₉	1	2	yes	—
232	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
240	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
243	SD₁₆	—	2	4	no	—
244	D₈	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	—	2	1	no	—
247	M₁₆	—	2	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	2	1	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	D₄×C₉	1	2	yes	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
258	C₂×C₈	D₄×C₉	1	2	yes	—
259	C₄⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	4	1	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	8	1	no	—
266	C₂²×C₃	—	8	2	no	—
267	C₂²×C₃	—	8	2	no	—
268	C₂²×C₃	—	8	2	no	—
269	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
270	C₂²×C₃	C₃×#121	1	1	yes	characteristic
271	C₂²×C₃	C₃×#123	1	1	yes	characteristic
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	8	1	no	—
274	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
275	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
276	C₃×C₄	C₃×#120	1	1	yes	characteristic
277	C₃×C₄	C₃×#121	1	1	yes	characteristic
278	C₉	#1903	1	1	yes	characteristic, direct product factor, Sylow
279	C₂³	—	2	1	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	8	2	no	—
284	C₂³	C₉×#41	1	1	yes	characteristic
285	Q₈	—	2	1	no	—
286	Q₈	C₉×C₂²⋊C₄	1	2	yes	—
287	D₄	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	C₉×C₂²⋊C₄	1	2	yes	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
302	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
304	C₂×C₄	C₉×#41	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	4	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	4	2	no	—
313	C₂×C₃	—	8	2	no	—
314	C₂×C₃	C₃×#386	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
323	C₂²	C₉×#121	1	1	yes	characteristic
324	C₂²	C₉×#123	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	8	1	no	—
327	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
328	C₄	C₄×D₄×C₉	1	1	yes	characteristic
329	C₄	C₉×#120	1	1	yes	characteristic
330	C₄	C₉×#121	1	1	yes	characteristic
331	C₃	C₃×#1903	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	4	2	no	—
336	C₂	—	8	2	no	—
337	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#386
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃
LCS	G ▷ C₂×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×#568	C₂	semidirect
3. C₉×#421	C₂	semidirect
4. C₉×#418	C₂	semidirect
5. C₂×C₉×#100	C₂	semidirect
6. C₉×#344	C₂	semidirect
7. C₉×#338	C₂	semidirect
13. C₂×C₉×SD₁₆	C₄	semidirect
14. C₂×C₉×D₈	C₄	semidirect
68. #1903	C₉	direct (N × H)
117. #568	C₂×C₉	semidirect
118. #421	C₂×C₉	semidirect
119. #418	C₂×C₉	semidirect
120. C₂×#100	C₂×C₉	semidirect
121. #344	C₂×C₉	semidirect
122. #338	C₂×C₉	semidirect
178. C₂×SD₁₆	C₄×C₉	semidirect
179. C₂×D₈	C₄×C₉	semidirect
278. C₉	#1903	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×#568	2
C₉×#421	2
C₉×#418	2
C₂×C₉×#100	2
C₉×#344	2
C₉×#338	2
C₃×#1903	3

Subgroups — C9×#1903 #11766624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1903 #11766624