Subgroups — #11787625

Order 1152.

465 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#575	C₂	1	1	yes	maximal, characteristic
2	#10589921	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588917	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588914	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588912	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588232	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588231	C₂	1	1	yes	maximal, characteristic, complemented
8	#72116	—	3	1	no	maximal
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂×C₉×#41	C₂²	1	1	yes	characteristic
11	C₉×#137	—	2	1	no	—
12	C₉×#137	C₄	1	2	yes	complemented
13	C₉×AGL(1,ℤ/8)	—	2	1	no	—
14	C₉×AGL(1,ℤ/8)	C₄	1	2	yes	complemented
15	C₉×#135	—	2	2	no	—
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	#63510	—	2	2	no	—
18	#63510	—	2	2	no	—
19	#63488	C₂²	1	1	yes	characteristic
20	C₂×#3667	—	2	2	no	—
21	C₂×#3667	C₂²	1	1	yes	characteristic
22	C₂×#3667	C₂²	1	1	yes	characteristic
23	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
24	#63396	—	2	2	no	—
25	#63395	—	2	1	no	—
26	#63392	—	2	1	no	—
27	#63389	—	2	2	no	—
28	C₃×#575	D₃	1	1	yes	characteristic
29	#5819	—	3	1	no	—
30	#5236	—	3	1	no	—
31	#5234	—	3	1	no	—
32	#5233	—	3	1	no	—
33	#4559	—	3	1	no	—
34	#4558	—	3	1	no	—
35	C₉×#41	—	2	1	no	—
36	C₉×#41	—	2	1	no	—
37	C₉×#41	—	2	1	no	—
38	C₉×#41	—	2	1	no	—
39	C₉×#41	—	2	1	no	—
40	C₉×#41	—	2	1	no	—
41	C₉×#41	C₂×C₄	1	2	yes	—
42	C₉×#41	D₄	1	2	yes	—
43	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
44	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
45	C₂×D₄×C₉	—	2	1	no	—
46	C₂×D₄×C₉	—	2	1	no	—
47	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
48	C₂×D₄×C₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₉×Q₁₆	—	2	2	no	—
53	C₉×SD₁₆	—	2	2	no	—
54	C₉×SD₁₆	—	2	2	no	—
55	C₉×D₈	—	2	2	no	—
56	C₉×M₁₆	—	2	1	no	—
57	C₉×M₁₆	C₂×C₄	1	2	yes	—
58	C₂×C₈×C₉	—	2	1	no	—
59	#3676	—	2	1	no	—
60	C₄⋊Dic₉	—	2	1	no	—
61	C₄×Dic₉	—	2	1	no	—
62	#3667	—	2	4	no	—
63	#3667	—	2	1	no	—
64	#3667	—	2	1	no	—
65	#3667	—	2	1	no	—
66	#3667	D₄	1	2	yes	—
67	C₂×C₉⋊C₈	—	2	4	no	—
68	C₂×C₉⋊C₈	—	2	1	no	—
69	C₂×C₉⋊C₈	—	2	1	no	—
70	C₂×C₉⋊C₈	—	2	1	no	—
71	C₂×C₉⋊C₈	—	2	1	no	—
72	C₂×C₉⋊C₈	D₄	1	2	yes	—
73	#1881	—	9	1	no	Sylow
74	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
75	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
76	C₃×#137	—	2	1	no	—
77	C₃×#137	Dic₃	1	2	yes	—
78	C₃×AGL(1,ℤ/8)	—	2	1	no	—
79	C₃×AGL(1,ℤ/8)	Dic₃	1	2	yes	—
80	C₃×#135	—	2	2	no	—
81	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
82	#948	—	6	2	no	—
83	#948	—	6	2	no	—
84	#926	—	3	1	no	—
85	C₂×#208	—	3	1	no	—
86	C₂×#208	—	3	1	no	—
87	C₂×#208	—	6	2	no	—
88	C₂²×C₃⋊C₈	—	3	1	no	—
89	#837	—	6	2	no	—
90	#835	—	6	1	no	—
91	#832	—	6	1	no	—
92	#830	—	6	2	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	—	2	1	no	—
95	C₂³×C₉	#41	1	1	yes	characteristic
96	Q₈×C₉	—	2	1	no	—
97	Q₈×C₉	—	2	1	no	—
98	Q₈×C₉	—	2	1	no	—
99	Q₈×C₉	C₂²⋊C₄	1	2	yes	—
100	D₄×C₉	—	2	2	no	—
101	D₄×C₉	—	2	2	no	—
102	D₄×C₉	—	2	1	no	—
103	D₄×C₉	—	2	1	no	—
104	D₄×C₉	—	2	1	no	—
105	D₄×C₉	—	4	1	no	—
106	D₄×C₉	—	4	1	no	—
107	D₄×C₉	C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	4	1	no	—
113	C₂×C₄×C₉	—	4	1	no	—
114	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₉	#41	1	1	yes	characteristic
120	C₂×Dic₉	—	4	1	no	—
121	C₈×C₉	—	2	2	no	—
122	C₉⋊C₈	—	2	2	no	—
123	C₉⋊C₈	—	2	2	no	—
124	C₉⋊C₈	—	4	1	no	—
125	#575	D₉	1	1	yes	characteristic
126	#568	—	9	1	no	—
127	#421	—	9	1	no	—
128	#416	—	9	1	no	—
129	#408	—	9	1	no	—
130	#344	—	9	1	no	—
131	#341	—	9	1	no	—
132	C₃×#41	—	2	1	no	—
133	C₃×#41	—	2	1	no	—
134	C₃×#41	—	2	1	no	—
135	C₃×#41	—	2	1	no	—
136	C₃×#41	—	2	1	no	—
137	C₃×#41	—	2	1	no	—
138	C₃×#41	C₂×Dic₃	1	2	yes	—
139	C₃×#41	#67	1	2	yes	—
140	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
141	C₂×C₃×Q₈	#67	1	1	yes	characteristic
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	—	2	1	no	—
144	C₂×C₃×D₄	C₂×Dic₃	1	1	yes	characteristic
145	C₂×C₃×D₄	#67	1	1	yes	characteristic
146	C₂²×C₃×C₄	—	2	1	no	—
147	C₂²×C₃×C₄	—	2	1	no	—
148	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
149	C₃×Q₁₆	—	2	2	no	—
150	C₃×SD₁₆	—	2	2	no	—
151	C₃×SD₁₆	—	2	2	no	—
152	C₃×D₈	—	2	2	no	—
153	C₃×M₁₆	—	2	1	no	—
154	C₃×M₁₆	C₂×Dic₃	1	2	yes	—
155	C₂×C₃×C₈	—	2	1	no	—
156	#217	—	6	1	no	—
157	C₄⋊Dic₃	—	6	1	no	—
158	C₄×Dic₃	—	6	1	no	—
159	#208	—	3	2	no	—
160	#208	—	6	4	no	—
161	#208	—	6	1	no	—
162	#208	—	6	1	no	—
163	#208	—	6	1	no	—
164	C₂×C₃⋊C₈	—	3	2	no	—
165	C₂×C₃⋊C₈	—	6	4	no	—
166	C₂×C₃⋊C₈	—	6	1	no	—
167	C₂×C₃⋊C₈	—	6	1	no	—
168	C₂×C₃⋊C₈	—	6	1	no	—
169	C₂×C₃⋊C₈	—	6	1	no	—
170	C₂²×C₉	—	2	2	no	—
171	C₂²×C₉	—	4	1	no	—
172	C₂²×C₉	—	4	2	no	—
173	C₂²×C₉	—	4	1	no	—
174	C₂²×C₉	—	8	1	no	—
175	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
176	C₂²×C₉	#121	1	1	yes	characteristic
177	C₂²×C₉	#123	1	1	yes	characteristic
178	C₄×C₉	—	2	2	no	—
179	C₄×C₉	—	4	1	no	—
180	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₄×C₉	C₄×D₄	1	1	yes	characteristic
182	C₄×C₉	#120	1	1	yes	characteristic
183	C₄×C₉	#121	1	1	yes	characteristic
184	Dic₉	—	8	1	no	—
185	C₂×#41	C₂×D₉	1	1	yes	characteristic
186	C₂×#41	C₂×D₉	1	1	yes	characteristic
187	#137	—	2	1	no	—
188	#137	Dic₉	1	2	yes	complemented
189	AGL(1,ℤ/8)	—	2	1	no	—
190	AGL(1,ℤ/8)	Dic₉	1	2	yes	complemented
191	#135	—	2	2	no	—
192	#131	—	18	2	no	—
193	#131	—	18	2	no	—
194	C₂×M₁₆	—	9	1	no	—
195	C₂×M₁₆	—	9	1	no	—
196	C₂×M₁₆	—	18	2	no	—
197	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
198	C₂²×C₈	—	9	1	no	—
199	#117	—	9	1	no	—
200	#104	—	18	2	no	—
201	#103	—	18	1	no	—
202	#102	—	18	1	no	—
203	C₂²⋊C₈	—	18	2	no	—
204	C₂³×C₃	—	2	1	no	—
205	C₂³×C₃	—	2	1	no	—
206	C₂³×C₃	#237	1	1	yes	characteristic
207	C₃×Q₈	—	2	1	no	—
208	C₃×Q₈	—	2	1	no	—
209	C₃×Q₈	—	2	1	no	—
210	C₃×Q₈	#217	1	2	yes	—
211	C₃×D₄	—	2	2	no	—
212	C₃×D₄	—	2	2	no	—
213	C₃×D₄	—	2	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₃×D₄	—	4	1	no	—
217	C₃×D₄	—	4	1	no	—
218	C₃×D₄	#217	1	2	yes	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	—	4	1	no	—
225	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	#237	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
231	C₂×Dic₃	—	12	1	no	—
232	C₃×C₈	—	2	2	no	—
233	C₃⋊C₈	—	6	2	no	—
234	C₃⋊C₈	—	6	2	no	—
235	C₃⋊C₈	—	12	1	no	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	2	1	no	—
239	C₂×C₉	—	4	2	no	—
240	C₂×C₉	—	8	1	no	—
241	C₂×C₉	#386	1	1	yes	characteristic
242	#41	—	2	1	no	—
243	#41	—	2	1	no	—
244	#41	—	2	1	no	—
245	#41	—	2	1	no	—
246	#41	—	2	1	no	—
247	#41	—	2	1	no	—
248	#41	C₂×Dic₉	1	2	yes	—
249	#41	#611	1	2	yes	—
250	C₂×Q₈	C₂×Dic₉	1	1	yes	characteristic
251	C₂×Q₈	#611	1	1	yes	characteristic
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	C₂×Dic₉	1	1	yes	characteristic
255	C₂×D₄	#611	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
259	Q₁₆	—	2	2	no	—
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	2	no	—
263	M₁₆	—	2	1	no	—
264	M₁₆	—	9	2	no	—
265	M₁₆	—	18	4	no	—
266	M₁₆	—	18	1	no	—
267	M₁₆	—	18	1	no	—
268	M₁₆	—	18	1	no	—
269	M₁₆	C₂×Dic₉	1	2	yes	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	9	2	no	—
272	C₂×C₈	—	18	4	no	—
273	C₂×C₈	—	18	1	no	—
274	C₂×C₈	—	18	1	no	—
275	C₂×C₈	—	18	1	no	—
276	C₂×C₈	—	18	1	no	—
277	C₄⋊C₄	—	18	1	no	—
278	C₂²⋊C₄	—	18	1	no	—
279	C₄²	—	18	1	no	—
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	4	1	no	—
282	C₂²×C₃	—	4	2	no	—
283	C₂²×C₃	—	4	1	no	—
284	C₂²×C₃	—	8	1	no	—
285	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
286	C₂²×C₃	#935	1	1	yes	characteristic
287	C₂²×C₃	#939	1	1	yes	characteristic
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
291	C₃×C₄	#937	1	1	yes	characteristic
292	C₃×C₄	#939	1	1	yes	characteristic
293	C₃×C₄	C₂×#217	1	1	yes	characteristic
294	Dic₃	—	24	1	no	—
295	C₉	#1881	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	#3699	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	#3676	1	2	yes	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	1	no	—
310	D₄	#3676	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	36	1	no	—
318	C₂×C₄	D₄×D₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₉	1	1	yes	characteristic
320	C₂×C₄	#3699	1	1	yes	characteristic
321	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
322	C₂×C₄	C₂×#611	1	1	yes	characteristic
323	C₂×C₄	C₂×#611	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	18	2	no	—
326	C₈	—	18	2	no	—
327	C₈	—	36	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	4	2	no	—
332	C₂×C₃	—	8	1	no	—
333	C₂×C₃	#5220	1	1	yes	characteristic
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	D₄×Dic₉	1	1	yes	characteristic
340	C₂²	#63497	1	1	yes	characteristic
341	C₂²	#63501	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	72	1	no	—
345	C₄	D₄×Dic₉	1	1	yes	characteristic
346	C₄	#63499	1	1	yes	characteristic
347	C₄	#63501	1	1	yes	characteristic
348	C₄	C₂×#3676	1	1	yes	characteristic
349	C₃	#72116	1	1	yes	characteristic
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	4	2	no	—
354	C₂	—	8	1	no	—
355	C₂	#10588900	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63499
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#575	C₂
socle	C₂×C₃	#5220
prime radical	C₃×#575	D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#575	D₉
hypocenter	C₉	#1881
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

Complements

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

N	H ≅ G/N	type
2. #10589921	C₂	semidirect
3. #10588917	C₂	semidirect
4. #10588914	C₂	semidirect
5. #10588912	C₂	semidirect
6. #10588232	C₂	semidirect
7. #10588231	C₂	semidirect
12. C₉×#137	C₄	semidirect
14. C₉×AGL(1,ℤ/8)	C₄	semidirect
188. #137	Dic₉	semidirect
190. AGL(1,ℤ/8)	Dic₉	semidirect
295. C₉	#1881	semidirect

Maximal subgroups

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

subgroup	index
C₉×#575	2
#10589921	2
#10588917	2
#10588914	2
#10588912	2
#10588232	2
#10588231	2
#72116	3