Subgroups — #11871313

Order 1152.

666 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	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
3	#10588772	C₂	1	2	yes	maximal, complemented
4	#10588768	C₂	1	2	yes	maximal, complemented
5	#10588729	C₂	1	2	yes	maximal
6	#10588724	C₂	1	2	yes	maximal, complemented
7	#10588709	C₂	1	4	yes	maximal, complemented
8	#10588606	C₂	1	1	yes	maximal, characteristic, complemented
9	#83353	—	3	1	no	maximal
10	C₂²×D₃₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₉	—	2	1	no	—
12	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₉×#125	—	2	1	no	—
15	C₉×C₂²⋊Q₈	—	2	1	no	—
16	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₄×Q₈×C₉	—	2	1	no	—
18	C₄×Q₈×C₉	C₂²	1	2	yes	complemented
19	C₉×#117	—	2	1	no	—
20	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₂×#3675	C₂²	1	4	yes	—
22	C₂×(C₂×D₉)⋊C₄	—	2	1	no	—
23	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
24	C₄×D₃₆	—	2	1	no	—
25	C₄×D₃₆	—	2	1	no	—
26	C₄×D₃₆	C₂²	1	2	yes	—
27	C₄×D₃₆	C₂²	1	2	yes	—
28	#63395	C₂²	1	4	yes	complemented
29	#63389	C₂²	1	2	yes	—
30	#63369	C₂²	1	4	yes	—
31	#63366	C₂²	1	4	yes	complemented
32	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
33	#63362	C₂²	1	2	yes	complemented
34	C₃×#533	D₃	1	1	yes	characteristic
35	C₂×C₄×D₁₂	—	3	1	no	—
36	#5087	—	3	2	no	—
37	#5083	—	3	2	no	—
38	#5046	—	3	2	no	—
39	#5044	—	3	2	no	—
40	#5026	—	3	4	no	—
41	#4926	—	3	1	no	—
42	C₂³×D₉	—	2	1	no	—
43	C₂×Q₈×C₉	—	2	1	no	—
44	C₂×Q₈×C₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×Dic₉	—	2	1	no	—
49	C₂×D₃₆	—	2	1	no	—
50	C₂×D₃₆	—	2	4	no	—
51	C₂×D₃₆	C₂³	1	2	yes	—
52	C₂×C₄×D₉	—	2	2	no	—
53	C₂×C₄×D₉	—	2	2	no	—
54	C₂×C₄×D₉	—	4	4	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	—	2	1	no	—
57	C₉×C₄⋊C₄	—	2	1	no	—
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	C₂³	1	2	yes	—
61	C₉×C₂²⋊C₄	—	2	2	no	—
62	C₄²×C₉	—	2	2	no	—
63	C₄²×C₉	D₄	1	1	yes	characteristic
64	C₄²×C₉	D₄	1	1	yes	characteristic
65	C₄²×C₉	C₂³	1	1	yes	characteristic
66	C₄²×C₉	C₂³	1	1	yes	characteristic
67	#3675	—	2	8	no	—
68	(C₂×D₉)⋊C₄	—	2	2	no	—
69	(C₂×D₉)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₉	—	2	1	no	—
71	C₄⋊Dic₉	C₂³	1	2	yes	—
72	C₂×C₉⋊C₈	C₂³	1	4	yes	—
73	#3110	—	9	1	no	Sylow
74	C₂²×D₁₂	—	3	1	no	—
75	C₂²×C₄×D₃	—	6	1	no	—
76	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
77	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
78	C₃×#125	—	2	1	no	—
79	C₃×C₂²⋊Q₈	—	2	1	no	—
80	C₃×C₂²⋊Q₈	C₂×D₃	1	2	yes	—
81	C₃×C₄×Q₈	—	2	1	no	—
82	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
83	C₃×#117	—	2	1	no	—
84	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
85	C₂×#215	—	3	4	no	—
86	C₂×(C₂×D₃)⋊C₄	—	6	1	no	—
87	C₂×C₄⋊Dic₃	—	3	1	no	—
88	C₄×D₁₂	—	3	2	no	—
89	C₄×D₁₂	—	3	2	no	—
90	C₄×D₁₂	—	6	1	no	—
91	C₄×D₁₂	—	6	1	no	—
92	#835	—	3	4	no	—
93	#830	—	3	2	no	—
94	#809	—	3	4	no	—
95	#807	—	3	4	no	—
96	C₄⋊(C₃⋊C₈)	—	3	2	no	—
97	#803	—	3	2	no	—
98	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×D₉	—	2	4	no	—
100	C₂²×D₉	—	2	2	no	—
101	C₂²×D₉	—	4	2	no	—
102	C₂²×D₉	—	4	2	no	—
103	Q₈×C₉	—	2	4	no	—
104	Q₈×C₉	—	4	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₉	—	2	2	no	—
117	C₂×Dic₉	—	4	2	no	—
118	D₃₆	—	2	4	no	—
119	D₃₆	—	2	2	no	—
120	D₃₆	#41	1	4	yes	—
121	C₄×D₉	—	4	4	no	—
122	C₄×D₉	—	4	4	no	—
123	C₉⋊C₈	—	2	4	no	—
124	#533	D₉	1	1	yes	characteristic, complemented
125	C₂×C₄×D₄	—	9	1	no	—
126	#484	—	9	2	no	—
127	#474	—	9	2	no	—
128	#461	—	9	2	no	—
129	#450	—	9	2	no	—
130	#440	—	9	4	no	—
131	#432	—	9	1	no	—
132	C₂³×D₃	—	6	1	no	—
133	C₂×C₃×Q₈	—	2	1	no	—
134	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
135	C₂²×C₃×C₄	#67	1	1	yes	characteristic
136	C₂²×C₃×C₄	#67	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
138	C₂²×Dic₃	—	6	1	no	—
139	C₂×D₁₂	—	3	2	no	—
140	C₂×D₁₂	—	6	4	no	—
141	C₂×D₁₂	—	6	1	no	—
142	C₂×C₄×D₃	—	6	2	no	—
143	C₂×C₄×D₃	—	6	2	no	—
144	C₂×C₄×D₃	—	12	4	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
151	C₃×C₂²⋊C₄	—	2	2	no	—
152	C₃×C₄²	—	2	2	no	—
153	C₃×C₄²	#67	1	1	yes	characteristic
154	C₃×C₄²	#67	1	1	yes	characteristic
155	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
156	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
157	#215	—	6	8	no	—
158	(C₂×D₃)⋊C₄	—	6	2	no	—
159	(C₂×D₃)⋊C₄	—	6	2	no	—
160	C₄⋊Dic₃	—	3	2	no	—
161	C₄⋊Dic₃	—	6	1	no	—
162	C₂×C₃⋊C₈	—	3	4	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	2	1	no	—
165	C₂²×C₉	#137	1	2	yes	—
166	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
167	C₂×D₉	—	2	4	no	—
168	C₂×D₉	—	4	8	no	—
169	C₂×D₉	—	4	2	no	—
170	C₂×D₉	—	4	2	no	—
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	—	2	1	no	—
173	C₄×C₉	—	2	1	no	—
174	C₄×C₉	—	2	1	no	—
175	C₄×C₉	—	4	2	no	—
176	C₄×C₉	—	4	2	no	—
177	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
178	C₄×C₉	C₂×#41	1	2	yes	—
179	Dic₉	—	4	2	no	—
180	C₂²×D₄	—	9	1	no	—
181	C₂³×C₄	—	18	1	no	—
182	C₂×SD₁₆	—	9	4	no	—
183	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic, complemented
184	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic, complemented
185	#125	—	2	1	no	—
186	C₂²⋊Q₈	—	2	1	no	—
187	C₂²⋊Q₈	C₂×D₉	1	2	yes	—
188	C₄×Q₈	—	2	1	no	—
189	C₄×Q₈	C₂×D₉	1	2	yes	complemented
190	C₄×D₄	—	9	2	no	—
191	C₄×D₄	—	9	2	no	—
192	C₄×D₄	—	18	1	no	—
193	C₄×D₄	—	18	1	no	—
194	#117	—	2	1	no	—
195	C₂×C₄⋊C₄	—	9	1	no	—
196	C₂×C₂²⋊C₄	—	18	1	no	—
197	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
198	#107	—	9	4	no	—
199	C₄⋊C₈	—	9	2	no	—
200	#103	—	9	4	no	—
201	#102	—	9	4	no	—
202	C₂²⋊C₈	—	9	2	no	—
203	#97	—	9	2	no	—
204	C₂³×C₃	C₂×#67	1	1	yes	characteristic
205	C₂²×D₃	—	6	4	no	—
206	C₂²×D₃	—	6	2	no	—
207	C₂²×D₃	—	12	2	no	—
208	C₂²×D₃	—	12	2	no	—
209	C₃×Q₈	—	2	4	no	—
210	C₃×Q₈	—	4	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
222	C₂×Dic₃	—	6	2	no	—
223	C₂×Dic₃	—	12	2	no	—
224	D₁₂	—	3	4	no	—
225	D₁₂	—	6	4	no	—
226	D₁₂	—	6	2	no	—
227	C₄×D₃	—	12	4	no	—
228	C₄×D₃	—	12	4	no	—
229	C₃⋊C₈	—	6	4	no	—
230	C₂×C₉	—	2	2	no	—
231	C₂×C₉	#525	1	1	yes	characteristic
232	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₂×C₉	C₂×#137	1	1	yes	characteristic
234	D₉	—	4	4	no	—
235	C₂⁴	—	18	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	C₂²×D₉	1	2	yes	—
238	C₂×D₄	—	9	2	no	—
239	C₂×D₄	—	18	4	no	—
240	C₂×D₄	—	18	1	no	—
241	C₂²×C₄	—	18	1	no	—
242	C₂²×C₄	—	18	2	no	—
243	C₂²×C₄	—	18	2	no	—
244	C₂²×C₄	—	36	4	no	—
245	C₂²×C₄	#611	1	1	yes	characteristic
246	C₂²×C₄	#611	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
248	SD₁₆	—	18	8	no	—
249	C₂×C₈	—	9	4	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	9	2	no	—
256	C₄⋊C₄	—	18	1	no	—
257	C₄⋊C₄	C₂²×D₉	1	2	yes	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	18	2	no	—
260	C₂²⋊C₄	—	18	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	#611	1	1	yes	characteristic
263	C₄²	#611	1	1	yes	characteristic
264	C₄²	C₂²×D₉	1	1	yes	characteristic
265	C₄²	C₂²×D₉	1	1	yes	characteristic
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	#942	1	2	yes	—
269	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
270	C₂×D₃	—	6	4	no	—
271	C₂×D₃	—	12	8	no	—
272	C₂×D₃	—	12	2	no	—
273	C₂×D₃	—	12	2	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	—	4	2	no	—
280	C₃×C₄	#949	1	2	yes	—
281	C₃×C₄	D₃×#41	1	2	yes	—
282	Dic₃	—	12	2	no	—
283	C₉	#3110	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	18	4	no	—
285	C₂³	—	18	2	no	—
286	C₂³	—	36	2	no	—
287	C₂³	—	36	2	no	—
288	C₂³	C₂×#611	1	1	yes	characteristic
289	Q₈	—	2	4	no	—
290	Q₈	—	4	2	no	—
291	D₄	—	9	4	no	—
292	D₄	—	18	4	no	—
293	D₄	—	18	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	18	2	no	—
300	C₂×C₄	—	36	4	no	—
301	C₂×C₄	—	36	2	no	—
302	C₂×C₄	—	36	4	no	—
303	C₂×C₄	C₂×#611	1	1	yes	characteristic
304	C₂×C₄	C₂×#611	1	1	yes	characteristic
305	C₂×C₄	C₂×#611	1	1	yes	characteristic
306	C₂×C₄	C₂×#611	1	1	yes	characteristic
307	C₂×C₄	C₂×#611	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
309	C₈	—	18	4	no	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	C₂×#942	1	1	yes	characteristic
312	C₂×C₃	C₂×#949	1	1	yes	characteristic
313	C₂×C₃	#5828	1	1	yes	characteristic
314	D₃	—	12	4	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	18	4	no	—
318	C₂²	—	36	8	no	—
319	C₂²	—	36	2	no	—
320	C₂²	—	36	2	no	—
321	C₂²	#63505	1	2	yes	—
322	C₂²	C₂²×#611	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	36	2	no	—
330	C₄	#63512	1	2	yes	—
331	C₄	#41×D₉	1	2	yes	—
332	C₃	#83353	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	—	36	4	no	—
335	C₂	C₂×#63505	1	1	yes	characteristic
336	C₂	C₂×#63512	1	1	yes	characteristic
337	C₂	#10589930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#533	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂²×D₃₆	C₂²
hypercenter	#533	D₉
hypocenter	C₉	#3110
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₉×#533	C₂	semidirect
3. #10588772	C₂	semidirect
4. #10588768	C₂	semidirect
6. #10588724	C₂	semidirect
7. #10588709	C₂	semidirect
8. #10588606	C₂	semidirect
12. C₉×C₄⋊Q₈	C₂²	semidirect
13. C₉×C₄⋊Q₈	C₂²	semidirect
18. C₄×Q₈×C₉	C₂²	semidirect
28. #63395	C₂²	semidirect
31. #63366	C₂²	semidirect
32. C₄⋊(C₉⋊C₈)	C₂²	semidirect
33. #63362	C₂²	semidirect
124. #533	D₉	semidirect
183. C₄⋊Q₈	C₂×D₉	semidirect
184. C₄⋊Q₈	C₂×D₉	semidirect
189. C₄×Q₈	C₂×D₉	semidirect
283. C₉	#3110	semidirect

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₂×C₄×D₃₆	2
#10588772	2
#10588768	2
#10588729	2
#10588724	2
#10588709	2
#10588606	2
#83353	3