Subgroups — #11895519

Order 1152.

504 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	#10589686	C₂	1	1	yes	maximal, characteristic
3	#10588772	C₂	1	1	yes	maximal, characteristic
4	#10588766	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588749	C₂	1	2	yes	maximal
6	#10588745	C₂	1	2	yes	maximal, complemented
7	#10588711	C₂	1	2	yes	maximal
8	C₄×#3675	C₂	1	1	yes	maximal, characteristic
9	#10588707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#3674	C₂	1	2	yes	maximal, complemented
11	Q₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#83689	—	3	1	no	maximal
13	C₉×C₄⋊Q₈	—	2	1	no	—
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₉×#125	—	2	2	no	—
16	C₉×#125	C₂²	1	2	yes	—
17	C₄×Q₈×C₉	—	2	1	no	—
18	C₄×Q₈×C₉	C₂²	1	2	yes	—
19	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
21	#63508	—	2	1	no	—
22	C₂×#3675	C₂²	1	1	yes	characteristic
23	C₂×#3674	C₂²	1	2	yes	—
24	#63460	—	2	2	no	—
25	#63454	—	2	1	no	—
26	#63437	C₂²	1	2	yes	—
27	C₄×D₃₆	C₂²	1	1	yes	characteristic
28	#63431	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₈	C₂²	1	2	yes	—
30	#63395	C₂²	1	1	yes	characteristic
31	#63369	C₂²	1	1	yes	characteristic
32	#63369	C₂²	1	2	yes	—
33	#63368	C₂²	1	2	yes	—
34	#63368	C₂²	1	2	yes	—
35	#63368	C₂²	1	2	yes	—
36	#63367	C₂²	1	1	yes	characteristic
37	#63366	C₂²	1	2	yes	—
38	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
40	C₄×C₉⋊C₈	C₂²	1	2	yes	—
41	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×#557	D₃	1	1	yes	characteristic
43	#5584	—	3	1	no	—
44	#5093	—	3	1	no	—
45	#5087	—	3	1	no	—
46	#5068	—	3	2	no	—
47	#5063	—	3	2	no	—
48	#5031	—	3	1	no	—
49	C₄×#216	—	3	2	no	—
50	#5028	—	3	2	no	—
51	C₄×#215	—	3	1	no	—
52	Q₈×C₃⋊C₈	—	3	1	no	—
53	C₂×Q₈×C₉	—	2	1	no	—
54	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
55	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
56	C₂×D₃₆	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₉	—	2	1	no	—
58	C₂×Dic₁₈	C₂³	1	2	yes	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	2	1	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	2	no	—
64	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₉×C₄⋊C₄	D₄	1	2	yes	—
66	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₉×C₄⋊C₄	C₂³	1	2	yes	—
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	—	2	1	no	—
70	C₄²×C₉	C₂³	1	2	yes	—
71	C₄²×C₉	C₂³	1	1	yes	characteristic
72	#3675	—	2	2	no	—
73	#3674	—	2	4	no	—
74	(C₂×D₉)⋊C₄	—	2	2	no	—
75	(C₂×D₉)⋊C₄	—	2	1	no	—
76	(C₂×D₉)⋊C₄	—	2	2	no	—
77	C₄⋊Dic₉	—	2	1	no	—
78	C₄⋊Dic₉	—	2	1	no	—
79	C₄⋊Dic₉	C₂³	1	1	yes	characteristic
80	Dic₉⋊C₄	—	2	2	no	—
81	C₄×Dic₉	—	2	2	no	—
82	C₂×C₉⋊C₈	C₂³	1	2	yes	—
83	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
84	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
85	#3364	—	9	1	no	Sylow
86	C₃×C₄⋊Q₈	—	2	1	no	—
87	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
88	C₃×#125	—	2	2	no	—
89	C₃×#125	C₂×D₃	1	2	yes	—
90	C₃×C₄×Q₈	—	2	1	no	—
91	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
92	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
93	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
94	#946	—	6	1	no	—
95	C₂×#216	—	3	2	no	—
96	C₂×#215	—	3	1	no	—
97	#898	—	6	2	no	—
98	#892	—	6	1	no	—
99	#875	—	3	2	no	—
100	C₄×D₁₂	—	3	1	no	—
101	#869	—	3	1	no	—
102	C₄×Dic₆	—	3	2	no	—
103	#835	—	3	1	no	—
104	#810	—	3	2	no	—
105	#810	—	3	2	no	—
106	#810	—	3	2	no	—
107	#809	—	3	1	no	—
108	#809	—	3	2	no	—
109	#808	—	3	1	no	—
110	#807	—	3	2	no	—
111	C₄⋊(C₃⋊C₈)	—	3	2	no	—
112	C₄⋊(C₃⋊C₈)	—	3	1	no	—
113	C₄×C₃⋊C₈	—	3	1	no	—
114	C₄×C₃⋊C₈	—	3	2	no	—
115	C₂²×D₉	—	2	1	no	—
116	Q₈×C₉	—	2	2	no	—
117	Q₈×C₉	—	2	1	no	—
118	Q₈×C₉	—	4	2	no	—
119	Q₈×C₉	#41	1	2	yes	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
128	C₂×Dic₉	—	2	1	no	—
129	C₂×Dic₉	—	2	2	no	—
130	D₃₆	—	2	1	no	—
131	D₃₆	—	2	1	no	—
132	C₄×D₉	—	4	2	no	—
133	Dic₁₈	—	2	2	no	—
134	Dic₁₈	—	2	2	no	—
135	C₉⋊C₈	—	2	2	no	—
136	C₉⋊C₈	—	2	1	no	—
137	C₉⋊C₈	—	2	1	no	—
138	#557	D₉	1	1	yes	characteristic, complemented
139	#556	—	9	1	no	—
140	#488	—	9	2	no	—
141	#487	—	9	1	no	—
142	#479	—	9	2	no	—
143	#474	—	9	1	no	—
144	#464	—	9	2	no	—
145	#462	—	9	1	no	—
146	C₈×Q₈	—	9	1	no	—
147	C₄×Q₁₆	—	9	2	no	—
148	C₄×SD₁₆	—	9	1	no	—
149	C₂×C₃×Q₈	—	2	1	no	—
150	C₂×C₃×Q₈	#67	1	1	yes	characteristic
151	C₂×C₃×Q₈	C₂²×D₃	1	1	yes	characteristic
152	C₂×D₁₂	—	3	1	no	—
153	C₂×C₄×D₃	—	6	1	no	—
154	C₂×Dic₆	—	3	2	no	—
155	C₃×C₄⋊C₄	—	2	2	no	—
156	C₃×C₄⋊C₄	—	2	1	no	—
157	C₃×C₄⋊C₄	—	2	2	no	—
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	—	2	2	no	—
160	C₃×C₄⋊C₄	#67	1	1	yes	characteristic
161	C₃×C₄⋊C₄	#67	1	2	yes	—
162	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
163	C₃×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
164	C₃×C₄²	—	2	2	no	—
165	C₃×C₄²	—	2	1	no	—
166	C₃×C₄²	C₂²×D₃	1	2	yes	—
167	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
168	#216	—	6	4	no	—
169	#215	—	6	2	no	—
170	(C₂×D₃)⋊C₄	—	6	1	no	—
171	(C₂×D₃)⋊C₄	—	6	2	no	—
172	(C₂×D₃)⋊C₄	—	6	2	no	—
173	C₄⋊Dic₃	—	3	1	no	—
174	C₄⋊Dic₃	—	6	1	no	—
175	C₄⋊Dic₃	—	6	1	no	—
176	Dic₃⋊C₄	—	6	2	no	—
177	C₄×Dic₃	—	6	2	no	—
178	C₂×C₃⋊C₈	—	3	1	no	—
179	C₂×C₃⋊C₈	—	3	2	no	—
180	C₂×C₃⋊C₈	—	3	1	no	—
181	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
182	C₂×D₉	—	4	1	no	—
183	C₂×D₉	—	4	1	no	—
184	C₂×D₉	—	4	1	no	—
185	C₄×C₉	—	2	2	no	—
186	C₄×C₉	—	2	2	no	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	2	1	no	—
189	C₄×C₉	—	4	2	no	—
190	C₄×C₉	—	4	1	no	—
191	C₄×C₉	#135	1	2	yes	—
192	C₄×C₉	C₂×#41	1	1	yes	characteristic
193	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
194	Dic₉	—	4	2	no	—
195	Dic₉	—	4	1	no	—
196	C₂×Q₁₆	—	9	2	no	—
197	C₂×SD₁₆	—	9	1	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	9	1	no	—
200	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
201	#126	—	18	2	no	—
202	#125	—	2	2	no	—
203	#125	C₂×D₉	1	2	yes	—
204	#124	—	9	2	no	—
205	C₂²⋊Q₈	—	18	1	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	9	2	no	—
208	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
209	C₄×Q₈	C₂×D₉	1	2	yes	—
210	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
211	C₄×D₄	—	9	1	no	—
212	#117	—	18	1	no	—
213	#107	—	9	2	no	—
214	#106	—	9	1	no	—
215	C₄⋊C₈	—	9	1	no	—
216	C₄⋊C₈	—	9	2	no	—
217	#103	—	9	1	no	—
218	#103	—	9	2	no	—
219	#103	—	9	2	no	—
220	#103	—	9	2	no	—
221	#102	—	9	1	no	—
222	#102	—	9	2	no	—
223	C₄×C₈	—	9	1	no	—
224	C₄×C₈	—	9	2	no	—
225	C₂²×D₃	—	6	1	no	—
226	C₃×Q₈	—	2	2	no	—
227	C₃×Q₈	—	2	1	no	—
228	C₃×Q₈	—	4	2	no	—
229	C₃×Q₈	#239	1	2	yes	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	C₂×#67	1	2	yes	—
234	C₂×C₃×C₄	C₂×#67	1	2	yes	—
235	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
238	C₂×Dic₃	—	6	1	no	—
239	C₂×Dic₃	—	6	2	no	—
240	D₁₂	—	6	1	no	—
241	D₁₂	—	6	1	no	—
242	C₄×D₃	—	12	2	no	—
243	Dic₆	—	6	2	no	—
244	Dic₆	—	6	2	no	—
245	C₃⋊C₈	—	6	1	no	—
246	C₃⋊C₈	—	6	2	no	—
247	C₃⋊C₈	—	6	1	no	—
248	C₂×C₉	#548	1	1	yes	characteristic
249	C₂×C₉	C₂×#135	1	1	yes	characteristic
250	C₂×C₉	#578	1	1	yes	characteristic
251	D₉	—	8	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	9	2	no	—
254	C₂×Q₈	#611	1	1	yes	characteristic
255	C₂×Q₈	C₂²×D₉	1	1	yes	characteristic
256	C₂×D₄	—	9	1	no	—
257	C₂²×C₄	—	18	1	no	—
258	Q₁₆	—	18	4	no	—
259	SD₁₆	—	18	2	no	—
260	C₂×C₈	—	9	1	no	—
261	C₂×C₈	—	9	2	no	—
262	C₂×C₈	—	9	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	9	1	no	—
269	C₄⋊C₄	—	18	1	no	—
270	C₄⋊C₄	—	18	2	no	—
271	C₄⋊C₄	—	18	1	no	—
272	C₄⋊C₄	#611	1	1	yes	characteristic
273	C₄⋊C₄	#611	1	2	yes	—
274	C₄⋊C₄	C₂²×D₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₉	1	2	yes	—
276	C₂²⋊C₄	—	18	2	no	—
277	C₂²⋊C₄	—	18	1	no	—
278	C₂²⋊C₄	—	18	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	18	2	no	—
282	C₄²	C₂²×D₉	1	1	yes	characteristic
283	C₄²	C₂²×D₉	1	2	yes	—
284	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
285	C₂×D₃	—	12	1	no	—
286	C₂×D₃	—	12	1	no	—
287	C₂×D₃	—	12	1	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	4	2	no	—
293	C₃×C₄	—	4	1	no	—
294	C₃×C₄	#950	1	2	yes	—
295	C₃×C₄	C₂×#239	1	1	yes	characteristic
296	C₃×C₄	#1007	1	1	yes	characteristic
297	Dic₃	—	12	2	no	—
298	Dic₃	—	12	1	no	—
299	C₉	#3364	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	18	1	no	—
301	Q₈	—	2	2	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	18	2	no	—
305	Q₈	—	18	2	no	—
306	Q₈	#3701	1	2	yes	—
307	D₄	—	18	1	no	—
308	D₄	—	18	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	18	1	no	—
313	C₂×C₄	—	18	2	no	—
314	C₂×C₄	—	36	2	no	—
315	C₂×C₄	C₂×#611	1	2	yes	—
316	C₂×C₄	C₂×#611	1	1	yes	characteristic
317	C₂×C₄	C₂×#611	1	2	yes	—
318	C₂×C₄	C₂×#611	1	1	yes	characteristic
319	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
320	C₈	—	18	1	no	—
321	C₈	—	18	2	no	—
322	C₈	—	18	1	no	—
323	C₂×C₃	#5817	1	1	yes	characteristic
324	C₂×C₃	C₂×#950	1	1	yes	characteristic
325	C₂×C₃	#5838	1	1	yes	characteristic
326	D₃	—	24	1	no	—
327	C₂²	—	36	1	no	—
328	C₂²	—	36	1	no	—
329	C₂²	—	36	1	no	—
330	C₂²	C₂²×#611	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	36	2	no	—
338	C₄	—	36	1	no	—
339	C₄	#63513	1	2	yes	—
340	C₄	C₂×#3701	1	1	yes	characteristic
341	C₄	#63713	1	1	yes	characteristic
342	C₃	#83689	1	1	yes	characteristic
343	C₂	—	72	1	no	—
344	C₂	#10589919	1	1	yes	characteristic
345	C₂	C₂×#63513	1	1	yes	characteristic
346	C₂	#10589940	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×#557	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂×D₃₆	C₂³
hypercenter	#557	D₉
hypocenter	C₉	#3364
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

Complements

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

N	H ≅ G/N	type
1. C₉×#557	C₂	semidirect
4. #10588766	C₂	semidirect
6. #10588745	C₂	semidirect
9. #10588707	C₂	semidirect
10. C₄×#3674	C₂	semidirect
11. Q₈×C₉⋊C₈	C₂	semidirect
138. #557	D₉	semidirect
299. C₉	#3364	semidirect

Maximal subgroups

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

subgroup	index
C₉×#557	2
#10589686	2
#10588772	2
#10588766	2
#10588749	2
#10588745	2
#10588711	2
C₄×#3675	2
#10588707	2
C₄×#3674	2
Q₈×C₉⋊C₈	2
#83689	3