Subgroups — #11785307

Order 1152.

468 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	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#63396	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588208	C₂	1	2	yes	maximal, complemented
6	#71646	—	3	1	no	maximal
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₄×Q₈×C₉	C₄	1	1	yes	characteristic, complemented
9	C₄×Q₈×C₉	C₄	1	1	yes	characteristic, complemented
10	C₄×D₄×C₉	—	2	1	no	—
11	C₄×D₄×C₉	—	2	1	no	—
12	C₄×D₄×C₉	C₄	1	1	yes	characteristic, complemented
13	C₄×D₄×C₉	C₄	1	1	yes	characteristic, complemented
14	C₉×#117	—	2	1	no	—
15	C₉×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉	—	2	1	no	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₉	—	2	1	no	—
19	C₂×C₄×Dic₉	C₂²	1	2	yes	—
20	C₂×#3667	C₂²	1	2	yes	—
21	#63396	—	2	4	no	—
22	C₉⋊#95	—	2	1	no	—
23	C₄⋊(C₉⋊C₈)	—	2	1	no	—
24	C₄⋊(C₉⋊C₈)	—	2	1	no	—
25	C₃×C₄×#41	D₃	1	1	yes	characteristic
26	C₂×#837	—	3	2	no	—
27	C₄×C₄⋊Dic₃	—	3	1	no	—
28	#4924	—	3	1	no	—
29	#4532	—	3	2	no	—
30	C₉×#41	—	2	1	no	—
31	C₉×#41	—	2	1	no	—
32	C₉×#41	D₄	1	2	yes	—
33	C₉×#41	Q₈	1	2	yes	—
34	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₉	—	2	1	no	—
36	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂²×Dic₉	—	2	2	no	—
44	C₉×C₄⋊C₄	—	2	1	no	—
45	C₉×C₄⋊C₄	—	2	1	no	—
46	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₉×C₂²⋊C₄	—	2	1	no	—
49	C₉×C₂²⋊C₄	—	2	1	no	—
50	C₉×C₂²⋊C₄	—	2	1	no	—
51	C₄²×C₉	—	2	1	no	—
52	C₄²×C₉	—	2	1	no	—
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₉	D₄	1	1	yes	characteristic
57	C₄²×C₉	D₄	1	1	yes	characteristic
58	C₄⋊Dic₉	—	2	4	no	—
59	C₄×Dic₉	—	2	2	no	—
60	C₄×Dic₉	—	2	2	no	—
61	#3667	—	2	2	no	—
62	#3667	—	2	2	no	—
63	C₂×C₉⋊C₈	—	2	2	no	—
64	#1806	—	9	1	no	Sylow
65	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
66	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
67	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
68	C₃×C₄×D₄	—	2	1	no	—
69	C₃×C₄×D₄	—	2	1	no	—
70	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
71	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
72	C₃×#117	—	2	1	no	—
73	C₃×#117	C₂×D₃	1	1	yes	characteristic
74	C₂×C₃×C₄²	—	2	1	no	—
75	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
76	C₂×C₄⋊Dic₃	—	6	1	no	—
77	C₂×C₄×Dic₃	—	3	2	no	—
78	C₂×#208	—	3	2	no	—
79	#837	—	6	4	no	—
80	C₃⋊#95	—	6	1	no	—
81	C₄⋊(C₃⋊C₈)	—	6	1	no	—
82	C₄⋊(C₃⋊C₈)	—	6	1	no	—
83	C₂³×C₉	—	2	1	no	—
84	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₉	—	2	1	no	—
86	Q₈×C₉	C₄⋊C₄	1	2	yes	complemented
87	D₄×C₉	—	2	2	no	—
88	D₄×C₉	—	2	2	no	—
89	D₄×C₉	—	2	1	no	—
90	D₄×C₉	C₄⋊C₄	1	2	yes	complemented
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	4	1	no	—
98	C₂×C₄×C₉	—	4	2	no	—
99	C₂×C₄×C₉	—	4	2	no	—
100	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂²×C₄	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×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₉	#41	1	1	yes	characteristic
109	C₂×C₄×C₉	#41	1	1	yes	characteristic
110	C₂×Dic₉	—	2	4	no	—
111	C₂×Dic₉	—	4	2	no	—
112	C₂×Dic₉	—	4	2	no	—
113	C₉⋊C₈	—	4	2	no	—
114	C₄×#41	D₉	1	1	yes	characteristic
115	#423	—	9	1	no	—
116	C₂×#104	—	9	2	no	—
117	C₄×C₄⋊C₄	—	9	1	no	—
118	#339	—	9	2	no	—
119	C₃×#41	—	2	1	no	—
120	C₃×#41	—	2	1	no	—
121	C₃×#41	Dic₆	1	2	yes	—
122	C₃×#41	D₁₂	1	2	yes	—
123	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
124	C₂×C₃×D₄	—	2	1	no	—
125	C₂×C₃×D₄	C₂×Dic₃	1	1	yes	characteristic
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	#67	1	1	yes	characteristic
130	C₂²×C₃×C₄	#67	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂²×Dic₃	—	6	2	no	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
136	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
137	C₃×C₂²⋊C₄	—	2	1	no	—
138	C₃×C₂²⋊C₄	—	2	1	no	—
139	C₃×C₂²⋊C₄	—	2	1	no	—
140	C₃×C₄²	—	2	1	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₃×C₄²	—	2	1	no	—
143	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
144	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
145	C₃×C₄²	#67	1	1	yes	characteristic
146	C₃×C₄²	#67	1	1	yes	characteristic
147	C₄⋊Dic₃	—	6	4	no	—
148	C₄×Dic₃	—	6	2	no	—
149	C₄×Dic₃	—	6	2	no	—
150	#208	—	6	2	no	—
151	#208	—	6	2	no	—
152	C₂×C₃⋊C₈	—	6	2	no	—
153	C₂²×C₉	—	2	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	2	no	—
156	C₂²×C₉	—	4	1	no	—
157	C₂²×C₉	—	4	1	no	—
158	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₉	#121	1	1	yes	characteristic
160	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	4	1	no	—
166	C₄×C₉	—	4	1	no	—
167	C₄×C₉	#104	1	2	yes	complemented
168	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₉	#117	1	1	yes	characteristic
170	C₄×C₉	#121	1	1	yes	characteristic
171	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
172	Dic₉	—	4	4	no	—
173	C₂×#41	C₂×D₉	1	1	yes	characteristic
174	C₂×M₁₆	—	9	2	no	—
175	C₄×Q₈	Dic₉	1	1	yes	characteristic, complemented
176	C₄×Q₈	Dic₉	1	1	yes	characteristic, complemented
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	Dic₉	1	1	yes	characteristic, complemented
180	C₄×D₄	Dic₉	1	1	yes	characteristic, complemented
181	#117	—	2	1	no	—
182	#117	C₂×D₉	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	18	1	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	—	9	2	no	—
186	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
187	C₄⋊C₈	—	18	1	no	—
188	C₄⋊C₈	—	18	1	no	—
189	#104	—	18	4	no	—
190	#95	—	18	1	no	—
191	C₂³×C₃	—	2	1	no	—
192	C₂³×C₃	C₂×#67	1	1	yes	characteristic
193	C₃×Q₈	—	2	1	no	—
194	C₃×Q₈	C₄⋊Dic₃	1	2	yes	—
195	C₃×D₄	—	2	2	no	—
196	C₃×D₄	—	2	2	no	—
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	C₄⋊Dic₃	1	2	yes	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	2	no	—
207	C₂×C₃×C₄	—	4	2	no	—
208	C₂×C₃×C₄	#217	1	1	yes	characteristic
209	C₂×C₃×C₄	#217	1	1	yes	characteristic
210	C₂×C₃×C₄	#217	1	1	yes	characteristic
211	C₂×C₃×C₄	#217	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
214	C₂×C₃×C₄	#235	1	1	yes	characteristic
215	C₂×C₃×C₄	#235	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
218	C₂×Dic₃	—	6	4	no	—
219	C₂×Dic₃	—	12	2	no	—
220	C₂×Dic₃	—	12	2	no	—
221	C₃⋊C₈	—	12	2	no	—
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	4	2	no	—
225	C₂×C₉	#380	1	1	yes	characteristic
226	C₂×C₉	C₂×#104	1	1	yes	characteristic
227	C₂×C₉	#421	1	1	yes	characteristic
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	Dic₁₈	1	2	yes	—
231	#41	D₃₆	1	2	yes	—
232	C₂×Q₈	C₂×Dic₉	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂×Dic₉	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	18	2	no	—
239	C₂²×C₄	#611	1	1	yes	characteristic
240	C₂²×C₄	#611	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
242	M₁₆	—	18	2	no	—
243	M₁₆	—	18	2	no	—
244	C₂×C₈	—	18	2	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	18	4	no	—
248	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	18	2	no	—
257	C₄²	—	18	2	no	—
258	C₄²	C₂×Dic₉	1	1	yes	characteristic
259	C₄²	C₂×Dic₉	1	1	yes	characteristic
260	C₄²	#611	1	1	yes	characteristic
261	C₄²	#611	1	1	yes	characteristic
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	#924	1	1	yes	characteristic
268	C₂²×C₃	#930	1	1	yes	characteristic
269	C₂²×C₃	C₂×#217	1	1	yes	characteristic
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	4	1	no	—
275	C₃×C₄	—	4	1	no	—
276	C₃×C₄	#837	1	2	yes	—
277	C₃×C₄	#924	1	1	yes	characteristic
278	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
279	C₃×C₄	#926	1	1	yes	characteristic
280	C₃×C₄	#930	1	1	yes	characteristic
281	Dic₃	—	12	4	no	—
282	C₉	#1806	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	2	1	no	—
284	C₂³	C₂×#611	1	1	yes	characteristic
285	Q₈	—	2	1	no	—
286	Q₈	C₄⋊Dic₉	1	2	yes	complemented
287	D₄	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	1	no	—
290	D₄	C₄⋊Dic₉	1	2	yes	complemented
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	18	4	no	—
301	C₂×C₄	—	36	2	no	—
302	C₂×C₄	—	36	2	no	—
303	C₂×C₄	#3676	1	1	yes	characteristic
304	C₂×C₄	#3676	1	1	yes	characteristic
305	C₂×C₄	#3676	1	1	yes	characteristic
306	C₂×C₄	#3676	1	1	yes	characteristic
307	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
308	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
309	C₂×C₄	#3697	1	1	yes	characteristic
310	C₂×C₄	#3697	1	1	yes	characteristic
311	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
312	C₂×C₄	C₂×#611	1	1	yes	characteristic
313	C₈	—	36	2	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	4	2	no	—
317	C₂×C₃	#5212	1	1	yes	characteristic
318	C₂×C₃	C₂×#837	1	1	yes	characteristic
319	C₂×C₃	#5236	1	1	yes	characteristic
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#63486	1	1	yes	characteristic
326	C₂²	#63492	1	1	yes	characteristic
327	C₂²	C₂×#3676	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	36	4	no	—
335	C₄	#63396	1	2	yes	complemented
336	C₄	#63486	1	1	yes	characteristic
337	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
338	C₄	#63488	1	1	yes	characteristic
339	C₄	#63492	1	1	yes	characteristic
340	C₃	#71646	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	2	no	—
344	C₂	#10588892	1	1	yes	characteristic
345	C₂	C₂×#63396	1	1	yes	characteristic
346	C₂	#10588914	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₄×C₉×#41	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₄×#41	D₉
hypocenter	C₉	#1806
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊Dic₉	C₂	semidirect
4. #10588604	C₂	semidirect
5. #10588208	C₂	semidirect
8. C₄×Q₈×C₉	C₄	semidirect
9. C₄×Q₈×C₉	C₄	semidirect
12. C₄×D₄×C₉	C₄	semidirect
13. C₄×D₄×C₉	C₄	semidirect
86. Q₈×C₉	C₄⋊C₄	semidirect
90. D₄×C₉	C₄⋊C₄	semidirect
167. C₄×C₉	#104	semidirect
175. C₄×Q₈	Dic₉	semidirect
176. C₄×Q₈	Dic₉	semidirect
179. C₄×D₄	Dic₉	semidirect
180. C₄×D₄	Dic₉	semidirect
282. C₉	#1806	semidirect
286. Q₈	C₄⋊Dic₉	semidirect
290. D₄	C₄⋊Dic₉	semidirect
335. C₄	#63396	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×#41	2
C₂×#63396	2
C₄×C₄⋊Dic₉	2
#10588604	2
#10588208	2
#71646	3