Subgroups — #10589765

Order 576.

360 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×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#63502	C₂	1	1	yes	maximal, characteristic, complemented
3	#63501	C₂	1	1	yes	maximal, characteristic, complemented
4	#63492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#611	C₂	1	1	yes	maximal, characteristic, complemented
6	#63457	C₂	1	1	yes	maximal, characteristic, complemented
7	#63456	C₂	1	1	yes	maximal, characteristic, complemented
8	#63454	C₂	1	1	yes	maximal, characteristic, complemented
9	#63451	C₂	1	1	yes	maximal, characteristic, complemented
10	#63448	C₂	1	1	yes	maximal, characteristic, complemented
11	#63446	C₂	1	1	yes	maximal, characteristic, complemented
12	#63446	C₂	1	1	yes	maximal, characteristic, complemented
13	#63446	C₂	1	1	yes	maximal, characteristic, complemented
14	#63445	C₂	1	1	yes	maximal, characteristic, complemented
15	#611⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
16	#5663	—	3	1	no	maximal
17	C₂×D₄×C₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₉	C₂²	1	1	yes	characteristic
19	C₂×#611	C₂²	1	1	yes	characteristic
20	C₂×#611	C₂²	1	1	yes	characteristic
21	C₂×#611	C₂²	1	1	yes	characteristic
22	C₂×#611	C₂²	1	1	yes	characteristic
23	C₂×#611	C₂²	1	1	yes	characteristic
24	C₂×#611	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉	C₂²	1	1	yes	characteristic
26	C₂×D₃₆	C₂²	1	1	yes	characteristic
27	C₂×D₃₆	C₂²	1	1	yes	characteristic
28	C₂×D₃₆	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₉	C₂²	1	1	yes	characteristic
32	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
35	C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
36	C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
37	#3676	C₂²	1	1	yes	characteristic
38	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
39	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
40	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
41	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
42	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
43	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
44	C₄⋊Dic₉	C₂²	1	1	yes	characteristic
45	C₄⋊Dic₉	C₂²	1	1	yes	characteristic
46	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
47	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
48	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
49	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
50	C₄×Dic₉	C₂²	1	1	yes	characteristic
51	C₄×Dic₉	C₂²	1	1	yes	characteristic
52	C₃×#123	D₃	1	1	yes	characteristic
53	#940	—	3	1	no	—
54	#939	—	3	1	no	—
55	#930	—	3	1	no	—
56	C₄×#67	—	3	1	no	—
57	#895	—	3	1	no	—
58	#894	—	3	1	no	—
59	#892	—	3	1	no	—
60	#889	—	3	1	no	—
61	#886	—	3	1	no	—
62	#884	—	3	1	no	—
63	#884	—	3	1	no	—
64	#884	—	3	1	no	—
65	#883	—	3	1	no	—
66	#67⋊C₄	—	3	1	no	—
67	C₂³×C₉	C₂³	1	1	yes	characteristic
68	C₂³×C₉	C₂³	1	1	yes	characteristic
69	C₂²×D₉	C₂³	1	1	yes	characteristic
70	C₂²×D₉	C₂³	1	1	yes	characteristic
71	C₂²×D₉	C₂³	1	1	yes	characteristic
72	D₄×C₉	—	4	1	no	—
73	C₂×C₄×C₉	—	4	1	no	—
74	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
79	#611	—	2	2	no	—
80	#611	—	2	2	no	—
81	#611	—	4	1	no	—
82	#611	—	4	1	no	—
83	#611	—	4	1	no	—
84	#611	—	4	1	no	—
85	C₂×Dic₉	—	4	1	no	—
86	C₂×Dic₉	C₂³	1	1	yes	characteristic
87	C₂×Dic₉	C₂³	1	1	yes	characteristic
88	C₂×Dic₉	C₂³	1	1	yes	characteristic
89	C₂×Dic₉	C₂³	1	1	yes	characteristic
90	C₂×Dic₉	C₂³	1	1	yes	characteristic
91	D₃₆	—	4	1	no	—
92	D₃₆	—	4	1	no	—
93	D₃₆	—	4	1	no	—
94	C₄×D₉	—	2	2	no	—
95	C₄×D₉	—	4	1	no	—
96	C₄×D₉	—	4	1	no	—
97	#540	—	9	1	no	Sylow
98	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
100	C₂×#67	—	3	1	no	—
101	C₂×#67	—	3	1	no	—
102	C₂×#67	—	3	1	no	—
103	C₂×#67	—	3	1	no	—
104	C₂×#67	—	3	1	no	—
105	C₂×#67	—	3	1	no	—
106	C₂²×Dic₃	—	3	1	no	—
107	C₂×D₁₂	—	3	1	no	—
108	C₂×D₁₂	—	3	1	no	—
109	C₂×D₁₂	—	3	1	no	—
110	C₂×C₄×D₃	—	3	1	no	—
111	C₂×C₄×D₃	—	3	1	no	—
112	C₂×C₄×D₃	—	3	1	no	—
113	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
114	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
115	C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
116	C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
117	C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
118	#217	—	3	1	no	—
119	(C₂×D₃)⋊C₄	—	3	1	no	—
120	(C₂×D₃)⋊C₄	—	3	1	no	—
121	(C₂×D₃)⋊C₄	—	3	1	no	—
122	(C₂×D₃)⋊C₄	—	3	1	no	—
123	(C₂×D₃)⋊C₄	—	3	1	no	—
124	(C₂×D₃)⋊C₄	—	3	1	no	—
125	C₄⋊Dic₃	—	3	1	no	—
126	C₄⋊Dic₃	—	3	1	no	—
127	Dic₃⋊C₄	—	3	1	no	—
128	Dic₃⋊C₄	—	3	1	no	—
129	Dic₃⋊C₄	—	3	1	no	—
130	Dic₃⋊C₄	—	3	1	no	—
131	C₄×Dic₃	—	3	1	no	—
132	C₄×Dic₃	—	3	1	no	—
133	C₂²×C₉	—	2	1	no	—
134	C₂²×C₉	—	2	1	no	—
135	C₂²×C₉	—	2	1	no	—
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	—	2	1	no	—
138	C₂²×C₉	—	2	1	no	—
139	C₂²×C₉	C₂⁴	1	1	yes	characteristic
140	C₂×D₉	—	2	1	no	—
141	C₂×D₉	—	2	1	no	—
142	C₂×D₉	—	2	1	no	—
143	C₂×D₉	—	2	1	no	—
144	C₂×D₉	—	2	1	no	—
145	C₂×D₉	—	2	1	no	—
146	C₂×D₉	—	2	1	no	—
147	C₂×D₉	—	2	1	no	—
148	C₂×D₉	—	2	1	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	2	1	no	—
154	Dic₉	—	2	1	no	—
155	Dic₉	—	2	1	no	—
156	Dic₉	—	2	1	no	—
157	Dic₉	—	2	1	no	—
158	Dic₉	#41	1	2	yes	—
159	#127	—	9	1	no	—
160	#125	—	9	1	no	—
161	#123	—	9	1	no	—
162	#123	—	9	1	no	—
163	#123	—	9	1	no	—
164	#123	D₉	1	1	yes	characteristic, complemented
165	#121	—	9	1	no	—
166	#121	—	9	1	no	—
167	#121	—	9	1	no	—
168	#121	—	9	1	no	—
169	#121	—	9	1	no	—
170	#121	—	9	1	no	—
171	C₄×D₄	—	9	1	no	—
172	C₄×D₄	—	9	1	no	—
173	#117	—	9	1	no	—
174	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
175	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
176	C₂²×D₃	—	3	1	no	—
177	C₂²×D₃	—	3	1	no	—
178	C₂²×D₃	—	3	1	no	—
179	C₃×D₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
186	#67	—	6	2	no	—
187	#67	—	6	2	no	—
188	#67	—	12	1	no	—
189	#67	—	12	1	no	—
190	#67	—	12	1	no	—
191	#67	—	12	1	no	—
192	C₂×Dic₃	—	3	1	no	—
193	C₂×Dic₃	—	3	1	no	—
194	C₂×Dic₃	—	3	1	no	—
195	C₂×Dic₃	—	3	1	no	—
196	C₂×Dic₃	—	3	1	no	—
197	C₂×Dic₃	—	12	1	no	—
198	D₁₂	—	12	1	no	—
199	D₁₂	—	12	1	no	—
200	D₁₂	—	12	1	no	—
201	C₄×D₃	—	6	2	no	—
202	C₄×D₃	—	12	1	no	—
203	C₄×D₃	—	12	1	no	—
204	C₂×C₉	—	4	1	no	—
205	C₂×C₉	—	4	1	no	—
206	C₂×C₉	C₂×#41	1	1	yes	characteristic
207	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
209	D₉	—	4	1	no	—
210	D₉	—	4	1	no	—
211	D₉	—	4	1	no	—
212	C₂×D₄	—	9	1	no	—
213	C₂×D₄	—	9	1	no	—
214	C₂×D₄	—	9	1	no	—
215	C₂×D₄	—	9	1	no	—
216	C₂×D₄	—	9	1	no	—
217	C₂×D₄	—	9	1	no	—
218	C₂×D₄	—	9	1	no	—
219	C₂×D₄	—	9	1	no	—
220	C₂×D₄	—	9	1	no	—
221	C₂×D₄	C₂×D₉	1	1	yes	characteristic
222	C₂²×C₄	—	9	1	no	—
223	C₂²×C₄	—	9	1	no	—
224	C₂²×C₄	—	9	1	no	—
225	C₂²×C₄	—	9	1	no	—
226	C₂²×C₄	C₂×D₉	1	1	yes	characteristic
227	C₄⋊C₄	—	9	1	no	—
228	C₄⋊C₄	—	9	1	no	—
229	C₄⋊C₄	—	9	1	no	—
230	C₄⋊C₄	—	9	1	no	—
231	C₄⋊C₄	—	9	1	no	—
232	C₄⋊C₄	—	9	1	no	—
233	C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
234	C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
235	C₂²⋊C₄	—	9	1	no	—
236	C₂²⋊C₄	—	9	1	no	—
237	C₂²⋊C₄	—	9	1	no	—
238	C₂²⋊C₄	—	9	1	no	—
239	C₂²⋊C₄	—	9	1	no	—
240	C₂²⋊C₄	—	9	1	no	—
241	C₂²⋊C₄	—	9	1	no	—
242	C₂²⋊C₄	C₂×D₉	1	1	yes	characteristic
243	C₂²⋊C₄	C₂×D₉	1	1	yes	characteristic
244	C₂²⋊C₄	C₂×D₉	1	1	yes	characteristic
245	C₄²	—	9	1	no	—
246	C₄²	—	9	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
254	C₂×D₃	—	6	1	no	—
255	C₂×D₃	—	6	1	no	—
256	C₂×D₃	—	6	1	no	—
257	C₂×D₃	—	6	1	no	—
258	C₂×D₃	—	6	1	no	—
259	C₂×D₃	—	6	1	no	—
260	C₂×D₃	—	6	1	no	—
261	C₂×D₃	—	6	1	no	—
262	C₂×D₃	—	6	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	Dic₃	—	3	2	no	—
269	Dic₃	—	6	1	no	—
270	Dic₃	—	6	1	no	—
271	Dic₃	—	6	1	no	—
272	Dic₃	—	6	1	no	—
273	C₉	#540	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	9	1	no	—
275	C₂³	—	9	1	no	—
276	C₂³	—	9	1	no	—
277	C₂³	C₂²×D₉	1	1	yes	characteristic
278	C₂³	C₂²×D₉	1	1	yes	characteristic
279	D₄	—	4	1	no	—
280	D₄	—	18	2	no	—
281	D₄	—	18	2	no	—
282	D₄	—	36	1	no	—
283	D₄	—	36	1	no	—
284	D₄	—	36	1	no	—
285	D₄	—	36	1	no	—
286	D₄	—	36	1	no	—
287	D₄	—	36	1	no	—
288	D₄	—	36	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	9	1	no	—
291	C₂×C₄	—	9	1	no	—
292	C₂×C₄	—	9	1	no	—
293	C₂×C₄	—	9	1	no	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	18	2	no	—
296	C₂×C₄	—	36	1	no	—
297	C₂×C₄	—	36	1	no	—
298	C₂×C₄	—	36	1	no	—
299	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
304	C₂×C₃	—	4	1	no	—
305	C₂×C₃	—	4	1	no	—
306	C₂×C₃	#1004	1	1	yes	characteristic
307	C₂×C₃	D₃×#41	1	1	yes	characteristic
308	C₂×C₃	#1009	1	1	yes	characteristic
309	D₃	—	12	1	no	—
310	D₃	—	12	1	no	—
311	D₃	—	12	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	18	1	no	—
319	C₂²	—	18	1	no	—
320	C₂²	—	18	1	no	—
321	C₂²	—	18	1	no	—
322	C₂²	—	18	1	no	—
323	C₂²	—	18	1	no	—
324	C₂²	—	18	1	no	—
325	C₂²	—	18	1	no	—
326	C₂²	—	18	1	no	—
327	C₂²	C₂³×D₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	9	2	no	—
334	C₄	—	18	1	no	—
335	C₄	—	18	1	no	—
336	C₄	—	18	1	no	—
337	C₄	—	18	1	no	—
338	C₃	#5663	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	4	1	no	—
341	C₂	—	36	1	no	—
342	C₂	—	36	1	no	—
343	C₂	—	36	1	no	—
344	C₂	#63710	1	1	yes	characteristic
345	C₂	#41×D₉	1	1	yes	characteristic
346	C₂	#63715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴
Frattini, socle	C₂²×C₃	C₂³×D₃
Fitting	C₉×#123	C₂
hypercenter	#123	D₉
hypocenter	C₉	#540
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. C₉×#123	C₂	semidirect
2. #63502	C₂	semidirect
3. #63501	C₂	semidirect
4. #63492	C₂	semidirect
5. C₄×#611	C₂	semidirect
6. #63457	C₂	semidirect
7. #63456	C₂	semidirect
8. #63454	C₂	semidirect
9. #63451	C₂	semidirect
10. #63448	C₂	semidirect
11. #63446	C₂	semidirect
12. #63446	C₂	semidirect
13. #63446	C₂	semidirect
14. #63445	C₂	semidirect
15. #611⋊C₄	C₂	semidirect
164. #123	D₉	semidirect
273. C₉	#540	semidirect

Maximal subgroups

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

subgroup	index
C₉×#123	2
#63502	2
#63501	2
#63492	2
C₄×#611	2
#63457	2
#63456	2
#63454	2
#63451	2
#63448	2
#63446	2
#63445	2
#611⋊C₄	2
#5663	3