Subgroups — #11788950

Order 1152.

360 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₉×#336	C₂	1	1	yes	maximal, characteristic
5	#10588228	C₂	1	1	yes	maximal, characteristic
6	#10588228	C₂	1	1	yes	maximal, characteristic
7	#10588228	C₂	1	1	yes	maximal, characteristic
8	#68611	—	3	1	no	maximal
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₉⋊C₄	C₄	1	1	yes	characteristic
13	C₂×Dic₉⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₉⋊#95	C₄	1	1	yes	characteristic
19	C₉⋊#95	C₄	1	1	yes	characteristic
20	C₈×Dic₉	—	2	2	no	—
21	C₄⋊(C₉⋊C₈)	—	2	2	no	—
22	C₂×C₈×Dic₃	—	3	1	no	—
23	C₄×Dic₃⋊C₄	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
25	C₃×#336	D₃	1	1	yes	characteristic
26	#4550	—	3	1	no	—
27	#4550	—	3	1	no	—
28	#4550	—	3	1	no	—
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₉	—	2	1	no	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₂×C₈×C₉	—	2	1	no	—
39	C₂×C₈×C₉	—	2	1	no	—
40	C₂×C₈×C₉	—	2	1	no	—
41	C₂×C₈×C₉	Q₈	1	2	yes	—
42	C₄²×C₉	—	2	1	no	—
43	C₄²×C₉	—	2	1	no	—
44	Dic₉⋊C₄	—	2	2	no	—
45	Dic₉⋊C₄	C₈	1	4	yes	complemented
46	C₄×Dic₉	—	2	1	no	—
47	C₄×Dic₉	—	2	1	no	—
48	C₄×Dic₉	—	2	1	no	—
49	C₄×Dic₉	—	2	1	no	—
50	C₂×C₉⋊C₈	—	2	1	no	—
51	C₂×C₉⋊C₈	—	2	1	no	—
52	C₂×C₉⋊C₈	—	2	1	no	—
53	C₂×C₉⋊C₈	—	2	1	no	—
54	C₂×C₉⋊C₈	—	2	1	no	—
55	C₂×C₉⋊C₈	D₄	1	2	yes	—
56	#1916	—	9	1	no	Sylow
57	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×Dic₃⋊C₄	—	3	1	no	—
61	C₂×Dic₃⋊C₄	—	3	1	no	—
62	C₂×C₄×Dic₃	—	3	1	no	—
63	C₂×C₄×Dic₃	—	3	1	no	—
64	C₂²×C₃⋊C₈	—	3	1	no	—
65	C₂²×C₃⋊C₈	—	3	1	no	—
66	C₃⋊#95	—	3	1	no	—
67	C₃⋊#95	—	3	1	no	—
68	C₈×Dic₃	—	6	2	no	—
69	C₄⋊(C₃⋊C₈)	—	6	2	no	—
70	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	—	2	1	no	—
74	C₂×C₄×C₉	—	2	1	no	—
75	C₂×C₄×C₉	—	2	1	no	—
76	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₉	#41	1	1	yes	characteristic
80	C₂×C₄×C₉	#41	1	1	yes	characteristic
81	C₂×C₄×C₉	#41	1	1	yes	characteristic
82	C₂×C₄×C₉	#41	1	1	yes	characteristic
83	C₂×Dic₉	—	2	1	no	—
84	C₂×Dic₉	—	2	1	no	—
85	C₂×Dic₉	—	2	1	no	—
86	C₂×Dic₉	—	2	1	no	—
87	C₂×Dic₉	—	2	1	no	—
88	C₂×Dic₉	—	2	1	no	—
89	C₂×Dic₉	—	2	1	no	—
90	C₂×Dic₉	—	2	1	no	—
91	C₂×Dic₉	—	2	1	no	—
92	C₂×Dic₉	C₂×C₈	1	2	yes	—
93	C₂×Dic₉	C₂×C₈	1	2	yes	—
94	C₂×Dic₉	M₁₆	1	2	yes	—
95	C₈×C₉	—	2	2	no	—
96	C₈×C₉	—	4	1	no	—
97	C₉⋊C₈	—	2	2	no	—
98	C₉⋊C₈	—	4	1	no	—
99	C₂×C₄⋊C₈	—	9	1	no	—
100	C₂×C₄×C₈	—	9	1	no	—
101	C₄×C₄⋊C₄	—	9	1	no	—
102	#336	—	9	1	no	—
103	#336	—	9	1	no	—
104	#336	—	9	1	no	—
105	#336	D₉	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂²×Dic₃	—	3	1	no	—
110	C₂²×Dic₃	—	3	1	no	—
111	C₂²×Dic₃	—	3	1	no	—
112	C₂²×Dic₃	—	3	1	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	2	1	no	—
116	C₂×C₃×C₈	—	2	1	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	Dic₆	1	2	yes	—
119	C₃×C₄²	—	2	1	no	—
120	C₃×C₄²	—	2	1	no	—
121	Dic₃⋊C₄	—	3	4	no	—
122	Dic₃⋊C₄	—	6	2	no	—
123	C₄×Dic₃	—	6	1	no	—
124	C₄×Dic₃	—	6	1	no	—
125	C₄×Dic₃	—	6	1	no	—
126	C₄×Dic₃	—	6	1	no	—
127	C₂×C₃⋊C₈	—	3	2	no	—
128	C₂×C₃⋊C₈	—	6	1	no	—
129	C₂×C₃⋊C₈	—	6	1	no	—
130	C₂×C₃⋊C₈	—	6	1	no	—
131	C₂×C₃⋊C₈	—	6	1	no	—
132	C₂×C₃⋊C₈	—	6	1	no	—
133	C₂²×C₉	#117	1	1	yes	characteristic
134	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
135	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
136	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
137	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂²×C₉	#131	1	1	yes	characteristic
139	C₂²×C₉	#131	1	1	yes	characteristic
140	C₄×C₉	—	2	2	no	—
141	C₄×C₉	—	4	1	no	—
142	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₄×C₉	#123	1	1	yes	characteristic
144	C₄×C₉	#125	1	1	yes	characteristic
145	C₄×C₉	#126	1	1	yes	characteristic
146	Dic₉	—	2	2	no	—
147	Dic₉	—	2	2	no	—
148	Dic₉	—	2	2	no	—
149	Dic₉	—	4	1	no	—
150	C₂²×C₈	—	9	1	no	—
151	C₂²×C₈	—	9	1	no	—
152	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
153	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	9	1	no	—
155	C₂×C₄⋊C₄	—	9	1	no	—
156	C₂×C₄²	—	9	1	no	—
157	C₂×C₄²	—	9	1	no	—
158	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
159	C₄⋊C₈	—	18	2	no	—
160	C₄×C₈	—	18	2	no	—
161	#95	—	9	1	no	—
162	#95	—	9	1	no	—
163	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
170	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
171	C₂×C₃×C₄	#235	1	1	yes	characteristic
172	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	#237	1	1	yes	characteristic
174	C₂×C₃×C₄	#237	1	1	yes	characteristic
175	C₂×C₃×C₄	#239	1	1	yes	characteristic
176	C₂×Dic₃	—	3	2	no	—
177	C₂×Dic₃	—	3	2	no	—
178	C₂×Dic₃	—	3	2	no	—
179	C₂×Dic₃	—	6	1	no	—
180	C₂×Dic₃	—	6	1	no	—
181	C₂×Dic₃	—	6	1	no	—
182	C₂×Dic₃	—	6	1	no	—
183	C₂×Dic₃	—	6	1	no	—
184	C₂×Dic₃	—	6	1	no	—
185	C₂×Dic₃	—	6	1	no	—
186	C₂×Dic₃	—	6	1	no	—
187	C₂×Dic₃	—	6	1	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	4	1	no	—
190	C₃⋊C₈	—	6	2	no	—
191	C₃⋊C₈	—	12	1	no	—
192	C₂×C₉	#387	1	1	yes	characteristic
193	C₂×C₉	#431	1	1	yes	characteristic
194	C₂×C₉	#433	1	1	yes	characteristic
195	C₂×C₉	C₈×D₄	1	1	yes	characteristic
196	C₂×C₉	#435	1	1	yes	characteristic
197	C₂×C₉	C₈×Q₈	1	1	yes	characteristic
198	C₂×C₉	#446	1	1	yes	characteristic
199	C₂²×C₄	—	9	1	no	—
200	C₂²×C₄	—	9	1	no	—
201	C₂²×C₄	—	9	1	no	—
202	C₂²×C₄	—	9	1	no	—
203	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
204	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	9	2	no	—
212	C₂×C₈	—	18	1	no	—
213	C₂×C₈	—	18	1	no	—
214	C₂×C₈	—	18	1	no	—
215	C₂×C₈	—	18	1	no	—
216	C₂×C₈	—	18	1	no	—
217	C₂×C₈	Dic₁₈	1	2	yes	—
218	C₄⋊C₄	—	9	4	no	—
219	C₄⋊C₄	—	18	2	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	18	1	no	—
223	C₄²	—	18	1	no	—
224	C₄²	—	18	1	no	—
225	C₄²	—	18	1	no	—
226	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
227	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
228	C₂²×C₃	#892	1	1	yes	characteristic
229	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
230	C₂²×C₃	#901	1	1	yes	characteristic
231	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
232	C₂²×C₃	#907	1	1	yes	characteristic
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	#878	1	1	yes	characteristic
236	C₃×C₄	#879	1	1	yes	characteristic
237	C₃×C₄	#890	1	1	yes	characteristic
238	C₃×C₄	#894	1	1	yes	characteristic
239	Dic₃	—	6	2	no	—
240	Dic₃	—	6	2	no	—
241	Dic₃	—	6	2	no	—
242	Dic₃	—	12	1	no	—
243	C₉	#1916	1	1	yes	characteristic, complemented, Sylow
244	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	9	2	no	—
251	C₂×C₄	—	9	2	no	—
252	C₂×C₄	—	9	2	no	—
253	C₂×C₄	—	18	1	no	—
254	C₂×C₄	—	18	1	no	—
255	C₂×C₄	—	18	1	no	—
256	C₂×C₄	—	18	1	no	—
257	C₂×C₄	—	18	1	no	—
258	C₂×C₄	—	18	1	no	—
259	C₂×C₄	—	18	1	no	—
260	C₂×C₄	—	18	1	no	—
261	C₂×C₄	—	18	1	no	—
262	C₂×C₄	C₈×D₉	1	2	yes	—
263	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
264	C₂×C₄	#3697	1	1	yes	characteristic
265	C₂×C₄	D₄×D₉	1	1	yes	characteristic
266	C₂×C₄	#3699	1	1	yes	characteristic
267	C₂×C₄	#3699	1	1	yes	characteristic
268	C₂×C₄	#3701	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	4	1	no	—
271	C₈	—	18	2	no	—
272	C₈	—	36	1	no	—
273	C₂×C₃	#4655	1	1	yes	characteristic
274	C₂×C₃	C₈×Dic₆	1	1	yes	characteristic
275	C₂×C₃	#4701	1	1	yes	characteristic
276	C₂×C₃	#67⋊C₈	1	1	yes	characteristic
277	C₂×C₃	#4728	1	1	yes	characteristic
278	C₂×C₃	#4833	1	1	yes	characteristic
279	C₂×C₃	#4840	1	1	yes	characteristic
280	C₂²	C₄×Dic₁₈	1	1	yes	characteristic
281	C₂²	#611⋊C₄	1	1	yes	characteristic
282	C₂²	#63454	1	1	yes	characteristic
283	C₂²	C₂×C₈×D₉	1	1	yes	characteristic
284	C₂²	#63464	1	1	yes	characteristic
285	C₂²	M₁₆×D₉	1	1	yes	characteristic
286	C₂²	#63469	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	18	2	no	—
290	C₄	—	18	2	no	—
291	C₄	—	18	2	no	—
292	C₄	—	36	1	no	—
293	C₄	#63440	1	1	yes	characteristic
294	C₄	#63441	1	1	yes	characteristic
295	C₄	#63452	1	1	yes	characteristic
296	C₄	#63456	1	1	yes	characteristic
297	C₃	#68611	1	1	yes	characteristic
298	C₂	#10588335	1	1	yes	characteristic
299	C₂	C₈×Dic₁₈	1	1	yes	characteristic
300	C₂	#10588381	1	1	yes	characteristic
301	C₂	#611⋊C₈	1	1	yes	characteristic
302	C₂	#10588410	1	1	yes	characteristic
303	C₂	#10588517	1	1	yes	characteristic
304	C₂	#10588524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#336	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#336	D₉
hypocenter	C₉	#1916
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
45. Dic₉⋊C₄	C₈	semidirect
243. C₉	#1916	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉	2
C₄×Dic₉⋊C₄	2
C₂×C₄⋊(C₉⋊C₈)	2
C₉×#336	2
#10588228	2
#68611	3