Subgroups — #11790021

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₉×#336	C₂	1	1	yes	maximal, characteristic
7	#10588228	C₂	1	1	yes	maximal, characteristic
8	#68617	—	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	Dic₉⋊C₈	—	2	2	no	—
21	Dic₉⋊C₈	—	2	2	no	—
22	C₈×Dic₉	—	2	2	no	—
23	C₄⋊(C₉⋊C₈)	—	2	2	no	—
24	C₂×Dic₃⋊C₈	—	3	1	no	—
25	C₂×Dic₃⋊C₈	—	3	1	no	—
26	C₂×C₈×Dic₃	—	3	1	no	—
27	C₄×Dic₃⋊C₄	—	3	1	no	—
28	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
29	C₃×#336	D₃	1	1	yes	characteristic
30	#4550	—	3	1	no	—
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
42	C₂×C₈×C₉	—	2	1	no	—
43	C₂×C₈×C₉	Q₈	1	2	yes	—
44	C₄²×C₉	—	2	1	no	—
45	C₄²×C₉	—	2	1	no	—
46	Dic₉⋊C₄	—	2	2	no	—
47	Dic₉⋊C₄	C₈	1	4	yes	complemented
48	C₄×Dic₉	—	2	1	no	—
49	C₄×Dic₉	—	2	1	no	—
50	C₄×Dic₉	D₄	1	2	yes	—
51	C₄×Dic₉	Q₈	1	2	yes	—
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	—	2	1	no	—
56	C₂×C₉⋊C₈	—	2	1	no	—
57	C₂×C₉⋊C₈	D₄	1	2	yes	—
58	#1939	—	9	1	no	Sylow
59	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₂×Dic₃⋊C₄	—	3	1	no	—
63	C₂×Dic₃⋊C₄	—	3	1	no	—
64	C₂×C₄×Dic₃	—	3	1	no	—
65	C₂×C₄×Dic₃	—	3	1	no	—
66	C₂²×C₃⋊C₈	—	3	1	no	—
67	C₂²×C₃⋊C₈	—	3	1	no	—
68	C₃⋊#95	—	3	1	no	—
69	C₃⋊#95	—	3	1	no	—
70	Dic₃⋊C₈	—	6	2	no	—
71	Dic₃⋊C₈	—	6	2	no	—
72	C₈×Dic₃	—	6	2	no	—
73	C₄⋊(C₃⋊C₈)	—	6	2	no	—
74	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₉	—	2	1	no	—
76	C₂×C₄×C₉	—	2	1	no	—
77	C₂×C₄×C₉	—	2	1	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
81	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₉	#41	1	1	yes	characteristic
86	C₂×C₄×C₉	#41	1	1	yes	characteristic
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	—	2	1	no	—
93	C₂×Dic₉	—	2	1	no	—
94	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
95	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
96	C₂×Dic₉	C₂×C₈	1	2	yes	—
97	C₂×Dic₉	C₂×C₈	1	2	yes	—
98	C₂×Dic₉	M₁₆	1	2	yes	—
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	4	1	no	—
101	C₉⋊C₈	—	2	2	no	—
102	C₉⋊C₈	—	4	1	no	—
103	C₂×C₄⋊C₈	—	9	1	no	—
104	C₂×C₄⋊C₈	—	9	1	no	—
105	C₂×C₄⋊C₈	—	9	1	no	—
106	C₂×C₄×C₈	—	9	1	no	—
107	C₄×C₄⋊C₄	—	9	1	no	—
108	#336	—	9	1	no	—
109	#336	D₉	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂²×Dic₃	—	3	1	no	—
114	C₂²×Dic₃	—	3	1	no	—
115	C₂²×Dic₃	—	3	1	no	—
116	C₂²×Dic₃	—	3	1	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	Dic₆	1	2	yes	—
123	C₃×C₄²	—	2	1	no	—
124	C₃×C₄²	—	2	1	no	—
125	Dic₃⋊C₄	—	3	4	no	—
126	Dic₃⋊C₄	—	6	2	no	—
127	C₄×Dic₃	—	3	2	no	—
128	C₄×Dic₃	—	3	2	no	—
129	C₄×Dic₃	—	6	1	no	—
130	C₄×Dic₃	—	6	1	no	—
131	C₂×C₃⋊C₈	—	3	2	no	—
132	C₂×C₃⋊C₈	—	6	1	no	—
133	C₂×C₃⋊C₈	—	6	1	no	—
134	C₂×C₃⋊C₈	—	6	1	no	—
135	C₂×C₃⋊C₈	—	6	1	no	—
136	C₂×C₃⋊C₈	—	6	1	no	—
137	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
140	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
141	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
142	C₂²×C₉	#131	1	1	yes	characteristic
143	C₂²×C₉	#131	1	1	yes	characteristic
144	C₄×C₉	—	2	2	no	—
145	C₄×C₉	—	4	1	no	—
146	C₄×C₉	#121	1	1	yes	characteristic
147	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
148	C₄×C₉	#125	1	1	yes	characteristic
149	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
150	Dic₉	—	2	2	no	—
151	Dic₉	—	2	2	no	—
152	Dic₉	—	4	1	no	—
153	Dic₉	C₄⋊C₈	1	4	yes	complemented
154	C₂²×C₈	—	9	1	no	—
155	C₂²×C₈	—	9	1	no	—
156	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
157	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
158	C₂×C₄⋊C₄	—	9	1	no	—
159	C₂×C₄⋊C₄	—	9	1	no	—
160	C₂×C₄²	—	9	1	no	—
161	C₂×C₄²	—	9	1	no	—
162	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
163	C₄⋊C₈	—	18	2	no	—
164	C₄⋊C₈	—	18	2	no	—
165	C₄⋊C₈	—	18	2	no	—
166	C₄×C₈	—	18	2	no	—
167	#95	—	9	1	no	—
168	#95	—	9	1	no	—
169	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
176	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
177	C₂×C₃×C₄	#235	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	#237	1	1	yes	characteristic
181	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
182	C₂×Dic₃	—	3	2	no	—
183	C₂×Dic₃	—	3	2	no	—
184	C₂×Dic₃	—	3	2	no	—
185	C₂×Dic₃	—	3	2	no	—
186	C₂×Dic₃	—	3	2	no	—
187	C₂×Dic₃	—	6	1	no	—
188	C₂×Dic₃	—	6	1	no	—
189	C₂×Dic₃	—	6	1	no	—
190	C₂×Dic₃	—	6	1	no	—
191	C₂×Dic₃	—	6	1	no	—
192	C₂×Dic₃	—	6	1	no	—
193	C₂×Dic₃	—	6	1	no	—
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	—	4	1	no	—
196	C₃⋊C₈	—	6	2	no	—
197	C₃⋊C₈	—	12	1	no	—
198	C₂×C₉	#389	1	1	yes	characteristic
199	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
200	C₂×C₉	#424	1	1	yes	characteristic
201	C₂×C₉	C₈×D₄	1	1	yes	characteristic
202	C₂×C₉	#436	1	1	yes	characteristic
203	C₂×C₉	C₈×Q₈	1	1	yes	characteristic
204	C₂×C₉	#446	1	1	yes	characteristic
205	C₂²×C₄	—	9	1	no	—
206	C₂²×C₄	—	9	1	no	—
207	C₂²×C₄	—	9	1	no	—
208	C₂²×C₄	—	9	1	no	—
209	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
210	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	9	2	no	—
218	C₂×C₈	—	18	1	no	—
219	C₂×C₈	—	18	1	no	—
220	C₂×C₈	—	18	1	no	—
221	C₂×C₈	—	18	1	no	—
222	C₂×C₈	—	18	1	no	—
223	C₂×C₈	Dic₁₈	1	2	yes	—
224	C₄⋊C₄	—	9	4	no	—
225	C₄⋊C₄	—	18	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	9	2	no	—
229	C₄²	—	9	2	no	—
230	C₄²	—	18	1	no	—
231	C₄²	—	18	1	no	—
232	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
233	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
234	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
235	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
236	C₂²×C₃	#901	1	1	yes	characteristic
237	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
238	C₂²×C₃	#907	1	1	yes	characteristic
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	#878	1	1	yes	characteristic
242	C₃×C₄	#884	1	1	yes	characteristic
243	C₃×C₄	#888	1	1	yes	characteristic
244	C₃×C₄	#889	1	1	yes	characteristic
245	Dic₃	—	3	4	no	—
246	Dic₃	—	6	2	no	—
247	Dic₃	—	6	2	no	—
248	Dic₃	—	12	1	no	—
249	C₉	#1939	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	9	2	no	—
257	C₂×C₄	—	9	2	no	—
258	C₂×C₄	—	9	2	no	—
259	C₂×C₄	—	9	2	no	—
260	C₂×C₄	—	9	2	no	—
261	C₂×C₄	—	18	1	no	—
262	C₂×C₄	—	18	1	no	—
263	C₂×C₄	—	18	1	no	—
264	C₂×C₄	—	18	1	no	—
265	C₂×C₄	—	18	1	no	—
266	C₂×C₄	—	18	1	no	—
267	C₂×C₄	—	18	1	no	—
268	C₂×C₄	C₈×D₉	1	2	yes	—
269	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
270	C₂×C₄	#3697	1	1	yes	characteristic
271	C₂×C₄	D₄×D₉	1	1	yes	characteristic
272	C₂×C₄	D₄×D₉	1	1	yes	characteristic
273	C₂×C₄	#3699	1	1	yes	characteristic
274	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	1	no	—
277	C₈	—	18	2	no	—
278	C₈	—	36	1	no	—
279	C₂×C₃	#4647	1	1	yes	characteristic
280	C₂×C₃	C₈×Dic₆	1	1	yes	characteristic
281	C₂×C₃	#4701	1	1	yes	characteristic
282	C₂×C₃	#67⋊C₈	1	1	yes	characteristic
283	C₂×C₃	#4729	1	1	yes	characteristic
284	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
285	C₂×C₃	#4839	1	1	yes	characteristic
286	C₂²	C₄×Dic₁₈	1	1	yes	characteristic
287	C₂²	#611⋊C₄	1	1	yes	characteristic
288	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
289	C₂²	C₂×C₈×D₉	1	1	yes	characteristic
290	C₂²	#63464	1	1	yes	characteristic
291	C₂²	M₁₆×D₉	1	1	yes	characteristic
292	C₂²	#63469	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	9	4	no	—
296	C₄	—	18	2	no	—
297	C₄	—	18	2	no	—
298	C₄	—	36	1	no	—
299	C₄	#63440	1	1	yes	characteristic
300	C₄	#63446	1	1	yes	characteristic
301	C₄	#63450	1	1	yes	characteristic
302	C₄	#63451	1	1	yes	characteristic
303	C₃	#68617	1	1	yes	characteristic
304	C₂	#10588327	1	1	yes	characteristic
305	C₂	C₈×Dic₁₈	1	1	yes	characteristic
306	C₂	#10588381	1	1	yes	characteristic
307	C₂	#611⋊C₈	1	1	yes	characteristic
308	C₂	#10588411	1	1	yes	characteristic
309	C₂	D₉×C₄⋊C₈	1	1	yes	characteristic
310	C₂	#10588523	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	#1939
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
47. Dic₉⋊C₄	C₈	semidirect
153. Dic₉	C₄⋊C₈	semidirect
249. C₉	#1939	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₉⋊C₈	2
C₂×C₈×Dic₉	2
C₄×Dic₉⋊C₄	2
C₂×C₄⋊(C₉⋊C₈)	2
C₉×#336	2
#10588228	2
#68617	3