Subgroups — #103×Dic₉ #11783967

Order 1152.

498 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#103	C₂	1	1	yes	maximal, characteristic
3	C₂×#63395	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
6	#10588229	C₂	1	1	yes	maximal, characteristic
7	#10588206	C₂	1	1	yes	maximal, characteristic
8	Dic₃×#103	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	Q₈×Dic₉	—	2	2	no	—
13	Q₈×Dic₉	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₉	—	2	1	no	—
15	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₉	—	2	1	no	—
18	C₂×C₄×Dic₉	—	2	1	no	—
19	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
20	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₉×#103	C₄	1	4	yes	complemented
22	#63395	C₄	1	4	yes	complemented
23	C₉⋊#95	—	2	1	no	—
24	C₈×Dic₉	—	2	2	no	—
25	C₂×Q₈×Dic₃	—	3	1	no	—
26	C₂×C₃×#103	D₃	1	1	yes	characteristic
27	C₂×#835	—	3	1	no	—
28	C₂×C₈×Dic₃	—	3	1	no	—
29	Dic₃×C₄⋊C₄	—	3	1	no	—
30	#4551	—	3	1	no	—
31	#4529	—	3	1	no	—
32	C₂×Q₈×C₉	—	2	1	no	—
33	C₂×Q₈×C₉	—	2	1	no	—
34	C₂×Q₈×C₉	—	2	1	no	—
35	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
36	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
37	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₉	—	2	1	no	—
42	C₂²×Dic₉	—	2	1	no	—
43	C₂²×Dic₉	D₄	1	1	yes	characteristic
44	C₂²×Dic₉	D₄	1	1	yes	characteristic
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	—	2	1	no	—
47	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
48	C₉×C₄⋊C₄	—	2	1	no	—
49	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₉	—	2	1	no	—
51	C₄⋊Dic₉	—	2	1	no	—
52	C₄⋊Dic₉	—	2	4	no	—
53	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₉	—	2	4	no	—
56	C₄×Dic₉	—	4	2	no	—
57	C₄×Dic₉	C₂×C₄	1	2	yes	—
58	C₄×Dic₉	D₄	1	2	yes	—
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂×C₉⋊C₈	—	2	1	no	—
61	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×#103	—	9	1	no	Sylow
63	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	Q₈×Dic₃	—	3	4	no	—
67	Q₈×Dic₃	—	6	2	no	—
68	C₂×C₄⋊Dic₃	—	3	1	no	—
69	C₂×C₄⋊Dic₃	—	3	1	no	—
70	C₂×C₄⋊Dic₃	—	6	1	no	—
71	C₂×C₄×Dic₃	—	3	1	no	—
72	C₂×C₄×Dic₃	—	6	1	no	—
73	C₂×C₄×Dic₃	—	6	1	no	—
74	C₂²×C₃⋊C₈	—	3	1	no	—
75	C₃×#103	Dic₃	1	4	yes	—
76	#835	—	3	4	no	—
77	C₃⋊#95	—	6	1	no	—
78	C₈×Dic₃	—	6	2	no	—
79	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₉	—	2	2	no	—
81	Q₈×C₉	—	2	2	no	—
82	Q₈×C₉	—	2	2	no	—
83	Q₈×C₉	C₄²	1	4	yes	complemented
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	—	4	1	no	—
90	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉	#41	1	1	yes	characteristic
95	C₂×C₄×C₉	#41	1	1	yes	characteristic
96	C₂×Dic₉	—	2	1	no	—
97	C₂×Dic₉	—	2	1	no	—
98	C₂×Dic₉	—	2	2	no	—
99	C₂×Dic₉	—	2	2	no	—
100	C₂×Dic₉	—	4	1	no	—
101	C₂×Dic₉	—	4	1	no	—
102	C₂×Dic₉	—	4	1	no	—
103	C₂×Dic₉	—	4	1	no	—
104	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
105	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
106	C₂×Dic₉	SD₁₆	1	2	yes	—
107	C₂×Dic₉	Q₁₆	1	2	yes	—
108	C₈×C₉	—	2	2	no	—
109	C₉⋊C₈	—	2	2	no	—
110	C₂×C₄×Q₈	—	9	1	no	—
111	C₂×#103	—	9	1	no	—
112	C₂×#103	D₉	1	1	yes	characteristic
113	C₂×C₄×C₈	—	9	1	no	—
114	C₄×C₄⋊C₄	—	9	1	no	—
115	#340	—	9	1	no	—
116	#340	—	9	1	no	—
117	C₂×C₃×Q₈	—	2	1	no	—
118	C₂×C₃×Q₈	—	2	1	no	—
119	C₂×C₃×Q₈	—	2	1	no	—
120	C₂×C₃×Q₈	C₄×D₃	1	2	yes	—
121	C₂×C₃×Q₈	C₄×D₃	1	2	yes	—
122	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
123	C₂²×C₃×C₄	—	2	1	no	—
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
126	C₂²×Dic₃	—	3	1	no	—
127	C₂²×Dic₃	—	3	1	no	—
128	C₂²×Dic₃	—	6	1	no	—
129	C₂²×Dic₃	—	6	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
135	C₄⋊Dic₃	—	3	2	no	—
136	C₄⋊Dic₃	—	3	2	no	—
137	C₄⋊Dic₃	—	6	4	no	—
138	C₄⋊Dic₃	—	6	1	no	—
139	C₄⋊Dic₃	—	6	1	no	—
140	C₄×Dic₃	—	3	2	no	—
141	C₄×Dic₃	—	3	2	no	—
142	C₄×Dic₃	—	6	4	no	—
143	C₄×Dic₃	—	12	2	no	—
144	C₂×C₃⋊C₈	—	3	2	no	—
145	C₂×C₃⋊C₈	—	6	1	no	—
146	C₂×C₃⋊C₈	—	6	1	no	—
147	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
149	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
150	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
151	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
152	C₂²×C₉	#135	1	1	yes	characteristic
153	C₂²×C₉	#135	1	1	yes	characteristic
154	C₄×C₉	—	2	4	no	—
155	C₄×C₉	—	4	2	no	—
156	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
157	C₄×C₉	#117	1	1	yes	characteristic
158	C₄×C₉	C₄×D₄	1	1	yes	characteristic
159	C₄×C₉	C₄×D₄	1	1	yes	characteristic
160	Dic₉	—	2	2	no	—
161	Dic₉	—	4	2	no	—
162	Dic₉	—	4	2	no	—
163	Dic₉	#103	1	4	yes	direct product factor
164	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
165	C₂²×C₈	—	9	1	no	—
166	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
167	C₄×Q₈	—	9	4	no	—
168	C₄×Q₈	—	18	2	no	—
169	C₂×C₄⋊C₄	—	9	1	no	—
170	C₂×C₄⋊C₄	—	9	1	no	—
171	C₂×C₄⋊C₄	—	18	1	no	—
172	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
173	C₂×C₄²	—	9	1	no	—
174	C₂×C₄²	—	18	1	no	—
175	C₂×C₄²	—	18	1	no	—
176	#103	—	9	4	no	—
177	#103	Dic₉	1	4	yes	direct product factor
178	C₄×C₈	—	18	2	no	—
179	#95	—	18	1	no	—
180	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
181	C₃×Q₈	—	2	2	no	—
182	C₃×Q₈	—	2	2	no	—
183	C₃×Q₈	—	2	2	no	—
184	C₃×Q₈	C₄×Dic₃	1	4	yes	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	4	1	no	—
190	C₂×C₃×C₄	—	4	1	no	—
191	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	#237	1	1	yes	characteristic
195	C₂×C₃×C₄	#237	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
197	C₂×Dic₃	—	3	2	no	—
198	C₂×Dic₃	—	3	2	no	—
199	C₂×Dic₃	—	3	2	no	—
200	C₂×Dic₃	—	3	2	no	—
201	C₂×Dic₃	—	6	1	no	—
202	C₂×Dic₃	—	6	1	no	—
203	C₂×Dic₃	—	6	2	no	—
204	C₂×Dic₃	—	6	2	no	—
205	C₂×Dic₃	—	12	1	no	—
206	C₂×Dic₃	—	12	1	no	—
207	C₂×Dic₃	—	12	1	no	—
208	C₂×Dic₃	—	12	1	no	—
209	C₃×C₈	—	2	2	no	—
210	C₃⋊C₈	—	6	2	no	—
211	C₂×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₉	C₂×#103	1	1	yes	characteristic
213	C₂×C₉	#416	1	1	yes	characteristic
214	C₂×C₉	C₄×SD₁₆	1	1	yes	characteristic
215	C₂×C₉	C₄×SD₁₆	1	1	yes	characteristic
216	C₂×C₉	C₄×Q₁₆	1	1	yes	characteristic
217	C₂×C₉	C₄×Q₁₆	1	1	yes	characteristic
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	C₄×D₉	1	2	yes	—
222	C₂×Q₈	C₄×D₉	1	2	yes	—
223	C₂×Q₈	C₂×Dic₉	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	9	1	no	—
227	C₂²×C₄	—	9	1	no	—
228	C₂²×C₄	—	18	1	no	—
229	C₂²×C₄	—	18	1	no	—
230	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	9	2	no	—
234	C₂×C₈	—	18	1	no	—
235	C₂×C₈	—	18	1	no	—
236	C₂×C₈	C₂×Dic₉	1	2	yes	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	9	2	no	—
239	C₄⋊C₄	—	9	2	no	—
240	C₄⋊C₄	—	18	4	no	—
241	C₄⋊C₄	—	18	1	no	—
242	C₄⋊C₄	—	18	1	no	—
243	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
244	C₄²	—	9	2	no	—
245	C₄²	—	9	2	no	—
246	C₄²	—	18	4	no	—
247	C₄²	—	36	2	no	—
248	C₂²×C₃	D₃×C₂²⋊C₄	1	1	yes	characteristic
249	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
250	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
251	C₂²×C₃	#916	1	1	yes	characteristic
252	C₂²×C₃	D₃×Q₁₆	1	1	yes	characteristic
253	C₂²×C₃	#919	1	1	yes	characteristic
254	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	#877	1	1	yes	characteristic
258	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
259	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
260	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
261	Dic₃	—	3	4	no	—
262	Dic₃	—	6	2	no	—
263	Dic₃	—	12	2	no	—
264	Dic₃	—	12	2	no	—
265	C₉	C₄×#103	1	1	yes	characteristic, complemented, Sylow
266	C₂³	D₄×D₉	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	Q₈	—	2	2	no	—
269	Q₈	—	2	2	no	—
270	Q₈	C₄×Dic₉	1	4	yes	complemented
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	9	2	no	—
278	C₂×C₄	—	9	2	no	—
279	C₂×C₄	—	9	2	no	—
280	C₂×C₄	—	9	2	no	—
281	C₂×C₄	—	18	1	no	—
282	C₂×C₄	—	18	1	no	—
283	C₂×C₄	—	18	2	no	—
284	C₂×C₄	—	18	2	no	—
285	C₂×C₄	—	36	1	no	—
286	C₂×C₄	—	36	1	no	—
287	C₂×C₄	—	36	1	no	—
288	C₂×C₄	—	36	1	no	—
289	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
291	C₂×C₄	D₄×D₉	1	1	yes	characteristic
292	C₂×C₄	#3699	1	1	yes	characteristic
293	C₂×C₄	#3699	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	18	2	no	—
297	C₂×C₃	#4788	1	1	yes	characteristic
298	C₂×C₃	#4790	1	1	yes	characteristic
299	C₂×C₃	D₃×#103	1	1	yes	characteristic
300	C₂×C₃	#4804	1	1	yes	characteristic
301	C₂×C₃	Dic₃×C₂²⋊C₄	1	1	yes	characteristic
302	C₂×C₃	Dic₃×SD₁₆	1	1	yes	characteristic
303	C₂×C₃	Dic₃×Q₁₆	1	1	yes	characteristic
304	C₂²	C₂²⋊C₄×D₉	1	1	yes	characteristic
305	C₂²	#611⋊C₄	1	1	yes	characteristic
306	C₂²	SD₁₆×D₉	1	1	yes	characteristic
307	C₂²	#63478	1	1	yes	characteristic
308	C₂²	Q₁₆×D₉	1	1	yes	characteristic
309	C₂²	#63481	1	1	yes	characteristic
310	C₂²	D₄×Dic₉	1	1	yes	characteristic
311	C₄	—	2	4	no	—
312	C₄	—	4	2	no	—
313	C₄	—	9	4	no	—
314	C₄	—	18	2	no	—
315	C₄	—	36	2	no	—
316	C₄	—	36	2	no	—
317	C₄	#63439	1	1	yes	characteristic
318	C₄	#611⋊C₄	1	1	yes	characteristic
319	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
320	C₄	D₄×Dic₉	1	1	yes	characteristic
321	C₃	Dic₃×#103	1	1	yes	characteristic
322	C₂	#10588468	1	1	yes	characteristic
323	C₂	#10588470	1	1	yes	characteristic
324	C₂	D₉×#103	1	1	yes	characteristic
325	C₂	#10588484	1	1	yes	characteristic
326	C₂	C₂²⋊C₄×Dic₉	1	1	yes	characteristic
327	C₂	SD₁₆×Dic₉	1	1	yes	characteristic
328	C₂	Q₁₆×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#103	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#103	D₉
hypocenter	C₉	C₄×#103
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
13. Q₈×Dic₉	C₄	semidirect
21. C₉×#103	C₄	semidirect
22. #63395	C₄	semidirect
83. Q₈×C₉	C₄²	semidirect
163. Dic₉	#103	direct (N × H)
177. #103	Dic₉	direct (N × H)
265. C₉	C₄×#103	semidirect
270. Q₈	C₄×Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₉	2
C₂×C₉×#103	2
C₂×#63395	2
C₂×C₈×Dic₉	2
C₄⋊C₄×Dic₉	2
#10588229	2
#10588206	2
Dic₃×#103	3

Subgroups — #103×Dic9 #11783967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #103×Dic₉ #11783967