Subgroups — #11786449

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#380	C₂	1	1	yes	maximal, characteristic
3	C₂×#63389	C₂	1	1	yes	maximal, characteristic
4	C₂×#63367	C₂	1	2	yes	maximal, complemented
5	#10588206	C₂	1	2	yes	maximal, complemented
6	#69505	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂³×Dic₉	—	2	1	no	—
9	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
12	C₂×C₄⋊Dic₉	—	2	1	no	—
13	C₂×C₄⋊Dic₉	—	2	4	no	—
14	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
15	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
16	C₉×#95	—	2	1	no	—
17	#63389	C₄	1	4	yes	complemented
18	#63367	—	2	4	no	—
19	C₂²×C₄⋊Dic₃	—	3	1	no	—
20	C₃×#380	D₃	1	1	yes	characteristic
21	C₂×#830	—	3	1	no	—
22	C₂×#808	—	3	2	no	—
23	#4529	—	3	2	no	—
24	C₂⁴×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	1	no	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂²×Dic₉	—	2	4	no	—
34	C₂²×Dic₉	—	2	2	no	—
35	C₂²×Dic₉	—	4	2	no	—
36	C₂²×Dic₉	—	4	2	no	—
37	C₉×C₄⋊C₄	—	2	2	no	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₉×C₂²⋊C₄	—	4	2	no	—
40	C₄⋊Dic₉	—	2	4	no	—
41	C₄⋊Dic₉	—	2	2	no	—
42	C₄⋊Dic₉	D₄	1	4	yes	—
43	C₂×C₉⋊C₈	—	2	2	no	—
44	C₂×C₉⋊C₈	—	2	2	no	—
45	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	complemented
46	#1853	—	9	1	no	Sylow
47	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
48	C₂³×Dic₃	—	6	1	no	—
49	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
52	C₂×C₄⋊Dic₃	—	3	2	no	—
53	C₂×C₄⋊Dic₃	—	6	4	no	—
54	C₂×C₄⋊Dic₃	—	6	1	no	—
55	C₂²×C₃⋊C₈	—	3	2	no	—
56	C₃×#95	—	2	1	no	—
57	#830	—	3	4	no	—
58	#808	—	6	4	no	—
59	C₂³×C₉	—	2	1	no	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	—	2	1	no	—
62	C₂³×C₉	—	2	1	no	—
63	C₂³×C₉	C₄⋊C₄	1	2	yes	—
64	C₂³×C₉	SD₁₆	1	2	yes	—
65	C₂³×C₉	SD₁₆	1	2	yes	—
66	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	4	1	no	—
72	C₂×C₄×C₉	—	4	1	no	—
73	C₂×C₄×C₉	—	4	1	no	—
74	C₂×C₄×C₉	—	4	1	no	—
75	C₂×C₄×C₉	—	4	1	no	—
76	C₂×C₄×C₉	—	4	1	no	—
77	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₉	#41	1	2	yes	—
82	C₂×Dic₉	—	2	4	no	—
83	C₂×Dic₉	—	4	8	no	—
84	C₂×Dic₉	—	4	2	no	—
85	C₂×Dic₉	—	4	2	no	—
86	C₉⋊C₈	—	2	4	no	—
87	C₂²×C₄⋊C₄	—	9	1	no	—
88	C₂×#106	—	9	2	no	—
89	C₂×C₂²⋊C₈	—	9	1	no	—
90	#380	D₉	1	1	yes	characteristic
91	#340	—	9	2	no	—
92	C₂⁴×C₃	#67	1	1	yes	characteristic
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	Dic₆	1	2	yes	—
98	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
99	C₂²×C₃×C₄	#67	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	6	4	no	—
102	C₂²×Dic₃	—	6	2	no	—
103	C₂²×Dic₃	—	12	2	no	—
104	C₂²×Dic₃	—	12	2	no	—
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₂²⋊C₄	—	4	2	no	—
108	C₄⋊Dic₃	—	3	4	no	—
109	C₄⋊Dic₃	—	6	4	no	—
110	C₄⋊Dic₃	—	6	2	no	—
111	C₂×C₃⋊C₈	—	3	4	no	—
112	C₂×C₃⋊C₈	—	6	2	no	—
113	C₂×C₃⋊C₈	—	6	2	no	—
114	C₂²×C₉	—	2	2	no	—
115	C₂²×C₉	—	2	2	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	2	2	no	—
118	C₂²×C₉	—	2	2	no	—
119	C₂²×C₉	—	2	2	no	—
120	C₂²×C₉	#106	1	4	yes	complemented
121	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₉	#120	1	1	yes	characteristic
123	C₂²×C₉	#123	1	1	yes	characteristic
124	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×C₉	#137	1	1	yes	characteristic
128	C₄×C₉	—	2	2	no	—
129	C₄×C₉	—	8	1	no	—
130	C₄×C₉	—	8	1	no	—
131	C₄×C₉	C₄×D₄	1	2	yes	—
132	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
133	Dic₉	—	4	4	no	—
134	C₂³×C₄	—	18	1	no	—
135	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
136	C₂²×C₈	—	9	2	no	—
137	C₂×C₄⋊C₄	—	9	2	no	—
138	C₂×C₄⋊C₄	—	18	4	no	—
139	C₂×C₄⋊C₄	—	18	1	no	—
140	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	—	2	1	no	—
143	#106	—	18	4	no	—
144	C₂²⋊C₈	—	9	4	no	—
145	#95	—	2	1	no	—
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	—	2	1	no	—
148	C₂³×C₃	—	2	1	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
151	C₂³×C₃	#214	1	2	yes	—
152	C₂³×C₃	#215	1	2	yes	—
153	C₂³×C₃	C₂×#67	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	4	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	1	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
165	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
168	C₂×C₃×C₄	#237	1	2	yes	—
169	C₂×Dic₃	—	6	4	no	—
170	C₂×Dic₃	—	12	8	no	—
171	C₂×Dic₃	—	12	2	no	—
172	C₂×Dic₃	—	12	2	no	—
173	C₃⋊C₈	—	6	4	no	—
174	C₂×C₉	—	2	4	no	—
175	C₂×C₉	#385	1	1	yes	characteristic
176	C₂×C₉	C₂×#106	1	1	yes	characteristic
177	C₂×C₉	#428	1	1	yes	characteristic
178	C₂×C₉	#448	1	1	yes	characteristic
179	C₂×C₉	#450	1	1	yes	characteristic
180	C₂×C₉	#481	1	1	yes	characteristic
181	C₂×C₉	#483	1	1	yes	characteristic
182	C₂⁴	#611	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	18	4	no	—
188	C₂²×C₄	—	18	2	no	—
189	C₂²×C₄	—	36	2	no	—
190	C₂²×C₄	—	36	2	no	—
191	C₂²×C₄	Dic₁₈	1	2	yes	—
192	C₂²×C₄	C₄×D₉	1	2	yes	—
193	C₂²×C₄	#611	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
195	C₂×C₈	—	9	4	no	—
196	C₂×C₈	—	18	2	no	—
197	C₂×C₈	—	18	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	9	4	no	—
201	C₄⋊C₄	—	18	4	no	—
202	C₄⋊C₄	—	18	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	#808	1	4	yes	—
211	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
212	C₂²×C₃	C₂×#214	1	1	yes	characteristic
213	C₂²×C₃	#935	1	1	yes	characteristic
214	C₂²×C₃	#937	1	1	yes	characteristic
215	C₂²×C₃	C₂×#215	1	1	yes	characteristic
216	C₂²×C₃	#949	1	1	yes	characteristic
217	C₂²×C₃	#951	1	1	yes	characteristic
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	8	1	no	—
220	C₃×C₄	—	8	1	no	—
221	C₃×C₄	#878	1	2	yes	—
222	C₃×C₄	#67⋊C₄	1	2	yes	—
223	Dic₃	—	12	4	no	—
224	C₉	#1853	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	Dic₉⋊C₄	1	2	yes	—
230	C₂³	#3672	1	2	yes	—
231	C₂³	#3675	1	2	yes	—
232	C₂³	C₂×#611	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	18	4	no	—
244	C₂×C₄	—	36	8	no	—
245	C₂×C₄	—	36	2	no	—
246	C₂×C₄	—	36	2	no	—
247	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
248	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
250	C₂×C₄	D₄×D₉	1	2	yes	—
251	C₂×C₄	#3699	1	2	yes	—
252	C₈	—	18	4	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	#4946	1	1	yes	characteristic
255	C₂×C₃	C₂×#808	1	1	yes	characteristic
256	C₂×C₃	#4995	1	1	yes	characteristic
257	C₂×C₃	#5034	1	1	yes	characteristic
258	C₂×C₃	#5040	1	1	yes	characteristic
259	C₂×C₃	#5043	1	1	yes	characteristic
260	C₂×C₃	#5046	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	#63367	1	4	yes	complemented
268	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
269	C₂²	C₂×#3672	1	1	yes	characteristic
270	C₂²	#63497	1	1	yes	characteristic
271	C₂²	#63499	1	1	yes	characteristic
272	C₂²	C₂×#3675	1	1	yes	characteristic
273	C₂²	#63511	1	1	yes	characteristic
274	C₂²	#63512	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	1	no	—
278	C₄	—	36	4	no	—
279	C₄	#63440	1	2	yes	—
280	C₄	#611⋊C₄	1	2	yes	—
281	C₃	#69505	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	#10588626	1	1	yes	characteristic
284	C₂	C₂×#63367	1	1	yes	characteristic
285	C₂	#10588679	1	1	yes	characteristic
286	C₂	#10588714	1	1	yes	characteristic
287	C₂	#10588716	1	1	yes	characteristic
288	C₂	#10588723	1	1	yes	characteristic
289	C₂	#10588729	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#380	C₂
socle	C₂³×C₃	C₂×#67
prime radical	C₂⁴×C₃	#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#380	D₉
hypocenter	C₉	#1853
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

Complements

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

N	H ≅ G/N	type
4. C₂×#63367	C₂	semidirect
5. #10588206	C₂	semidirect
17. #63389	C₄	semidirect
45. C₂×C₉⋊C₈	C₂×C₄	semidirect
120. C₂²×C₉	#106	semidirect
224. C₉	#1853	semidirect
267. C₂²	#63367	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₉	2
C₉×#380	2
C₂×#63389	2
C₂×#63367	2
#10588206	2
#69505	3