Subgroups — #11789047

Order 1152.

345 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	#10588664	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#63366	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	1	yes	maximal, characteristic
5	#10588229	C₂	1	1	yes	maximal, characteristic
6	#10588228	C₂	1	1	yes	maximal, characteristic
7	#10588206	C₂	1	1	yes	maximal, characteristic
8	#68815	—	3	1	no	maximal
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₉	—	2	1	no	—
13	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₉	—	2	1	no	—
16	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
17	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₉⋊#95	—	2	1	no	—
19	C₉⋊#95	—	2	1	no	—
20	#63366	C₄	1	4	yes	complemented
21	#4984	—	3	1	no	—
22	Dic₃×C₄⋊C₄	—	3	1	no	—
23	C₂×#807	—	3	1	no	—
24	C₃×#340	D₃	1	1	yes	characteristic
25	#4551	—	3	1	no	—
26	#4550	—	3	1	no	—
27	#4529	—	3	1	no	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	—	2	1	no	—
32	C₂²×Dic₉	—	2	1	no	—
33	C₂²×Dic₉	D₄	1	1	yes	characteristic
34	C₂²×Dic₉	D₄	1	1	yes	characteristic
35	C₂×C₈×C₉	—	2	1	no	—
36	C₂×C₈×C₉	—	2	1	no	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₉×C₄⋊C₄	—	2	1	no	—
39	C₉×C₄⋊C₄	—	2	1	no	—
40	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₉×C₄⋊C₄	Q₈	1	2	yes	—
42	C₄⋊Dic₉	—	2	1	no	—
43	C₄⋊Dic₉	—	2	1	no	—
44	C₄⋊Dic₉	—	2	1	no	—
45	C₄⋊Dic₉	—	4	2	no	—
46	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
47	C₄×Dic₉	—	2	1	no	—
48	C₄×Dic₉	—	2	1	no	—
49	C₄×Dic₉	—	4	2	no	—
50	C₂×C₉⋊C₈	—	2	1	no	—
51	C₂×C₉⋊C₈	—	2	1	no	—
52	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
53	#1918	—	9	1	no	Sylow
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄⋊Dic₃	—	3	1	no	—
58	C₂×C₄⋊Dic₃	—	3	1	no	—
59	C₂×C₄⋊Dic₃	—	6	1	no	—
60	C₂×C₄×Dic₃	—	3	1	no	—
61	C₂×C₄×Dic₃	—	6	1	no	—
62	C₂²×C₃⋊C₈	—	3	1	no	—
63	C₃⋊#95	—	6	1	no	—
64	C₃⋊#95	—	6	1	no	—
65	#807	—	3	4	no	—
66	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	4	1	no	—
70	C₂×C₄×C₉	—	4	1	no	—
71	C₂×C₄×C₉	—	4	1	no	—
72	C₂×C₄×C₉	—	4	1	no	—
73	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₉	#41	1	1	yes	characteristic
76	C₂×C₄×C₉	#41	1	1	yes	characteristic
77	C₂×C₄×C₉	#41	1	1	yes	characteristic
78	C₂×C₄×C₉	#41	1	1	yes	characteristic
79	C₂×Dic₉	—	2	1	no	—
80	C₂×Dic₉	—	2	1	no	—
81	C₂×Dic₉	—	2	2	no	—
82	C₂×Dic₉	—	2	1	no	—
83	C₂×Dic₉	—	2	1	no	—
84	C₂×Dic₉	—	2	1	no	—
85	C₂×Dic₉	—	4	1	no	—
86	C₂×Dic₉	—	4	1	no	—
87	C₂×Dic₉	—	4	1	no	—
88	C₂×Dic₉	—	4	1	no	—
89	C₂×Dic₉	—	4	1	no	—
90	C₂×Dic₉	D₈	1	2	yes	—
91	C₈×C₉	—	4	1	no	—
92	C₉⋊C₈	—	2	2	no	—
93	C₂×#107	—	9	1	no	—
94	#389	—	9	1	no	—
95	C₄×C₄⋊C₄	—	9	1	no	—
96	#340	—	9	1	no	—
97	#340	—	9	1	no	—
98	#340	D₉	1	1	yes	characteristic
99	#336	—	9	1	no	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×Dic₃	—	3	1	no	—
104	C₂²×Dic₃	—	3	1	no	—
105	C₂²×Dic₃	—	6	1	no	—
106	C₂²×Dic₃	—	6	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₃×C₄⋊C₄	—	2	1	no	—
111	C₃×C₄⋊C₄	—	2	1	no	—
112	C₃×C₄⋊C₄	Dic₆	1	2	yes	—
113	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
114	C₄⋊Dic₃	—	3	2	no	—
115	C₄⋊Dic₃	—	6	1	no	—
116	C₄⋊Dic₃	—	6	1	no	—
117	C₄⋊Dic₃	—	6	1	no	—
118	C₄⋊Dic₃	—	12	2	no	—
119	C₄×Dic₃	—	6	1	no	—
120	C₄×Dic₃	—	6	1	no	—
121	C₄×Dic₃	—	12	2	no	—
122	C₂×C₃⋊C₈	—	3	2	no	—
123	C₂×C₃⋊C₈	—	6	1	no	—
124	C₂×C₃⋊C₈	—	6	1	no	—
125	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂²×C₉	#123	1	1	yes	characteristic
128	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
129	C₂²×C₉	#135	1	1	yes	characteristic
130	C₂²×C₉	#137	1	1	yes	characteristic
131	C₂²×C₉	#137	1	1	yes	characteristic
132	C₄×C₉	—	4	2	no	—
133	C₄×C₉	—	4	2	no	—
134	C₄×C₉	#117	1	1	yes	characteristic
135	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
136	C₄×C₉	#125	1	1	yes	characteristic
137	C₄×C₉	#126	1	1	yes	characteristic
138	Dic₉	—	2	2	no	—
139	Dic₉	—	4	2	no	—
140	Dic₉	—	4	1	no	—
141	Dic₉	—	8	1	no	—
142	C₂²×C₈	—	9	1	no	—
143	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
144	C₂×C₄⋊C₄	—	9	1	no	—
145	C₂×C₄⋊C₄	—	9	1	no	—
146	C₂×C₄⋊C₄	—	18	1	no	—
147	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
149	C₂×C₄²	—	9	1	no	—
150	C₂×C₄²	—	18	1	no	—
151	#107	—	9	4	no	—
152	#95	—	18	1	no	—
153	#95	—	18	1	no	—
154	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	4	1	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	C₂×Dic₆	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂×C₃×C₄	#235	1	1	yes	characteristic
164	C₂×C₃×C₄	#237	1	1	yes	characteristic
165	C₂×C₃×C₄	#237	1	1	yes	characteristic
166	C₂×C₃×C₄	#239	1	1	yes	characteristic
167	C₂×Dic₃	—	3	2	no	—
168	C₂×Dic₃	—	6	1	no	—
169	C₂×Dic₃	—	6	1	no	—
170	C₂×Dic₃	—	6	2	no	—
171	C₂×Dic₃	—	6	1	no	—
172	C₂×Dic₃	—	6	1	no	—
173	C₂×Dic₃	—	6	1	no	—
174	C₂×Dic₃	—	12	1	no	—
175	C₂×Dic₃	—	12	1	no	—
176	C₂×Dic₃	—	12	1	no	—
177	C₂×Dic₃	—	12	1	no	—
178	C₂×Dic₃	—	12	1	no	—
179	C₃×C₈	—	4	1	no	—
180	C₃⋊C₈	—	6	2	no	—
181	C₂×C₉	#387	1	1	yes	characteristic
182	C₂×C₉	C₄×D₈	1	1	yes	characteristic
183	C₂×C₉	#441	1	1	yes	characteristic
184	C₂×C₉	#474	1	1	yes	characteristic
185	C₂×C₉	#479	1	1	yes	characteristic
186	C₂×C₉	#480	1	1	yes	characteristic
187	C₂×C₉	#485	1	1	yes	characteristic
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	9	1	no	—
191	C₂²×C₄	—	9	1	no	—
192	C₂²×C₄	—	18	1	no	—
193	C₂²×C₄	—	18	1	no	—
194	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	9	2	no	—
199	C₂×C₈	—	18	1	no	—
200	C₂×C₈	—	18	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	9	2	no	—
204	C₄⋊C₄	—	18	1	no	—
205	C₄⋊C₄	—	18	1	no	—
206	C₄⋊C₄	—	18	1	no	—
207	C₄⋊C₄	—	36	2	no	—
208	C₄⋊C₄	Dic₁₈	1	2	yes	—
209	C₄⋊C₄	C₄×D₉	1	2	yes	—
210	C₄²	—	18	1	no	—
211	C₄²	—	18	1	no	—
212	C₄²	—	36	2	no	—
213	C₂²×C₃	#878	1	1	yes	characteristic
214	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
215	C₂²×C₃	#894	1	1	yes	characteristic
216	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
217	C₂²×C₃	#915	1	1	yes	characteristic
218	C₂²×C₃	#916	1	1	yes	characteristic
219	C₂²×C₃	#918	1	1	yes	characteristic
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
223	C₃×C₄	#879	1	1	yes	characteristic
224	C₃×C₄	#890	1	1	yes	characteristic
225	C₃×C₄	#892	1	1	yes	characteristic
226	Dic₃	—	6	2	no	—
227	Dic₃	—	12	1	no	—
228	Dic₃	—	12	2	no	—
229	Dic₃	—	24	1	no	—
230	C₉	#1918	1	1	yes	characteristic, complemented, Sylow
231	C₂³	D₄×D₉	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	9	2	no	—
239	C₂×C₄	—	18	1	no	—
240	C₂×C₄	—	18	1	no	—
241	C₂×C₄	—	18	2	no	—
242	C₂×C₄	—	18	1	no	—
243	C₂×C₄	—	18	1	no	—
244	C₂×C₄	—	18	1	no	—
245	C₂×C₄	—	36	1	no	—
246	C₂×C₄	—	36	1	no	—
247	C₂×C₄	—	36	1	no	—
248	C₂×C₄	—	36	1	no	—
249	C₂×C₄	—	36	1	no	—
250	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
252	C₂×C₄	#3697	1	1	yes	characteristic
253	C₂×C₄	#3699	1	1	yes	characteristic
254	C₂×C₄	#3699	1	1	yes	characteristic
255	C₂×C₄	#3701	1	1	yes	characteristic
256	C₈	—	4	1	no	—
257	C₈	—	18	2	no	—
258	C₂×C₃	#4655	1	1	yes	characteristic
259	C₂×C₃	#4756	1	1	yes	characteristic
260	C₂×C₃	#4759	1	1	yes	characteristic
261	C₂×C₃	#4789	1	1	yes	characteristic
262	C₂×C₃	#4799	1	1	yes	characteristic
263	C₂×C₃	#4868	1	1	yes	characteristic
264	C₂×C₃	#4882	1	1	yes	characteristic
265	C₂²	#63440	1	1	yes	characteristic
266	C₂²	#611⋊C₄	1	1	yes	characteristic
267	C₂²	#63456	1	1	yes	characteristic
268	C₂²	D₈×D₉	1	1	yes	characteristic
269	C₂²	#63477	1	1	yes	characteristic
270	C₂²	#63478	1	1	yes	characteristic
271	C₂²	#63480	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	18	2	no	—
275	C₄	—	36	2	no	—
276	C₄	—	36	1	no	—
277	C₄	—	72	1	no	—
278	C₄	C₄×Dic₁₈	1	1	yes	characteristic
279	C₄	#63441	1	1	yes	characteristic
280	C₄	#63452	1	1	yes	characteristic
281	C₄	#63454	1	1	yes	characteristic
282	C₃	#68815	1	1	yes	characteristic
283	C₂	#10588335	1	1	yes	characteristic
284	C₂	#10588436	1	1	yes	characteristic
285	C₂	#10588439	1	1	yes	characteristic
286	C₂	#10588469	1	1	yes	characteristic
287	C₂	#10588479	1	1	yes	characteristic
288	C₂	#10588548	1	1	yes	characteristic
289	C₂	#10588562	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#340	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#340	D₉
hypocenter	C₉	#1918
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
20. #63366	C₄	semidirect
230. C₉	#1918	semidirect

Maximal subgroups

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

subgroup	index
#10588664	2
C₄⋊C₄×Dic₉	2
C₂×#63366	2
C₉×#340	2
#10588229	2
#10588228	2
#10588206	2
#68815	3