Subgroups — #11786037

Order 1152.

589 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592634	C₂	1	1	yes	maximal, characteristic
2	C₃×#4935	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589164	C₂	1	4	yes	maximal
5	#69762	—	3	1	no	maximal
6	#69574	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
12	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
16	C₂×C₃×C₄×C₈	—	3	1	no	—
17	C₃×#384	—	3	1	no	—
18	#4935	—	3	1	no	—
19	#4935	—	3	1	no	—
20	#4935	—	6	1	no	—
21	#4935	D₃	1	1	yes	characteristic
22	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
23	#4551	—	3	4	no	—
24	#4529	—	3	4	no	—
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃⋊Dic₃	—	2	2	no	—
29	C₂²×C₃×Dic₃	—	2	2	no	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₃²×C₄²	Q₈	1	2	yes	—
32	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
33	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
34	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
35	C₃×C₄⋊Dic₃	—	2	4	no	—
36	C₃×C₄⋊Dic₃	—	4	2	no	—
37	C₃×C₄⋊Dic₃	—	4	2	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
41	#1834	—	9	1	no	Sylow
42	C₂²×C₃×C₈	—	3	2	no	—
43	C₂×C₃×C₄⋊C₄	—	3	2	no	—
44	C₂×C₃×C₄⋊C₄	—	6	1	no	—
45	C₂×C₃×C₄⋊C₄	—	6	1	no	—
46	C₂×C₃×C₄²	—	2	1	no	—
47	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂×C₄⋊Dic₃	—	2	1	no	—
50	C₂×C₄⋊Dic₃	—	2	1	no	—
51	C₂×C₄⋊Dic₃	—	3	2	no	—
52	C₂×C₄⋊Dic₃	—	3	2	no	—
53	C₂×C₄⋊Dic₃	—	6	2	no	—
54	C₂×C₄⋊Dic₃	—	6	2	no	—
55	C₂×C₄⋊Dic₃	—	6	2	no	—
56	C₂×C₄⋊Dic₃	—	6	1	no	—
57	C₂×C₄⋊Dic₃	—	6	1	no	—
58	C₂×C₄⋊Dic₃	—	6	1	no	—
59	C₂×C₄⋊Dic₃	—	6	1	no	—
60	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
61	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
62	C₃×C₄×C₈	—	3	4	no	—
63	C₄×C₃⋊C₈	Dic₃	1	4	yes	complemented
64	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
66	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
67	C₂×C₃²×C₄	D₈	1	2	yes	—
68	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
69	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
70	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
71	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₃²×C₄	#41	1	2	yes	—
74	C₂×C₃²×C₄	#41	1	2	yes	—
75	C₂×C₃⋊Dic₃	—	4	2	no	—
76	C₂×C₃⋊Dic₃	—	4	2	no	—
77	C₂×C₃⋊Dic₃	—	4	2	no	—
78	C₂×C₃×Dic₃	—	4	2	no	—
79	C₂×C₃×Dic₃	—	4	2	no	—
80	C₂×C₃×Dic₃	—	4	2	no	—
81	C₃×C₃⋊C₈	—	2	4	no	—
82	C₂×C₄×C₈	—	3	1	no	—
83	#384	—	3	1	no	—
84	#384	—	9	1	no	—
85	#340	—	9	4	no	—
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	6	2	no	—
90	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
91	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂²×C₃×C₄	#67	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂²×Dic₃	—	2	2	no	—
97	C₂²×Dic₃	—	6	4	no	—
98	C₂²×Dic₃	—	6	2	no	—
99	C₂²×Dic₃	—	6	2	no	—
100	C₂×C₃×C₈	—	3	4	no	—
101	C₂×C₃×C₈	—	6	2	no	—
102	C₂×C₃×C₈	—	6	2	no	—
103	C₃×C₄⋊C₄	—	6	4	no	—
104	C₃×C₄⋊C₄	—	12	2	no	—
105	C₃×C₄⋊C₄	—	12	2	no	—
106	C₃×C₄²	—	2	2	no	—
107	C₃×C₄²	—	2	2	no	—
108	C₃×C₄²	Dic₆	1	2	yes	—
109	C₃×C₄²	Dic₆	1	2	yes	—
110	C₃×C₄²	C₄×D₃	1	2	yes	—
111	C₃×C₄²	C₂×Dic₃	1	2	yes	—
112	C₄⋊Dic₃	—	2	4	no	—
113	C₄⋊Dic₃	—	4	2	no	—
114	C₄⋊Dic₃	—	4	2	no	—
115	C₄⋊Dic₃	—	6	4	no	—
116	C₄⋊Dic₃	—	6	4	no	—
117	C₄⋊Dic₃	—	12	4	no	—
118	C₄⋊Dic₃	—	12	4	no	—
119	C₄⋊Dic₃	—	12	4	no	—
120	C₄⋊Dic₃	—	12	2	no	—
121	C₄⋊Dic₃	—	12	2	no	—
122	C₄⋊Dic₃	—	12	2	no	—
123	C₄⋊Dic₃	—	12	2	no	—
124	C₂×C₃⋊C₈	—	2	2	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
127	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₃²	#124	1	1	yes	characteristic
129	C₂²×C₃²	#124	1	1	yes	characteristic
130	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
131	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
132	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
133	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
134	C₃²×C₄	#106	1	4	yes	—
135	C₃²×C₄	#107	1	4	yes	—
136	C₃²×C₄	#117	1	2	yes	—
137	C₃²×C₄	#125	1	2	yes	—
138	C₃⋊Dic₃	—	8	2	no	—
139	C₃×Dic₃	—	8	2	no	—
140	C₂²×C₈	—	3	2	no	—
141	C₂×C₄⋊C₄	—	3	2	no	—
142	C₂×C₄⋊C₄	—	6	1	no	—
143	C₂×C₄⋊C₄	—	6	1	no	—
144	C₂×C₄⋊C₄	—	9	2	no	—
145	C₂×C₄⋊C₄	—	18	1	no	—
146	C₂×C₄⋊C₄	—	18	1	no	—
147	C₂×C₄²	D₃²	1	1	yes	characteristic
148	C₄×C₈	—	3	4	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
151	C₂³×C₃	C₂×#67	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	12	2	no	—
163	C₂×C₃×C₄	—	12	2	no	—
164	C₂×C₃×C₄	—	12	2	no	—
165	C₂×C₃×C₄	#204	1	2	yes	—
166	C₂×C₃×C₄	#204	1	2	yes	—
167	C₂×C₃×C₄	D₂₄	1	2	yes	—
168	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
169	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
170	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
172	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
173	C₂×C₃×C₄	#213	1	2	yes	—
174	C₂×C₃×C₄	#214	1	2	yes	—
175	C₂×C₃×C₄	#215	1	2	yes	—
176	C₂×C₃×C₄	#216	1	2	yes	—
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	#235	1	2	yes	—
181	C₂×C₃×C₄	#235	1	2	yes	—
182	C₂×C₃×C₄	#237	1	2	yes	—
183	C₂×C₃×C₄	#239	1	2	yes	—
184	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
185	C₂×Dic₃	—	4	2	no	—
186	C₂×Dic₃	—	4	2	no	—
187	C₂×Dic₃	—	4	2	no	—
188	C₂×Dic₃	—	12	4	no	—
189	C₂×Dic₃	—	12	4	no	—
190	C₂×Dic₃	—	12	4	no	—
191	C₂×Dic₃	—	12	2	no	—
192	C₂×Dic₃	—	12	2	no	—
193	C₂×Dic₃	—	12	2	no	—
194	C₂×Dic₃	—	12	2	no	—
195	C₂×Dic₃	—	12	2	no	—
196	C₂×Dic₃	—	12	2	no	—
197	C₃×C₈	—	6	4	no	—
198	C₃⋊C₈	—	2	4	no	—
199	C₂×C₃²	#382	1	1	yes	characteristic
200	C₂×C₃²	C₂×#106	1	1	yes	characteristic
201	C₂×C₃²	C₂×#107	1	1	yes	characteristic
202	C₂×C₃²	#486	1	1	yes	characteristic
203	C₂×C₃²	#486	1	1	yes	characteristic
204	C₂×C₃²	#487	1	1	yes	characteristic
205	C₂×C₃²	#487	1	1	yes	characteristic
206	C₂²×C₄	—	6	2	no	—
207	C₂²×C₄	—	18	2	no	—
208	C₂²×C₄	#626	1	1	yes	characteristic
209	C₂²×C₄	#626	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
211	C₂×C₈	—	3	4	no	—
212	C₂×C₈	—	6	2	no	—
213	C₂×C₈	—	6	2	no	—
214	C₄⋊C₄	—	6	4	no	—
215	C₄⋊C₄	—	12	2	no	—
216	C₄⋊C₄	—	12	2	no	—
217	C₄⋊C₄	—	18	4	no	—
218	C₄⋊C₄	—	36	2	no	—
219	C₄⋊C₄	—	36	2	no	—
220	C₄²	D₃×Dic₃	1	2	yes	—
221	C₄²	#627	1	2	yes	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	#875	1	1	yes	characteristic
230	C₂²×C₃	#875	1	1	yes	characteristic
231	C₂²×C₃	C₂×#204	1	1	yes	characteristic
232	C₂²×C₃	C₂×#204	1	1	yes	characteristic
233	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
234	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
235	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
236	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
237	C₂²×C₃	C₂×#213	1	1	yes	characteristic
238	C₂²×C₃	C₂×#214	1	1	yes	characteristic
239	C₂²×C₃	#936	1	1	yes	characteristic
240	C₂²×C₃	C₂×#215	1	1	yes	characteristic
241	C₂²×C₃	C₂×#216	1	1	yes	characteristic
242	C₂²×C₃	#947	1	1	yes	characteristic
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	24	2	no	—
248	C₃×C₄	#807	1	4	yes	—
249	C₃×C₄	#808	1	4	yes	—
250	C₃×C₄	#817	1	4	yes	—
251	C₃×C₄	#818	1	4	yes	—
252	C₃×C₄	#870	1	2	yes	—
253	C₃×C₄	#890	1	2	yes	—
254	C₃×C₄	#892	1	2	yes	—
255	C₃×C₄	#926	1	2	yes	—
256	Dic₃	—	8	2	no	—
257	Dic₃	—	24	4	no	—
258	Dic₃	—	24	2	no	—
259	Dic₃	—	24	2	no	—
260	C₃²	#1834	1	1	yes	characteristic, complemented, Sylow
261	C₂³	C₂×#626	1	1	yes	characteristic
262	C₂×C₄	—	12	2	no	—
263	C₂×C₄	—	12	2	no	—
264	C₂×C₄	—	12	2	no	—
265	C₂×C₄	—	36	2	no	—
266	C₂×C₄	—	36	2	no	—
267	C₂×C₄	—	36	2	no	—
268	C₂×C₄	#3714	1	2	yes	—
269	C₂×C₄	#3716	1	2	yes	—
270	C₂×C₄	#3717	1	2	yes	—
271	C₂×C₄	#3719	1	2	yes	—
272	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
273	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
274	C₂×C₄	#3795	1	2	yes	—
275	C₂×C₄	#3799	1	2	yes	—
276	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#627	1	1	yes	characteristic
278	C₈	—	6	4	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	#4681	1	1	yes	characteristic
287	C₂×C₃	#4681	1	1	yes	characteristic
288	C₂×C₃	#4694	1	1	yes	characteristic
289	C₂×C₃	#4694	1	1	yes	characteristic
290	C₂×C₃	#4936	1	1	yes	characteristic
291	C₂×C₃	C₂×#807	1	1	yes	characteristic
292	C₂×C₃	C₂×#808	1	1	yes	characteristic
293	C₂×C₃	#4977	1	1	yes	characteristic
294	C₂×C₃	#5070	1	1	yes	characteristic
295	C₂×C₃	#5079	1	1	yes	characteristic
296	C₂×C₃	#5088	1	1	yes	characteristic
297	C₂×C₃	#5093	1	1	yes	characteristic
298	C₂×C₃	C₂×#817	1	1	yes	characteristic
299	C₂×C₃	C₂×#818	1	1	yes	characteristic
300	C₂²	C₂×#3714	1	1	yes	characteristic
301	C₂²	C₂×#3716	1	1	yes	characteristic
302	C₂²	C₂×#3717	1	1	yes	characteristic
303	C₂²	C₂×#3719	1	1	yes	characteristic
304	C₂²	#63860	1	1	yes	characteristic
305	C₂²	#63864	1	1	yes	characteristic
306	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
307	C₄	—	24	2	no	—
308	C₄	—	72	2	no	—
309	C₄	#63573	1	4	yes	—
310	C₄	#63575	1	4	yes	—
311	C₄	#63841	1	2	yes	—
312	C₄	#63869	1	2	yes	—
313	C₃	—	2	1	no	—
314	C₃	#69574	1	1	yes	characteristic, complemented
315	C₃	#69762	1	1	yes	characteristic, complemented
316	C₂	#10590273	1	1	yes	characteristic
317	C₂	#10590281	1	1	yes	characteristic
318	C₂	#10590287	1	1	yes	characteristic
319	C₂	#10590288	1	1	yes	characteristic
320	C₂	C₂×#63573	1	1	yes	characteristic
321	C₂	C₂×#63575	1	1	yes	characteristic
322	C₂	#10591224	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1834
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
15. C₃×C₄×C₃⋊C₈	C₄	semidirect
63. C₄×C₃⋊C₈	Dic₃	semidirect
260. C₃²	#1834	semidirect
314. C₃	#69574	semidirect
315. C₃	#69762	semidirect

Maximal subgroups

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

subgroup	index
#10592634	2
C₃×#4935	2
C₂×C₃×C₄×C₃⋊C₈	2
#10589164	2
#69762	3
#69574	3