Subgroups — #11776300

Order 1152.

543 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	#10592623	C₂	1	1	yes	maximal, characteristic
3	C₃×#4925	C₂	1	1	yes	maximal, characteristic
4	#10589092	C₂	1	4	yes	maximal, complemented
5	#71773	—	3	1	no	maximal
6	#70778	—	3	1	no	maximal
7	C₂²×C₃×Dic₆	—	2	1	no	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×#3748	—	2	1	no	—
10	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₃	C₄	1	2	yes	—
12	C₃×#869	—	2	1	no	—
13	C₃×#869	—	2	1	no	—
14	C₃×#869	C₄	1	2	yes	—
15	C₃×#869	C₂²	1	2	yes	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
18	C₃×#830	—	2	1	no	—
19	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₃×C₄×C₃⋊C₈	—	2	1	no	—
21	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
22	C₂×#869	D₃	1	1	yes	characteristic
23	C₃×#431	—	3	1	no	—
24	#4925	D₃	1	1	yes	characteristic
25	#4924	—	3	1	no	—
26	#4924	—	3	1	no	—
27	#4924	—	6	1	no	—
28	#4547	—	3	4	no	—
29	#4486	—	3	4	no	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	—	2	2	no	—
34	C₂×C₃×Dic₆	—	2	2	no	—
35	C₂×C₃×Dic₆	—	2	2	no	—
36	C₂×C₃×Dic₆	—	4	2	no	—
37	C₂×C₃×Dic₆	—	4	2	no	—
38	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₃²×C₄²	D₄	1	1	yes	characteristic
40	C₃²×C₄²	D₄	1	1	yes	characteristic
41	C₃²×C₄²	C₂³	1	1	yes	characteristic
42	#3748	—	4	2	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×C₄⋊Dic₃	—	2	2	no	—
47	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	#1546	—	9	1	no	Sylow
50	C₂²×C₃×Q₈	—	6	1	no	—
51	C₂²×Dic₆	—	2	1	no	—
52	C₃×C₄⋊Q₈	—	3	2	no	—
53	C₃×C₄⋊Q₈	—	3	2	no	—
54	C₃×C₄⋊Q₈	—	6	1	no	—
55	C₃×C₄⋊Q₈	—	6	1	no	—
56	C₂×C₃×C₄⋊C₄	—	3	2	no	—
57	C₂×C₃×C₄⋊C₄	—	6	1	no	—
58	C₂×C₃×C₄²	—	2	1	no	—
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	—	2	1	no	—
62	C₂×C₄⋊Dic₃	Dic₃	1	2	yes	—
63	C₂×#208	—	6	2	no	—
64	C₂×#208	—	6	1	no	—
65	C₂×#208	—	6	1	no	—
66	#869	—	2	1	no	—
67	#869	—	2	1	no	—
68	#869	Dic₃	1	2	yes	—
69	#869	C₂×D₃	1	2	yes	—
70	C₃×C₄⋊C₈	—	3	2	no	—
71	C₃×C₂²⋊C₈	—	6	1	no	—
72	C₃×C₄×C₈	—	6	1	no	—
73	#830	—	2	1	no	—
74	C₄⋊(C₃⋊C₈)	—	3	2	no	—
75	C₄⋊(C₃⋊C₈)	—	3	2	no	—
76	C₄⋊(C₃⋊C₈)	—	6	2	no	—
77	C₄⋊(C₃⋊C₈)	—	6	2	no	—
78	C₄⋊(C₃⋊C₈)	—	6	1	no	—
79	C₄⋊(C₃⋊C₈)	—	6	1	no	—
80	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
81	C₄×C₃⋊C₈	—	2	1	no	—
82	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	D₈	1	2	yes	—
89	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
90	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×Dic₃	—	2	4	no	—
94	C₂×C₃×Dic₃	—	4	4	no	—
95	C₃×Dic₆	—	4	4	no	—
96	C₃×Dic₆	—	4	2	no	—
97	C₃×Dic₆	—	4	2	no	—
98	C₃⋊(C₃⋊C₈)	—	4	2	no	—
99	C₃×C₃⋊C₈	—	4	2	no	—
100	C₂×C₄⋊Q₈	—	3	1	no	—
101	#431	—	3	1	no	—
102	#423	—	9	1	no	—
103	#332	—	9	4	no	—
104	C₂×C₃×Q₈	—	6	2	no	—
105	C₂×C₃×Q₈	—	6	2	no	—
106	C₂×C₃×Q₈	—	12	2	no	—
107	C₂×C₃×Q₈	—	12	2	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	6	2	no	—
112	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
115	C₂²×C₃×C₄	#67	1	1	yes	characteristic
116	C₂²×C₃×C₄	#67	1	1	yes	characteristic
117	C₂²×C₃×C₄	#67	1	1	yes	characteristic
118	C₂²×Dic₃	—	2	2	no	—
119	C₂×Dic₆	—	2	2	no	—
120	C₂×Dic₆	—	2	2	no	—
121	C₂×Dic₆	—	4	2	no	—
122	C₂×Dic₆	—	4	2	no	—
123	C₂×C₃×C₈	—	6	2	no	—
124	C₃×C₄⋊C₄	—	3	4	no	—
125	C₃×C₄⋊C₄	—	6	2	no	—
126	C₃×C₄⋊C₄	—	6	2	no	—
127	C₃×C₄⋊C₄	—	6	2	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
133	C₃×C₄²	D₁₂	1	1	yes	characteristic
134	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
135	C₃×C₄²	#67	1	1	yes	characteristic
136	C₃×C₄²	#67	1	1	yes	characteristic
137	C₃×C₄²	#67	1	1	yes	characteristic
138	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
139	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
140	C₄⋊Dic₃	—	2	2	no	—
141	C₄⋊Dic₃	—	2	2	no	—
142	C₄⋊Dic₃	—	2	2	no	—
143	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
144	#208	—	12	4	no	—
145	#208	—	12	2	no	—
146	#208	—	12	2	no	—
147	C₂×C₃⋊C₈	—	2	2	no	—
148	C₂×C₃⋊C₈	—	6	4	no	—
149	C₂×C₃⋊C₈	—	6	2	no	—
150	C₂×C₃⋊C₈	—	6	2	no	—
151	C₂²×C₃²	—	2	1	no	—
152	C₂²×C₃²	—	2	1	no	—
153	C₂²×C₃²	#102	1	2	yes	—
154	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₃²×C₄	—	2	1	no	—
156	C₃²×C₄	—	2	1	no	—
157	C₃²×C₄	#101	1	2	yes	—
158	C₃²×C₄	#102	1	2	yes	—
159	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
160	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
161	C₃²×C₄	#137	1	2	yes	—
162	C₃×Dic₃	—	4	4	no	—
163	C₂²×Q₈	—	6	1	no	—
164	C₂×M₁₆	—	18	1	no	—
165	C₄⋊Q₈	—	3	2	no	—
166	C₄⋊Q₈	—	3	2	no	—
167	C₄⋊Q₈	—	6	1	no	—
168	C₄⋊Q₈	—	6	1	no	—
169	C₂×C₄⋊C₄	—	3	2	no	—
170	C₂×C₄⋊C₄	—	6	1	no	—
171	C₂×C₄²	D₃²	1	1	yes	characteristic
172	C₄⋊C₈	—	3	2	no	—
173	C₄⋊C₈	—	9	2	no	—
174	C₄⋊C₈	—	18	1	no	—
175	C₂²⋊C₈	—	6	1	no	—
176	C₄×C₈	—	6	1	no	—
177	C₂³×C₃	—	2	1	no	—
178	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂³×C₃	#217	1	1	yes	characteristic
180	C₃×Q₈	—	12	4	no	—
181	C₃×Q₈	—	12	2	no	—
182	C₃×Q₈	—	12	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
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	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	4	2	no	—
197	C₂×C₃×C₄	—	6	4	no	—
198	C₂×C₃×C₄	—	12	4	no	—
199	C₂×C₃×C₄	#204	1	2	yes	—
200	C₂×C₃×C₄	D₂₄	1	2	yes	—
201	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₃×C₄	#213	1	2	yes	—
205	C₂×C₃×C₄	#214	1	2	yes	—
206	C₂×C₃×C₄	#217	1	1	yes	characteristic
207	C₂×C₃×C₄	#217	1	1	yes	characteristic
208	C₂×C₃×C₄	#217	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
215	C₂×Dic₃	—	2	4	no	—
216	C₂×Dic₃	—	4	4	no	—
217	Dic₆	—	4	4	no	—
218	Dic₆	—	4	2	no	—
219	Dic₆	—	4	2	no	—
220	C₃×C₈	—	12	2	no	—
221	C₃⋊C₈	—	4	2	no	—
222	C₃⋊C₈	—	12	4	no	—
223	C₃⋊C₈	—	12	2	no	—
224	C₃⋊C₈	—	12	2	no	—
225	C₂×C₃²	—	2	2	no	—
226	C₂×C₃²	C₂×#101	1	1	yes	characteristic
227	C₂×C₃²	C₂×#102	1	1	yes	characteristic
228	C₂×C₃²	#419	1	1	yes	characteristic
229	C₂×Q₈	—	6	2	no	—
230	C₂×Q₈	—	6	2	no	—
231	C₂×Q₈	—	12	2	no	—
232	C₂×Q₈	—	12	2	no	—
233	C₂²×C₄	—	6	2	no	—
234	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
235	C₂²×C₄	#625	1	1	yes	characteristic
236	C₂²×C₄	#626	1	1	yes	characteristic
237	M₁₆	—	36	2	no	—
238	C₂×C₈	—	6	2	no	—
239	C₂×C₈	—	18	2	no	—
240	C₄⋊C₄	—	3	4	no	—
241	C₄⋊C₄	—	6	2	no	—
242	C₄⋊C₄	—	6	2	no	—
243	C₄⋊C₄	—	6	2	no	—
244	C₄²	D₃×Dic₃	1	1	yes	characteristic
245	C₄²	#625	1	1	yes	characteristic
246	C₄²	#626	1	1	yes	characteristic
247	C₄²	C₂×D₃²	1	1	yes	characteristic
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	#821	1	2	yes	—
257	C₂²×C₃	#832	1	2	yes	—
258	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
259	C₂²×C₃	C₂×#217	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	4	1	no	—
270	C₃×C₄	—	4	1	no	—
271	C₃×C₄	—	12	4	no	—
272	C₃×C₄	#821	1	2	yes	—
273	C₃×C₄	#824	1	2	yes	—
274	C₃×C₄	#832	1	2	yes	—
275	C₃×C₄	#836	1	2	yes	—
276	C₃×C₄	C₂×#204	1	1	yes	characteristic
277	C₃×C₄	C₂×D₂₄	1	1	yes	characteristic
278	C₃×C₄	C₂×#213	1	1	yes	characteristic
279	C₃×C₄	C₂×#214	1	1	yes	characteristic
280	C₃×C₄	#942	1	2	yes	—
281	C₃×C₄	#951	1	2	yes	—
282	Dic₃	—	4	4	no	—
283	C₃²	#1546	1	1	yes	characteristic, complemented, Sylow
284	C₂³	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
285	Q₈	—	12	4	no	—
286	Q₈	—	12	2	no	—
287	Q₈	—	12	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	6	4	no	—
291	C₂×C₄	—	12	4	no	—
292	C₂×C₄	#3714	1	2	yes	—
293	C₂×C₄	#3716	1	2	yes	—
294	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
295	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
296	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
298	C₂×C₄	C₂×#625	1	1	yes	characteristic
299	C₂×C₄	C₂×#626	1	1	yes	characteristic
300	C₈	—	12	2	no	—
301	C₈	—	36	2	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	4	2	no	—
308	C₂×C₃	#5004	1	1	yes	characteristic
309	C₂×C₃	C₂×#821	1	1	yes	characteristic
310	C₂×C₃	C₂×#824	1	1	yes	characteristic
311	C₂×C₃	C₂×#832	1	1	yes	characteristic
312	C₂×C₃	#5225	1	1	yes	characteristic
313	C₂×C₃	C₂×#836	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	#63563	1	2	yes	—
317	C₂²	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	12	4	no	—
321	C₄	#63560	1	2	yes	—
322	C₄	#63563	1	2	yes	—
323	C₄	C₂×#3714	1	1	yes	characteristic
324	C₄	#63827	1	2	yes	—
325	C₄	C₂×#3716	1	1	yes	characteristic
326	C₃	—	2	1	no	—
327	C₃	#70778	1	1	yes	characteristic, complemented
328	C₃	#71773	1	1	yes	characteristic, complemented
329	C₂	—	2	2	no	—
330	C₂	C₂×#63560	1	1	yes	characteristic
331	C₂	C₂×#63563	1	1	yes	characteristic
332	C₂	#10591099	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃)⋊Dic₃
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1546
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #10589092	C₂	semidirect
283. C₃²	#1546	semidirect
327. C₃	#70778	semidirect
328. C₃	#71773	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
#10592623	2
C₃×#4925	2
#10589092	2
#71773	3
#70778	3