Subgroups — #11776363

Order 1152.

543 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#10589095	C₂	1	4	yes	maximal, complemented
5	#71774	—	3	1	no	maximal
6	#70779	—	3	1	no	maximal
7	C₂²×C₃×Dic₆	C₄	1	2	yes	—
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	—	2	1	no	—
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	#4546	—	3	4	no	—
29	#4481	—	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	—	4	4	no	—
36	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
37	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₃²×C₄²	D₄	1	1	yes	characteristic
39	C₃²×C₄²	D₄	1	1	yes	characteristic
40	C₃²×C₄²	C₂³	1	1	yes	characteristic
41	#3748	—	4	2	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
43	C₃×C₄⋊Dic₃	—	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	C₂×C₃×C₃⋊C₈	—	2	2	no	—
48	#1548	—	9	1	no	Sylow
49	C₂²×C₃×Q₈	—	3	2	no	—
50	C₂²×Dic₆	Dic₃	1	2	yes	—
51	C₃×C₄⋊Q₈	—	3	2	no	—
52	C₃×C₄⋊Q₈	—	3	2	no	—
53	C₃×C₄⋊Q₈	—	6	1	no	—
54	C₃×C₄⋊Q₈	—	6	1	no	—
55	C₂×C₃×C₄⋊C₄	—	6	1	no	—
56	C₂×C₃×C₄⋊C₄	—	6	1	no	—
57	C₂×C₃×C₄²	—	2	1	no	—
58	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	—	2	1	no	—
61	C₂×C₄⋊Dic₃	—	2	1	no	—
62	C₂×#208	—	6	2	no	—
63	C₂×#208	—	6	1	no	—
64	C₂×#208	—	6	1	no	—
65	#869	—	2	1	no	—
66	#869	—	2	1	no	—
67	#869	Dic₃	1	2	yes	—
68	#869	C₂×D₃	1	2	yes	—
69	C₃×C₄⋊C₈	—	3	2	no	—
70	C₃×C₂²⋊C₈	—	6	1	no	—
71	C₃×C₄×C₈	—	6	1	no	—
72	#830	—	2	1	no	—
73	C₄⋊(C₃⋊C₈)	—	3	2	no	—
74	C₄⋊(C₃⋊C₈)	—	3	2	no	—
75	C₄⋊(C₃⋊C₈)	—	6	2	no	—
76	C₄⋊(C₃⋊C₈)	—	6	2	no	—
77	C₄⋊(C₃⋊C₈)	—	6	1	no	—
78	C₄⋊(C₃⋊C₈)	—	6	1	no	—
79	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
80	C₄×C₃⋊C₈	—	2	1	no	—
81	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
88	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
89	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×Dic₃	—	2	4	no	—
93	C₂×C₃×Dic₃	—	4	4	no	—
94	C₃×Dic₆	—	4	4	no	—
95	C₃×Dic₆	—	4	4	no	—
96	C₃⋊(C₃⋊C₈)	—	4	2	no	—
97	C₃×C₃⋊C₈	—	4	2	no	—
98	C₂×C₄⋊Q₈	—	3	1	no	—
99	#431	—	3	1	no	—
100	#423	—	9	1	no	—
101	#333	—	9	4	no	—
102	C₂×C₃×Q₈	—	3	4	no	—
103	C₂×C₃×Q₈	—	6	2	no	—
104	C₂×C₃×Q₈	—	12	4	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	6	2	no	—
109	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	#67	1	1	yes	characteristic
113	C₂²×C₃×C₄	#67	1	1	yes	characteristic
114	C₂²×C₃×C₄	#67	1	1	yes	characteristic
115	C₂²×Dic₃	—	2	2	no	—
116	C₂×Dic₆	—	2	2	no	—
117	C₂×Dic₆	—	4	4	no	—
118	C₂×Dic₆	C₂×Dic₃	1	4	yes	—
119	C₂×C₃×C₈	—	6	2	no	—
120	C₃×C₄⋊C₄	—	6	2	no	—
121	C₃×C₄⋊C₄	—	6	2	no	—
122	C₃×C₄⋊C₄	—	6	2	no	—
123	C₃×C₄⋊C₄	—	6	2	no	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	—	2	1	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
129	C₃×C₄²	D₁₂	1	1	yes	characteristic
130	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
131	C₃×C₄²	#67	1	1	yes	characteristic
132	C₃×C₄²	#67	1	1	yes	characteristic
133	C₃×C₄²	#67	1	1	yes	characteristic
134	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
135	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
136	C₄⋊Dic₃	—	2	2	no	—
137	C₄⋊Dic₃	—	2	2	no	—
138	C₄⋊Dic₃	—	2	2	no	—
139	C₄⋊Dic₃	—	2	2	no	—
140	#208	—	12	4	no	—
141	#208	—	12	2	no	—
142	#208	—	12	2	no	—
143	C₂×C₃⋊C₈	—	2	2	no	—
144	C₂×C₃⋊C₈	—	6	4	no	—
145	C₂×C₃⋊C₈	—	6	2	no	—
146	C₂×C₃⋊C₈	—	6	2	no	—
147	C₂²×C₃²	—	2	1	no	—
148	C₂²×C₃²	—	2	1	no	—
149	C₂²×C₃²	#103	1	2	yes	—
150	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₃²×C₄	—	2	1	no	—
152	C₃²×C₄	—	2	1	no	—
153	C₃²×C₄	#100	1	2	yes	—
154	C₃²×C₄	#103	1	2	yes	—
155	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
157	C₃²×C₄	#137	1	2	yes	—
158	C₃×Dic₃	—	4	4	no	—
159	C₂²×Q₈	—	3	2	no	—
160	C₂×M₁₆	—	18	1	no	—
161	C₄⋊Q₈	—	3	2	no	—
162	C₄⋊Q₈	—	3	2	no	—
163	C₄⋊Q₈	—	6	1	no	—
164	C₄⋊Q₈	—	6	1	no	—
165	C₂×C₄⋊C₄	—	6	1	no	—
166	C₂×C₄⋊C₄	—	6	1	no	—
167	C₂×C₄²	D₃²	1	1	yes	characteristic
168	C₄⋊C₈	—	3	2	no	—
169	C₄⋊C₈	—	9	2	no	—
170	C₄⋊C₈	—	18	1	no	—
171	C₂²⋊C₈	—	6	1	no	—
172	C₄×C₈	—	6	1	no	—
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₂³×C₃	#217	1	1	yes	characteristic
176	C₃×Q₈	—	12	4	no	—
177	C₃×Q₈	—	12	4	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	6	4	no	—
193	C₂×C₃×C₄	—	12	4	no	—
194	C₂×C₃×C₄	#204	1	2	yes	—
195	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
196	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
198	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
199	C₂×C₃×C₄	#215	1	2	yes	—
200	C₂×C₃×C₄	#216	1	2	yes	—
201	C₂×C₃×C₄	#217	1	1	yes	characteristic
202	C₂×C₃×C₄	#217	1	1	yes	characteristic
203	C₂×C₃×C₄	#217	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
210	C₂×Dic₃	—	2	4	no	—
211	C₂×Dic₃	—	4	4	no	—
212	Dic₆	—	4	4	no	—
213	Dic₆	—	4	4	no	—
214	C₃×C₈	—	12	2	no	—
215	C₃⋊C₈	—	4	2	no	—
216	C₃⋊C₈	—	12	4	no	—
217	C₃⋊C₈	—	12	2	no	—
218	C₃⋊C₈	—	12	2	no	—
219	C₂×C₃²	—	2	2	no	—
220	C₂×C₃²	C₂×#100	1	1	yes	characteristic
221	C₂×C₃²	C₂×#103	1	1	yes	characteristic
222	C₂×C₃²	#419	1	1	yes	characteristic
223	C₂×Q₈	—	3	4	no	—
224	C₂×Q₈	—	6	2	no	—
225	C₂×Q₈	—	12	4	no	—
226	C₂²×C₄	—	6	2	no	—
227	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
228	C₂²×C₄	#625	1	1	yes	characteristic
229	C₂²×C₄	#626	1	1	yes	characteristic
230	M₁₆	—	36	2	no	—
231	C₂×C₈	—	6	2	no	—
232	C₂×C₈	—	18	2	no	—
233	C₄⋊C₄	—	6	2	no	—
234	C₄⋊C₄	—	6	2	no	—
235	C₄⋊C₄	—	6	2	no	—
236	C₄⋊C₄	—	6	2	no	—
237	C₄²	D₃×Dic₃	1	1	yes	characteristic
238	C₄²	#625	1	1	yes	characteristic
239	C₄²	#626	1	1	yes	characteristic
240	C₄²	C₂×D₃²	1	1	yes	characteristic
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	4	1	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	#816	1	2	yes	—
250	C₂²×C₃	#835	1	2	yes	—
251	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	C₂²×C₃	C₂×#217	1	1	yes	characteristic
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	12	4	no	—
265	C₃×C₄	#816	1	2	yes	—
266	C₃×C₄	#823	1	2	yes	—
267	C₃×C₄	#833	1	2	yes	—
268	C₃×C₄	#835	1	2	yes	—
269	C₃×C₄	C₂×#204	1	1	yes	characteristic
270	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
271	C₃×C₄	C₂×#215	1	1	yes	characteristic
272	C₃×C₄	#942	1	2	yes	—
273	C₃×C₄	C₂×#216	1	1	yes	characteristic
274	C₃×C₄	#951	1	2	yes	—
275	Dic₃	—	4	4	no	—
276	C₃²	#1548	1	1	yes	characteristic, complemented, Sylow
277	C₂³	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
278	Q₈	—	12	4	no	—
279	Q₈	—	12	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	6	4	no	—
283	C₂×C₄	—	12	4	no	—
284	C₂×C₄	#3717	1	2	yes	—
285	C₂×C₄	#3719	1	2	yes	—
286	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
287	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
288	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
289	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
290	C₂×C₄	C₂×#625	1	1	yes	characteristic
291	C₂×C₄	C₂×#626	1	1	yes	characteristic
292	C₈	—	12	2	no	—
293	C₈	—	36	2	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	4	2	no	—
300	C₂×C₃	#5004	1	1	yes	characteristic
301	C₂×C₃	C₂×#816	1	1	yes	characteristic
302	C₂×C₃	C₂×#823	1	1	yes	characteristic
303	C₂×C₃	C₂×#833	1	1	yes	characteristic
304	C₂×C₃	C₂×#835	1	1	yes	characteristic
305	C₂×C₃	#5225	1	1	yes	characteristic
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#63566	1	2	yes	—
309	C₂²	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	12	4	no	—
313	C₄	#63558	1	2	yes	—
314	C₄	#63566	1	2	yes	—
315	C₄	#63827	1	2	yes	—
316	C₄	C₂×#3717	1	1	yes	characteristic
317	C₄	C₂×#3719	1	1	yes	characteristic
318	C₃	—	2	1	no	—
319	C₃	#70779	1	1	yes	characteristic, complemented
320	C₃	#71774	1	1	yes	characteristic, complemented
321	C₂	—	2	2	no	—
322	C₂	C₂×#63558	1	1	yes	characteristic
323	C₂	C₂×#63566	1	1	yes	characteristic
324	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	#1548
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. #10589095	C₂	semidirect
276. C₃²	#1548	semidirect
319. C₃	#70779	semidirect
320. C₃	#71774	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
#10589095	2
#71774	3
#70779	3