Subgroups — #11776362

Order 1152.

625 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64078	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#4924	C₂	1	1	yes	maximal, characteristic
4	#10589096	C₂	1	4	yes	maximal, complemented
5	#70779	—	3	1	no	maximal
6	#70507	—	3	1	no	maximal
7	C₂²×#634	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
10	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
11	#64078	—	2	1	no	—
12	#64078	—	2	1	no	—
13	#64078	C₄	1	2	yes	—
14	#64078	C₂²	1	2	yes	—
15	C₂×C₃×#208	—	2	1	no	—
16	C₃×#830	—	2	1	no	—
17	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₃×C₄×C₃⋊C₈	—	2	1	no	—
21	C₂×#869	—	3	1	no	—
22	C₂×#869	—	3	1	no	—
23	C₂×#869	—	6	1	no	—
24	C₃×#431	—	3	1	no	—
25	C₃×#423	—	3	1	no	—
26	#4925	D₃	1	1	yes	characteristic
27	#4924	D₃	1	1	yes	characteristic
28	#4481	—	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×#634	—	2	2	no	—
35	C₂×#634	—	4	4	no	—
36	C₂×#634	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	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
42	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
43	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
44	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
45	C₃×#208	—	4	2	no	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	—	2	2	no	—
48	#1548	—	9	1	no	Sylow
49	C₂²×Dic₆	—	3	2	no	—
50	C₂²×Dic₆	—	3	2	no	—
51	C₂²×Dic₆	—	6	2	no	—
52	C₂×C₃×M₁₆	—	6	1	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄⋊Dic₃	—	6	2	no	—
57	C₂×C₄⋊Dic₃	—	6	2	no	—
58	C₂×C₄⋊Dic₃	—	6	1	no	—
59	C₂×C₄⋊Dic₃	—	6	1	no	—
60	C₂×C₄⋊Dic₃	—	6	1	no	—
61	C₂×C₄⋊Dic₃	—	6	1	no	—
62	C₂×#208	—	2	1	no	—
63	#869	—	3	2	no	—
64	#869	—	3	2	no	—
65	#869	—	3	2	no	—
66	#869	—	3	2	no	—
67	#869	—	6	2	no	—
68	#869	—	6	2	no	—
69	#869	—	6	2	no	—
70	#869	—	6	2	no	—
71	#869	—	6	1	no	—
72	#869	—	6	1	no	—
73	#869	—	6	1	no	—
74	#869	—	6	1	no	—
75	C₃×C₄⋊C₈	—	3	2	no	—
76	C₃×C₄⋊C₈	—	3	2	no	—
77	C₃×C₄⋊C₈	—	6	1	no	—
78	C₃×C₂²⋊C₈	—	6	1	no	—
79	C₃×C₄×C₈	—	6	1	no	—
80	#830	—	2	1	no	—
81	C₄⋊(C₃⋊C₈)	—	2	1	no	—
82	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
83	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
84	C₄×C₃⋊C₈	—	2	1	no	—
85	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
92	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
93	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃⋊Dic₃	—	2	4	no	—
97	C₂×C₃⋊Dic₃	—	4	4	no	—
98	#634	—	4	4	no	—
99	#634	—	4	4	no	—
100	C₃×C₃⋊C₈	—	4	2	no	—
101	C₃×C₃⋊C₈	—	4	2	no	—
102	C₂×C₄⋊Q₈	—	9	1	no	—
103	#431	—	3	1	no	—
104	#423	—	3	1	no	—
105	#333	—	9	4	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
112	C₂²×C₃×C₄	D₁₂	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	—	6	4	no	—
116	C₂²×Dic₃	—	6	2	no	—
117	C₂²×Dic₃	—	6	2	no	—
118	C₂×Dic₆	—	3	4	no	—
119	C₂×Dic₆	—	3	4	no	—
120	C₂×Dic₆	—	6	4	no	—
121	C₂×Dic₆	—	6	4	no	—
122	C₂×Dic₆	—	6	2	no	—
123	C₂×Dic₆	—	6	2	no	—
124	C₂×Dic₆	—	12	8	no	—
125	C₂×Dic₆	—	12	4	no	—
126	C₂×Dic₆	—	12	4	no	—
127	C₃×M₁₆	—	12	2	no	—
128	C₂×C₃×C₈	—	6	2	no	—
129	C₂×C₃×C₈	—	6	2	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₃×C₄²	—	2	1	no	—
134	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
135	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
136	C₃×C₄²	D₁₂	1	1	yes	characteristic
137	C₃×C₄²	D₁₂	1	1	yes	characteristic
138	C₃×C₄²	#67	1	1	yes	characteristic
139	C₃×C₄²	#67	1	1	yes	characteristic
140	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
141	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
142	C₄⋊Dic₃	—	6	4	no	—
143	C₄⋊Dic₃	—	6	4	no	—
144	C₄⋊Dic₃	—	6	4	no	—
145	C₄⋊Dic₃	—	6	4	no	—
146	C₄⋊Dic₃	—	6	2	no	—
147	C₄⋊Dic₃	—	6	2	no	—
148	C₄⋊Dic₃	—	6	2	no	—
149	C₄⋊Dic₃	—	6	2	no	—
150	C₄⋊Dic₃	—	6	2	no	—
151	C₄⋊Dic₃	—	6	2	no	—
152	C₄⋊Dic₃	—	6	2	no	—
153	C₄⋊Dic₃	—	6	2	no	—
154	#208	—	4	2	no	—
155	C₂×C₃⋊C₈	—	2	2	no	—
156	C₂×C₃⋊C₈	—	2	2	no	—
157	C₂²×C₃²	—	2	1	no	—
158	C₂²×C₃²	—	2	1	no	—
159	C₂²×C₃²	#103	1	2	yes	—
160	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₃²×C₄	—	2	1	no	—
162	C₃²×C₄	—	2	1	no	—
163	C₃²×C₄	#100	1	2	yes	—
164	C₃²×C₄	#103	1	2	yes	—
165	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
166	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
167	C₃²×C₄	#137	1	2	yes	—
168	C₃⋊Dic₃	—	4	4	no	—
169	C₂²×Q₈	—	9	2	no	—
170	C₂×M₁₆	—	6	1	no	—
171	C₄⋊Q₈	—	9	2	no	—
172	C₄⋊Q₈	—	9	2	no	—
173	C₄⋊Q₈	—	18	1	no	—
174	C₄⋊Q₈	—	18	1	no	—
175	C₂×C₄⋊C₄	—	18	1	no	—
176	C₂×C₄⋊C₄	—	18	1	no	—
177	C₂×C₄²	D₃²	1	1	yes	characteristic
178	C₄⋊C₈	—	3	2	no	—
179	C₄⋊C₈	—	3	2	no	—
180	C₄⋊C₈	—	6	1	no	—
181	C₂²⋊C₈	—	6	1	no	—
182	C₄×C₈	—	6	1	no	—
183	C₂³×C₃	—	2	1	no	—
184	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	#204	1	2	yes	—
201	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
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	(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
206	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	#214	1	2	yes	—
209	C₂×C₃×C₄	#216	1	2	yes	—
210	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
216	C₂×Dic₃	—	6	8	no	—
217	C₂×Dic₃	—	6	4	no	—
218	C₂×Dic₃	—	6	4	no	—
219	C₂×Dic₃	—	12	8	no	—
220	C₂×Dic₃	—	12	4	no	—
221	C₂×Dic₃	—	12	4	no	—
222	Dic₆	—	12	8	no	—
223	Dic₆	—	12	8	no	—
224	Dic₆	—	12	4	no	—
225	Dic₆	—	12	4	no	—
226	Dic₆	—	12	4	no	—
227	Dic₆	—	12	4	no	—
228	C₃×C₈	—	12	2	no	—
229	C₃×C₈	—	12	2	no	—
230	C₃⋊C₈	—	4	2	no	—
231	C₃⋊C₈	—	4	2	no	—
232	C₂×C₃²	—	2	2	no	—
233	C₂×C₃²	C₂×#100	1	1	yes	characteristic
234	C₂×C₃²	C₂×#103	1	1	yes	characteristic
235	C₂×C₃²	#419	1	1	yes	characteristic
236	C₂×Q₈	—	9	4	no	—
237	C₂×Q₈	—	18	2	no	—
238	C₂×Q₈	—	36	4	no	—
239	C₂²×C₄	—	18	2	no	—
240	C₂²×C₄	#624	1	1	yes	characteristic
241	C₂²×C₄	#626	1	1	yes	characteristic
242	C₂²×C₄	#626	1	1	yes	characteristic
243	M₁₆	—	12	2	no	—
244	C₂×C₈	—	6	2	no	—
245	C₂×C₈	—	6	2	no	—
246	C₄⋊C₄	—	18	2	no	—
247	C₄⋊C₄	—	18	2	no	—
248	C₄⋊C₄	—	18	2	no	—
249	C₄⋊C₄	—	18	2	no	—
250	C₄²	#624	1	1	yes	characteristic
251	C₄²	#626	1	1	yes	characteristic
252	C₄²	#626	1	1	yes	characteristic
253	C₄²	C₂×D₃²	1	1	yes	characteristic
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	#810	1	2	yes	—
263	C₂²×C₃	#816	1	2	yes	—
264	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
265	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	#810	1	2	yes	—
278	C₃×C₄	#816	1	2	yes	—
279	C₃×C₄	#823	1	2	yes	—
280	C₃×C₄	#823	1	2	yes	—
281	C₃×C₄	C₂×#204	1	1	yes	characteristic
282	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
283	C₃×C₄	#909	1	2	yes	—
284	C₃×C₄	C₂×#214	1	1	yes	characteristic
285	C₃×C₄	C₂×#216	1	1	yes	characteristic
286	C₃×C₄	#951	1	2	yes	—
287	Dic₃	—	12	8	no	—
288	Dic₃	—	12	4	no	—
289	Dic₃	—	12	4	no	—
290	C₃²	#1548	1	1	yes	characteristic, complemented, Sylow
291	C₂³	#3722	1	1	yes	characteristic
292	Q₈	—	36	4	no	—
293	Q₈	—	36	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	18	4	no	—
297	C₂×C₄	—	36	4	no	—
298	C₂×C₄	#3716	1	2	yes	—
299	C₂×C₄	#3719	1	2	yes	—
300	C₂×C₄	#3722	1	1	yes	characteristic
301	C₂×C₄	#3722	1	1	yes	characteristic
302	C₂×C₄	#3722	1	1	yes	characteristic
303	C₂×C₄	C₂×#624	1	1	yes	characteristic
304	C₂×C₄	C₂×#626	1	1	yes	characteristic
305	C₂×C₄	C₂×#626	1	1	yes	characteristic
306	C₈	—	12	2	no	—
307	C₈	—	12	2	no	—
308	C₂×C₃	—	2	4	no	—
309	C₂×C₃	—	2	2	no	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	C₂×#810	1	1	yes	characteristic
315	C₂×C₃	#5004	1	1	yes	characteristic
316	C₂×C₃	C₂×#816	1	1	yes	characteristic
317	C₂×C₃	#5120	1	1	yes	characteristic
318	C₂×C₃	C₂×#823	1	1	yes	characteristic
319	C₂×C₃	C₂×#823	1	1	yes	characteristic
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	#63567	1	2	yes	—
323	C₂²	C₂×#3722	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	36	4	no	—
327	C₄	#63559	1	2	yes	—
328	C₄	#63567	1	2	yes	—
329	C₄	C₂×#3716	1	1	yes	characteristic
330	C₄	#63831	1	2	yes	—
331	C₄	C₂×#3719	1	1	yes	characteristic
332	C₃	—	2	1	no	—
333	C₃	#70507	1	1	yes	characteristic, complemented
334	C₃	#70779	1	1	yes	characteristic, complemented
335	C₂	—	2	2	no	—
336	C₂	C₂×#63559	1	1	yes	characteristic
337	C₂	C₂×#63567	1	1	yes	characteristic
338	C₂	#10591107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3722
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. #10589096	C₂	semidirect
290. C₃²	#1548	semidirect
333. C₃	#70507	semidirect
334. C₃	#70779	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64078	2
C₃×#4925	2
C₃×#4924	2
#10589096	2
#70779	3
#70507	3