Subgroups — #10948743

Order 768.

518 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2966	C₂	1	1	yes	maximal, characteristic
2	#70949	C₂	1	2	yes	maximal, complemented
3	#70857	C₂	1	2	yes	maximal, complemented
4	#70857	C₂	1	2	yes	maximal, complemented
5	D₄×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	#69883	C₂	1	2	yes	maximal, complemented
7	#69881	C₂	1	2	yes	maximal, complemented
8	#69744	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#4461	C₂	1	1	yes	maximal, characteristic, complemented
10	#33482	—	3	1	no	maximal, Sylow
11	C₃×#529	C₄	1	1	yes	characteristic
12	C₃×#446	C₄	1	1	yes	characteristic
13	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
14	C₃×#436	C₄	1	1	yes	characteristic
15	C₃×#435	C₄	1	2	yes	—
16	C₃×C₈×D₄	C₂²	1	2	yes	—
17	C₃×C₈×D₄	C₂²	1	1	yes	characteristic
18	C₃×#433	C₄	1	2	yes	—
19	C₃×#432	C₄	1	1	yes	characteristic
20	C₃×#405	C₂²	1	2	yes	—
21	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×#812	C₂²	1	2	yes	—
23	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
24	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
25	#4549	C₂²	1	1	yes	characteristic
26	#4549	C₂²	1	1	yes	characteristic
27	#4549	C₂²	1	2	yes	—
28	#4549	C₂²	1	2	yes	—
29	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
30	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
31	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
32	#4461	C₂²	1	2	yes	—
33	#4461	C₂²	1	2	yes	—
34	#4461	C₂²	1	1	yes	characteristic
35	#4461	C₂²	1	1	yes	characteristic
36	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
37	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
38	#2966	D₃	1	1	yes	characteristic
39	#2183	—	3	2	no	—
40	#2168	—	3	2	no	—
41	#2168	—	3	2	no	—
42	D₄×C₁₆	—	3	2	no	—
43	#2164	—	3	2	no	—
44	#2162	—	3	1	no	—
45	#2108	—	3	2	no	—
46	C₂×#346	—	3	1	no	—
47	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
48	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂³	1	2	yes	—
51	C₃×#126	C₈	1	2	yes	—
52	C₃×#125	C₈	1	1	yes	characteristic
53	C₃×#124	C₈	1	1	yes	characteristic
54	C₃×#123	C₈	1	2	yes	—
55	C₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
56	C₃×#121	C₈	1	1	yes	characteristic
57	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
58	C₃×C₄×D₄	C₂×C₄	1	2	yes	—
59	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
60	C₃×#117	C₂×C₄	1	2	yes	—
61	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
62	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
65	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
67	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
68	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
69	C₃×#97	C₂×C₄	1	2	yes	—
70	C₃×#97	C₂×C₄	1	2	yes	—
71	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂³	1	2	yes	—
75	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
76	#812	—	2	4	no	—
77	C₂×C₃⋊C₁₆	—	2	4	no	—
78	C₂×C₃⋊C₁₆	—	2	4	no	—
79	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
80	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
81	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
82	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
83	#529	Dic₃	1	1	yes	characteristic
84	C₂×M₃₂	—	3	2	no	—
85	C₂²×C₁₆	—	3	2	no	—
86	C₂²×C₁₆	—	3	2	no	—
87	#446	Dic₃	1	1	yes	characteristic
88	C₈×Q₈	C₂×D₃	1	1	yes	characteristic
89	#436	Dic₃	1	1	yes	characteristic
90	#435	Dic₃	1	2	yes	—
91	C₈×D₄	C₂×D₃	1	2	yes	—
92	C₈×D₄	C₂×D₃	1	1	yes	characteristic
93	#433	Dic₃	1	2	yes	—
94	#432	Dic₃	1	1	yes	characteristic
95	#405	C₂×D₃	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
97	C₄⋊C₁₆	—	3	2	no	—
98	C₄⋊C₁₆	—	3	2	no	—
99	C₄⋊C₁₆	—	3	2	no	—
100	C₂²⋊C₁₆	—	3	1	no	—
101	C₂²⋊C₁₆	—	3	1	no	—
102	C₂²⋊C₁₆	—	3	2	no	—
103	C₂²⋊C₁₆	—	3	2	no	—
104	#346	—	3	1	no	—
105	#346	—	3	1	no	—
106	#346	—	3	2	no	—
107	#346	—	3	2	no	—
108	C₄×C₁₆	—	3	2	no	—
109	C₄×C₁₆	—	3	2	no	—
110	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
111	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
112	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
117	C₃×M₁₆	—	2	4	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
128	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
129	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
130	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
131	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
132	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
133	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
134	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
135	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
136	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
137	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
138	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
139	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
140	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
141	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
142	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
143	C₃×C₄²	C₂²×C₄	1	2	yes	—
144	C₃⋊C₁₆	—	2	2	no	—
145	C₃⋊C₁₆	—	2	2	no	—
146	C₃⋊C₁₆	—	2	2	no	—
147	C₃⋊C₁₆	#41	1	4	yes	—
148	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
149	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₃	1	2	yes	—
152	#126	C₃⋊C₈	1	2	yes	—
153	#125	C₃⋊C₈	1	1	yes	characteristic
154	#124	C₃⋊C₈	1	1	yes	characteristic
155	#123	C₃⋊C₈	1	2	yes	—
156	C₂²⋊Q₈	C₃⋊C₈	1	1	yes	characteristic
157	#121	C₃⋊C₈	1	1	yes	characteristic
158	C₄×Q₈	C₂×Dic₃	1	1	yes	characteristic
159	C₄×D₄	C₂×Dic₃	1	2	yes	—
160	C₄×D₄	C₂×Dic₃	1	1	yes	characteristic
161	#117	C₂×Dic₃	1	2	yes	—
162	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
163	M₃₂	—	6	4	no	—
164	C₂×C₁₆	—	3	2	no	—
165	C₂×C₁₆	—	3	2	no	—
166	C₂×C₁₆	—	3	2	no	—
167	C₂×C₁₆	—	3	2	no	—
168	C₂×C₁₆	—	6	4	no	—
169	C₂×C₁₆	—	6	4	no	—
170	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
171	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
172	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
173	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
174	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
175	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
176	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
177	#97	C₂×Dic₃	1	2	yes	—
178	#97	C₂×Dic₃	1	2	yes	—
179	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
181	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₃	1	2	yes	—
183	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
184	C₂³×C₃	C₂²×C₈	1	2	yes	—
185	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
186	C₃×Q₈	—	4	1	no	—
187	C₃×D₄	—	2	4	no	—
188	C₃×D₄	—	4	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	4	1	no	—
195	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
201	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
202	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
203	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	—	2	2	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	2	2	no	—
208	C₃×C₈	#131	1	2	yes	—
209	C₃×C₈	C₂×#41	1	2	yes	—
210	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
211	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
212	C₂×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×Dic₃	1	2	yes	—
225	C₂×C₈	C₂²×Dic₃	1	2	yes	—
226	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
227	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
228	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
229	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
230	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
231	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
232	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
233	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
234	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
235	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
236	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
237	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
238	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
239	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
240	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
241	C₄²	C₂²×Dic₃	1	1	yes	characteristic
242	C₄²	C₂²×Dic₃	1	1	yes	characteristic
243	C₄²	C₂²×Dic₃	1	2	yes	—
244	C₁₆	—	3	4	no	—
245	C₁₆	—	6	2	no	—
246	C₁₆	—	6	2	no	—
247	C₁₆	—	6	2	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	#504	1	2	yes	—
254	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	C₄×#41	1	1	yes	characteristic
264	C₃×C₄	C₂×#131	1	1	yes	characteristic
265	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
266	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
267	Q₈	—	4	1	no	—
268	D₄	—	2	4	no	—
269	D₄	—	4	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
282	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
283	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
284	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	2	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	2	no	—
289	C₈	#948	1	2	yes	—
290	C₈	D₃×#41	1	2	yes	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	4	2	no	—
293	C₂×C₃	C₈×#41	1	1	yes	characteristic
294	C₂×C₃	C₂×#504	1	1	yes	characteristic
295	C₂×C₃	#3407	1	1	yes	characteristic
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#5140	1	2	yes	—
302	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	C₂×#948	1	1	yes	characteristic
312	C₄	Dic₃×#41	1	1	yes	characteristic
313	C₃	#33482	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	#41×C₃⋊C₈	1	1	yes	characteristic
317	C₂	C₂×#5140	1	1	yes	characteristic
318	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2966	C₂
prime radical	C₃×#121	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2966	D₃
hypocenter	C₃	#33482
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
2. #70949	C₂	semidirect
3. #70857	C₂	semidirect
4. #70857	C₂	semidirect
5. D₄×C₃⋊C₁₆	C₂	semidirect
6. #69883	C₂	semidirect
7. #69881	C₂	semidirect
8. #69744	C₂	semidirect
9. C₂×#4461	C₂	semidirect
313. C₃	#33482	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2966	2
#70949	2
#70857	2
D₄×C₃⋊C₁₆	2
#69883	2
#69881	2
#69744	2
C₂×#4461	2
#33482	3