Subgroups — #10701028

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#82062	C₂	1	1	yes	maximal, characteristic
3	C₂×#4924	C₂	1	1	yes	maximal, characteristic
4	C₂×#4556	C₂	1	4	yes	maximal
5	#10848	—	3	1	no	maximal, Sylow
6	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#926	C₂²	1	2	yes	—
9	C₂²×#208	C₂²	1	2	yes	—
10	C₃×C₄×M₁₆	—	2	2	no	—
11	C₃×C₄×M₁₆	—	2	2	no	—
12	C₂×C₃×#97	C₄	1	2	yes	complemented
13	C₂×C₃×C₄×C₈	C₄	1	2	yes	complemented
14	#5212	C₄	1	4	yes	—
15	#4936	—	2	1	no	—
16	#4935	—	2	1	no	—
17	C₄×C₄⋊Dic₃	—	2	2	no	—
18	#4924	—	2	1	no	—
19	#4924	—	2	1	no	—
20	#4924	C₄	1	4	yes	complemented
21	C₂×C₄⋊(C₃⋊C₈)	—	2	2	no	—
22	#4556	—	2	8	no	—
23	C₂×#423	—	3	1	no	—
24	C₂×C₄×M₁₆	D₃	1	1	yes	characteristic
25	#2285	—	3	1	no	—
26	C₂×#344	—	3	4	no	—
27	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃×M₁₆	—	2	2	no	—
30	C₂×C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×M₁₆	—	2	2	no	—
32	C₂×C₃×M₁₆	—	2	2	no	—
33	C₂×C₃×M₁₆	D₄	1	4	yes	—
34	C₂×C₃×M₁₆	Q₈	1	4	yes	—
35	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	D₄	1	2	yes	—
42	C₂×C₃×C₄²	D₄	1	2	yes	—
43	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
44	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
45	C₂×#217	C₂×C₄	1	4	yes	—
46	#926	—	2	4	no	—
47	#926	—	2	4	no	—
48	C₂×C₄⋊Dic₃	—	2	4	no	—
49	C₂×C₄⋊Dic₃	—	2	2	no	—
50	C₂×C₄×Dic₃	—	2	2	no	—
51	C₂×#208	—	2	2	no	—
52	C₂×#208	—	2	4	no	—
53	C₂×#208	—	2	4	no	—
54	C₂×#208	C₂×C₄	1	4	yes	—
55	C₂²×C₃⋊C₈	—	2	2	no	—
56	C₃×#97	—	2	4	no	—
57	C₃×C₄×C₈	—	2	4	no	—
58	C₃⋊#95	—	2	4	no	—
59	C₄⋊(C₃⋊C₈)	—	2	4	no	—
60	C₄⋊(C₃⋊C₈)	—	2	4	no	—
61	C₂²×M₁₆	—	3	2	no	—
62	C₂²×M₁₆	C₂×D₃	1	2	yes	—
63	C₂×#117	—	3	2	no	—
64	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
65	#423	—	3	4	no	—
66	#423	—	6	1	no	—
67	#423	—	6	1	no	—
68	C₂×C₄⋊C₈	—	6	2	no	—
69	C₄×M₁₆	—	2	2	no	—
70	C₄×M₁₆	—	2	2	no	—
71	C₂×#97	Dic₃	1	2	yes	complemented
72	C₂×C₄×C₈	Dic₃	1	2	yes	complemented
73	#384	—	6	1	no	—
74	#382	—	6	1	no	—
75	#380	—	3	4	no	—
76	C₄×C₄⋊C₄	—	6	2	no	—
77	#344	—	6	8	no	—
78	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	4	4	no	—
80	C₂²×C₃×C₄	C₄²	1	2	yes	—
81	C₂²×C₃×C₄	C₄²	1	2	yes	—
82	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
85	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
86	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
94	C₂²×C₃×C₄	#41	1	2	yes	—
95	C₂²×C₃×C₄	#41	1	2	yes	—
96	C₂²×Dic₃	—	2	4	no	—
97	C₃×M₁₆	—	2	4	no	—
98	C₃×M₁₆	—	2	4	no	—
99	C₃×M₁₆	—	2	4	no	—
100	C₃×M₁₆	—	2	4	no	—
101	C₂×C₃×C₈	—	2	4	no	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	complemented
104	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
105	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
106	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
107	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
108	C₃×C₄²	C₄⋊C₄	1	2	yes	—
109	C₃×C₄²	C₄⋊C₄	1	2	yes	—
110	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
113	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
114	#217	—	2	8	no	—
115	C₄⋊Dic₃	—	4	8	no	—
116	C₄⋊Dic₃	—	4	2	no	—
117	C₄⋊Dic₃	—	4	2	no	—
118	C₄×Dic₃	—	4	4	no	—
119	#208	—	2	8	no	—
120	#208	—	4	2	no	—
121	#208	—	4	2	no	—
122	C₂×C₃⋊C₈	—	2	4	no	—
123	C₂×C₃⋊C₈	—	4	4	no	—
124	C₂³×C₄	C₄×D₃	1	2	yes	—
125	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	3	4	no	—
131	C₂×M₁₆	—	6	2	no	—
132	C₂×M₁₆	—	6	4	no	—
133	C₂×M₁₆	—	6	4	no	—
134	C₂×M₁₆	Dic₆	1	4	yes	—
135	C₂×M₁₆	D₁₂	1	4	yes	—
136	C₂²×C₈	—	6	2	no	—
137	C₂²×C₈	C₂×Dic₃	1	4	yes	—
138	#117	—	6	4	no	—
139	#117	—	6	4	no	—
140	C₂×C₄⋊C₄	—	6	4	no	—
141	C₂×C₄⋊C₄	—	6	2	no	—
142	C₂×C₂²⋊C₄	—	3	4	no	—
143	C₂×C₄²	—	6	2	no	—
144	C₂×C₄²	Dic₆	1	1	yes	characteristic
145	C₂×C₄²	Dic₆	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₃	1	2	yes	—
147	C₂×C₄²	D₁₂	1	1	yes	characteristic
148	C₂×C₄²	D₁₂	1	1	yes	characteristic
149	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
151	C₂×C₄²	#67	1	2	yes	—
152	C₂×C₄²	#67	1	2	yes	—
153	C₄⋊C₈	—	6	4	no	—
154	C₄⋊C₈	—	6	4	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	#95	—	6	4	no	—
158	C₂³×C₃	—	4	2	no	—
159	C₂³×C₃	C₂×C₄²	1	1	yes	characteristic
160	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂³×C₃	C₄×D₄	1	2	yes	—
163	C₂³×C₃	C₄×Q₈	1	2	yes	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	4	2	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	—	4	2	no	—
170	C₂×C₃×C₄	#95	1	2	yes	—
171	C₂×C₃×C₄	#95	1	2	yes	—
172	C₂×C₃×C₄	#95	1	2	yes	—
173	C₂×C₃×C₄	#95	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	#117	1	2	yes	—
180	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
182	C₂×C₃×C₄	#121	1	2	yes	—
183	C₂×C₃×C₄	#121	1	2	yes	—
184	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
185	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
186	C₂×C₃×C₄	#124	1	1	yes	characteristic
187	C₂×C₃×C₄	#124	1	1	yes	characteristic
188	C₂×C₃×C₄	#125	1	2	yes	—
189	C₂×C₃×C₄	#127	1	1	yes	characteristic
190	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
191	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
192	C₂×Dic₃	—	4	8	no	—
193	C₂×Dic₃	—	4	4	no	—
194	C₃×C₈	—	2	8	no	—
195	C₃⋊C₈	—	4	4	no	—
196	C₂⁴	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	6	4	no	—
199	C₂²×C₄	C₄×Dic₃	1	2	yes	—
200	C₂²×C₄	C₄×Dic₃	1	2	yes	—
201	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
202	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
203	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
204	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
205	C₂²×C₄	C₂×Dic₆	1	2	yes	—
206	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₁₂	1	2	yes	—
209	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
210	C₂²×C₄	#235	1	2	yes	—
211	C₂²×C₄	#235	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×#67	1	1	yes	characteristic
214	C₂²×C₄	C₂×#67	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	6	8	no	—
220	M₁₆	—	12	2	no	—
221	M₁₆	—	12	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	6	4	no	—
225	C₂×C₈	—	12	4	no	—
226	C₂×C₈	C₄⋊Dic₃	1	8	yes	complemented
227	C₄⋊C₄	—	12	8	no	—
228	C₄⋊C₄	—	12	2	no	—
229	C₄⋊C₄	—	12	2	no	—
230	C₂²⋊C₄	—	6	8	no	—
231	C₄²	—	12	4	no	—
232	C₄²	Dic₃⋊C₄	1	2	yes	—
233	C₄²	Dic₃⋊C₄	1	2	yes	—
234	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
235	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
236	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
237	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
238	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
239	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
240	C₄²	#217	1	2	yes	—
241	C₄²	#217	1	2	yes	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	4	2	no	—
246	C₂²×C₃	—	4	2	no	—
247	C₂²×C₃	C₂×#95	1	1	yes	characteristic
248	C₂²×C₃	C₄×C₄⋊C₄	1	2	yes	—
249	C₂²×C₃	#389	1	2	yes	—
250	C₂²×C₃	#390	1	1	yes	characteristic
251	C₂²×C₃	#391	1	1	yes	characteristic
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	#344	1	4	yes	complemented
256	C₃×C₄	#380	1	2	yes	—
257	C₃×C₄	#382	1	1	yes	characteristic
258	C₃×C₄	#384	1	1	yes	characteristic
259	C₃×C₄	#389	1	2	yes	—
260	C₃×C₄	#390	1	1	yes	characteristic
261	C₃×C₄	#391	1	1	yes	characteristic
262	Dic₃	—	8	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₄×Dic₆	1	2	yes	—
265	C₂³	C₄×D₁₂	1	2	yes	—
266	C₂³	C₂×C₄×Dic₃	1	1	yes	characteristic
267	C₂³	C₂×Dic₃⋊C₄	1	1	yes	characteristic
268	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	12	8	no	—
276	C₂×C₄	—	12	4	no	—
277	C₂×C₄	C₃⋊#95	1	2	yes	—
278	C₂×C₄	C₃⋊#95	1	2	yes	—
279	C₂×C₄	C₃⋊#95	1	2	yes	—
280	C₂×C₄	C₃⋊#95	1	2	yes	—
281	C₂×C₄	C₄×Dic₆	1	2	yes	—
282	C₂×C₄	#869	1	2	yes	—
283	C₂×C₄	#869	1	1	yes	characteristic
284	C₂×C₄	#870	1	2	yes	—
285	C₂×C₄	C₄×D₁₂	1	2	yes	—
286	C₂×C₄	#874	1	1	yes	characteristic
287	C₂×C₄	#875	1	1	yes	characteristic
288	C₂×C₄	#875	1	1	yes	characteristic
289	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
290	C₂×C₄	#924	1	2	yes	—
291	C₂×C₄	#924	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
294	C₂×C₄	#926	1	2	yes	—
295	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	#930	1	2	yes	—
297	C₂×C₄	#930	1	2	yes	—
298	C₂×C₄	C₂×#217	1	1	yes	characteristic
299	C₈	—	2	8	no	—
300	C₈	—	12	4	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	C₄⋊#95	1	1	yes	characteristic
306	C₂×C₃	C₂×#344	1	2	yes	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	#4928	1	2	yes	—
313	C₂²	#4929	1	1	yes	characteristic
314	C₂²	C₄×C₄⋊Dic₃	1	2	yes	—
315	C₂²	#4939	1	1	yes	characteristic
316	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	24	4	no	—
321	C₄	#4556	1	4	yes	complemented
322	C₄	#4928	1	2	yes	—
323	C₄	#4929	1	1	yes	characteristic
324	C₄	#4935	1	1	yes	characteristic
325	C₄	#4936	1	1	yes	characteristic
326	C₄	#4939	1	1	yes	characteristic
327	C₄	#5212	1	2	yes	—
328	C₃	#10848	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	#71535	1	1	yes	characteristic
334	C₂	C₂×#4556	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#217
commutator	C₂³×C₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄×M₁₆	C₂
socle	C₂²×C₃	C₂×#95
prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂³×C₃×C₄	C₂³
hypercenter	C₂×C₄×M₁₆	D₃
hypocenter	C₃	#10848
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
12. C₂×C₃×#97	C₄	semidirect
13. C₂×C₃×C₄×C₈	C₄	semidirect
20. #4924	C₄	semidirect
71. C₂×#97	Dic₃	semidirect
72. C₂×C₄×C₈	Dic₃	semidirect
103. C₂×C₃×C₈	C₄⋊C₄	semidirect
226. C₂×C₈	C₄⋊Dic₃	semidirect
255. C₃×C₄	#344	semidirect
321. C₄	#4556	semidirect
328. C₃	#10848	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×M₁₆	2
#82062	2
C₂×#4924	2
C₂×#4556	2
#10848	3