Subgroups — #10694051

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	#71614	C₂	1	2	yes	maximal, complemented
3	C₂×D₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×#4458	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#9657	—	3	1	no	maximal, Sylow
7	C₂²×C₃×D₈	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄×D₄	C₂²	1	2	yes	—
9	C₃×C₄×D₈	—	2	2	no	—
10	C₃×C₄×D₈	—	2	2	no	—
11	C₂×C₃×#107	C₄	1	1	yes	characteristic
12	C₂×C₃×#102	C₄	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×#830	—	2	4	no	—
15	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	D₄⋊(C₃⋊C₈)	—	2	4	no	—
19	#4458	—	2	2	no	—
20	C₈×C₃⋊C₈	—	2	2	no	—
21	C₂×C₄×D₈	D₃	1	1	yes	characteristic
22	#1954	—	3	2	no	—
23	C₂×#335	—	3	1	no	—
24	C₂×D₄⋊C₈	—	3	2	no	—
25	C₂×C₈²	—	3	1	no	—
26	C₂²×C₃×D₄	C₂×C₄	1	2	yes	—
27	C₂³×C₃×C₄	—	2	2	no	—
28	C₂×C₃×D₈	—	2	2	no	—
29	C₂×C₃×D₈	—	2	2	no	—
30	C₂×C₃×D₈	C₈	1	4	yes	complemented
31	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×D₄	—	2	2	no	—
34	C₃×C₄×D₄	—	2	2	no	—
35	C₃×C₄×D₄	—	2	2	no	—
36	C₃×C₄×D₄	—	2	2	no	—
37	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
39	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
40	C₂²×C₃⋊C₈	—	2	1	no	—
41	C₂²×C₃⋊C₈	—	2	2	no	—
42	C₂²×C₃⋊C₈	D₄	1	2	yes	—
43	C₃×#107	—	2	2	no	—
44	C₃×#102	—	2	4	no	—
45	C₃×C₄×C₈	D₄	1	2	yes	—
46	C₃×C₄×C₈	D₄	1	2	yes	—
47	#830	—	4	8	no	—
48	C₄⋊(C₃⋊C₈)	—	2	2	no	—
49	C₄⋊(C₃⋊C₈)	—	2	2	no	—
50	C₄×C₃⋊C₈	—	2	1	no	—
51	C₄×C₃⋊C₈	—	2	1	no	—
52	C₄×C₃⋊C₈	—	2	1	no	—
53	C₄×C₃⋊C₈	D₄	1	2	yes	—
54	C₂²×D₈	Dic₃	1	1	yes	characteristic
55	C₂×C₄×D₄	C₂×D₃	1	2	yes	—
56	C₄×D₈	—	2	2	no	—
57	C₄×D₈	—	2	2	no	—
58	C₂×#107	Dic₃	1	1	yes	characteristic
59	C₂×C₄⋊C₈	—	3	2	no	—
60	C₂×#102	Dic₃	1	2	yes	—
61	C₂×C₂²⋊C₈	—	6	4	no	—
62	C₂×C₄×C₈	—	3	1	no	—
63	C₂×C₄×C₈	—	3	1	no	—
64	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
65	#335	—	6	2	no	—
66	D₄⋊C₈	—	6	4	no	—
67	C₈²	—	6	2	no	—
68	C₂⁴×C₃	—	2	2	no	—
69	C₂×C₃×D₄	—	2	2	no	—
70	C₂×C₃×D₄	—	2	2	no	—
71	C₂×C₃×D₄	—	2	2	no	—
72	C₂×C₃×D₄	—	2	2	no	—
73	C₂×C₃×D₄	—	2	2	no	—
74	C₂×C₃×D₄	C₂×C₈	1	4	yes	—
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	—	4	2	no	—
77	C₂²×C₃×C₄	—	4	2	no	—
78	C₂²×C₃×C₄	—	4	4	no	—
79	C₂²×C₃×C₄	—	4	4	no	—
80	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	#41	1	1	yes	characteristic
83	C₃×D₈	—	2	8	no	—
84	C₂×C₃×C₈	—	2	1	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
89	C₂×C₃×C₈	M₁₆	1	2	yes	—
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄⋊C₄	—	2	2	no	—
92	C₃×C₂²⋊C₄	—	4	2	no	—
93	C₃×C₂²⋊C₄	—	4	2	no	—
94	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₃×C₄²	#41	1	1	yes	characteristic
98	C₂×C₃⋊C₈	—	2	4	no	—
99	C₂×C₃⋊C₈	—	2	2	no	—
100	C₂×C₃⋊C₈	—	2	2	no	—
101	C₂×C₃⋊C₈	—	2	2	no	—
102	C₂×C₃⋊C₈	—	4	2	no	—
103	C₂×C₃⋊C₈	—	4	4	no	—
104	C₂×C₃⋊C₈	D₈	1	4	yes	complemented
105	C₂²×D₄	C₂×Dic₃	1	2	yes	—
106	C₂³×C₄	—	2	2	no	—
107	C₂×D₈	—	2	2	no	—
108	C₂×D₈	—	2	2	no	—
109	C₂×D₈	C₃⋊C₈	1	4	yes	complemented
110	C₂²×C₈	—	3	2	no	—
111	C₂²×C₈	—	6	1	no	—
112	C₂²×C₈	—	6	2	no	—
113	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
115	C₄×D₄	—	2	2	no	—
116	C₄×D₄	—	2	2	no	—
117	C₄×D₄	—	2	2	no	—
118	C₄×D₄	—	2	2	no	—
119	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
120	C₂×C₂²⋊C₄	—	2	2	no	—
121	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
122	#107	—	2	2	no	—
123	C₄⋊C₈	—	6	2	no	—
124	C₄⋊C₈	—	6	2	no	—
125	#102	—	2	4	no	—
126	C₂²⋊C₈	—	12	8	no	—
127	C₄×C₈	—	3	2	no	—
128	C₄×C₈	—	6	1	no	—
129	C₄×C₈	—	6	1	no	—
130	C₄×C₈	—	6	1	no	—
131	C₄×C₈	#67	1	2	yes	—
132	C₄×C₈	#67	1	2	yes	—
133	C₂³×C₃	—	2	4	no	—
134	C₂³×C₃	—	4	2	no	—
135	C₂³×C₃	—	4	2	no	—
136	C₂³×C₃	—	4	4	no	—
137	C₂³×C₃	—	4	4	no	—
138	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
139	C₃×D₄	—	2	4	no	—
140	C₃×D₄	—	2	4	no	—
141	C₃×D₄	—	4	2	no	—
142	C₃×D₄	—	4	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	—	8	4	no	—
147	C₂×C₃×C₄	—	8	2	no	—
148	C₂×C₃×C₄	—	8	2	no	—
149	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	#121	1	1	yes	characteristic
152	C₂×C₃×C₄	#121	1	1	yes	characteristic
153	C₂×C₃×C₄	#124	1	1	yes	characteristic
154	C₂×C₃×C₄	#127	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
157	C₂×C₃×C₄	#131	1	1	yes	characteristic
158	C₂×C₃×C₄	#131	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
160	C₂×C₃×C₄	#135	1	2	yes	—
161	C₃×C₈	—	2	2	no	—
162	C₃×C₈	C₂²⋊C₈	1	4	yes	complemented
163	C₃⋊C₈	—	2	4	no	—
164	C₃⋊C₈	—	2	4	no	—
165	C₃⋊C₈	—	8	2	no	—
166	C₂⁴	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	C₂×C₃⋊C₈	1	4	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
179	C₂²×C₄	#237	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
181	D₈	—	2	8	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	3	4	no	—
185	C₂×C₈	—	6	4	no	—
186	C₂×C₈	—	6	2	no	—
187	C₂×C₈	—	6	2	no	—
188	C₂×C₈	—	6	2	no	—
189	C₂×C₈	—	12	4	no	—
190	C₂×C₈	—	12	2	no	—
191	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
192	C₂×C₈	#208	1	2	yes	—
193	C₂×C₈	#217	1	2	yes	—
194	C₂×C₈	#217	1	2	yes	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₄²	D₃×D₄	1	1	yes	characteristic
200	C₄²	#237	1	1	yes	characteristic
201	C₄²	C₂×#67	1	1	yes	characteristic
202	C₄²	C₂×#67	1	1	yes	characteristic
203	C₂²×C₃	—	4	8	no	—
204	C₂²×C₃	—	4	4	no	—
205	C₂²×C₃	—	8	4	no	—
206	C₂²×C₃	—	8	2	no	—
207	C₂²×C₃	—	8	2	no	—
208	C₂²×C₃	#390	1	1	yes	characteristic
209	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
210	C₂²×C₃	#436	1	1	yes	characteristic
211	C₂²×C₃	C₄×D₈	1	2	yes	—
212	C₂²×C₃	#443	1	2	yes	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	8	2	no	—
216	C₃×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
217	C₃×C₄	#408	1	1	yes	characteristic
218	C₃×C₄	C₈×D₄	1	1	yes	characteristic
219	C₃×C₄	#436	1	1	yes	characteristic
220	C₃×C₄	#466	1	2	yes	—
221	C₃×C₄	#493	1	1	yes	characteristic
222	C₃×C₄	#495	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	D₄×Dic₃	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	2	4	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	D₃×D₈	1	2	yes	—
240	C₂×C₄	#912	1	2	yes	—
241	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
242	C₂×C₄	C₂×#208	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
244	C₂×C₄	#936	1	1	yes	characteristic
245	C₂×C₄	#938	1	1	yes	characteristic
246	C₂×C₄	#938	1	1	yes	characteristic
247	C₂×C₄	#940	1	1	yes	characteristic
248	C₂×C₄	#948	1	1	yes	characteristic
249	C₂×C₄	#948	1	1	yes	characteristic
250	C₂×C₄	C₂×#217	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	6	4	no	—
253	C₈	—	6	4	no	—
254	C₈	—	24	2	no	—
255	C₈	#830	1	4	yes	complemented
256	C₂×C₃	—	8	4	no	—
257	C₂×C₃	C₈×D₈	1	2	yes	—
258	C₂×C₃	#1589	1	2	yes	—
259	C₂×C₃	#1954	1	1	yes	characteristic
260	C₂×C₃	#1970	1	1	yes	characteristic
261	C₂×C₃	#1974	1	1	yes	characteristic
262	C₂²	—	4	4	no	—
263	C₂²	—	4	8	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	D₄×C₃⋊C₈	1	1	yes	characteristic
268	C₂²	#5012	1	1	yes	characteristic
269	C₂²	Dic₃×D₈	1	2	yes	—
270	C₂²	#5196	1	2	yes	—
271	C₂²	#5221	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	8	2	no	—
275	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
276	C₄	#5012	1	1	yes	characteristic
277	C₄	#5151	1	1	yes	characteristic
278	C₄	#5155	1	1	yes	characteristic
279	C₄	#5157	1	2	yes	—
280	C₄	C₂×#830	1	1	yes	characteristic
281	C₄	#5234	1	1	yes	characteristic
282	C₃	#9657	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	8	4	no	—
284	C₂	D₈×C₃⋊C₈	1	2	yes	—
285	C₂	#70881	1	2	yes	—
286	C₂	#71614	1	1	yes	characteristic
287	C₂	#72038	1	1	yes	characteristic
288	C₂	#72084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₄×D₈	C₂
socle	C₂³×C₃	C₄×D₄
prime radical	C₂²×C₃×D₈	C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₄×D₈	D₃
hypocenter	C₃	#9657
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #71614	C₂	semidirect
3. C₂×D₄⋊(C₃⋊C₈)	C₂	semidirect
4. C₂×#4458	C₂	semidirect
5. C₂×C₈×C₃⋊C₈	C₂	semidirect
30. C₂×C₃×D₈	C₈	semidirect
104. C₂×C₃⋊C₈	D₈	semidirect
109. C₂×D₈	C₃⋊C₈	semidirect
162. C₃×C₈	C₂²⋊C₈	semidirect
255. C₈	#830	semidirect
282. C₃	#9657	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×D₈	2
#71614	2
C₂×D₄⋊(C₃⋊C₈)	2
C₂×#4458	2
C₂×C₈×C₃⋊C₈	2
#9657	3