Subgroups — #10695211

Order 768.

506 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	C₃×C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#69394	C₂	1	1	yes	maximal, characteristic
3	#69388	C₂	1	1	yes	maximal, characteristic
4	#67619	C₂	1	4	yes	maximal
5	#9874	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Q₈	—	2	1	no	—
8	C₃×#391	—	2	1	no	—
9	C₃×#389	—	2	1	no	—
10	C₃×#389	C₄	1	2	yes	—
11	C₃×#384	C₄	1	1	yes	characteristic
12	C₃×C₄×C₄⋊C₄	—	2	1	no	—
13	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₄³	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
16	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
17	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
19	#4524	—	2	1	no	—
20	#4524	—	2	1	no	—
21	C₄×C₄⋊Q₈	D₃	1	1	yes	characteristic
22	#1842	—	3	1	no	—
23	#1744	—	3	1	no	—
24	#1279	—	3	4	no	—
25	C₂²×C₃×Q₈	—	2	1	no	—
26	C₃×C₄⋊Q₈	—	2	4	no	—
27	C₃×C₄×Q₈	—	4	2	no	—
28	C₃×C₄×Q₈	—	4	2	no	—
29	C₂×C₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄²	—	2	2	no	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄²	D₄	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	1	yes	characteristic
42	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
43	C₂²×C₃⋊C₈	—	2	2	no	—
44	C₂²×C₃⋊C₈	—	2	2	no	—
45	C₃×#95	—	2	2	no	—
46	C₃×#95	—	2	2	no	—
47	C₄⋊(C₃⋊C₈)	—	2	4	no	—
48	C₄⋊(C₃⋊C₈)	—	4	2	no	—
49	C₄⋊(C₃⋊C₈)	—	4	2	no	—
50	C₄×C₃⋊C₈	—	2	2	no	—
51	C₄×C₃⋊C₈	Q₈	1	4	yes	complemented
52	C₂×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
53	C₂×C₄×Q₈	—	2	1	no	—
54	C₂×C₄⋊C₈	—	3	2	no	—
55	C₂×C₄⋊C₈	—	6	1	no	—
56	C₂×C₄⋊C₈	—	6	1	no	—
57	C₂×C₄×C₈	—	3	2	no	—
58	#391	—	2	1	no	—
59	#389	—	2	1	no	—
60	#389	Dic₃	1	2	yes	—
61	#384	Dic₃	1	1	yes	characteristic
62	C₄×C₄⋊C₄	—	2	1	no	—
63	C₄×C₄⋊C₄	C₂×D₃	1	2	yes	—
64	C₄³	C₂×D₃	1	1	yes	characteristic
65	#336	—	6	1	no	—
66	#336	—	6	1	no	—
67	C₂×C₃×Q₈	—	2	4	no	—
68	C₂×C₃×Q₈	—	4	2	no	—
69	C₂×C₃×Q₈	—	4	2	no	—
70	C₂²×C₃×C₄	—	2	2	no	—
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄⋊C₄	—	2	4	no	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	4	no	—
82	C₃×C₄⋊C₄	—	4	2	no	—
83	C₃×C₄⋊C₄	—	4	4	no	—
84	C₃×C₄⋊C₄	—	4	2	no	—
85	C₃×C₄⋊C₄	M₁₆	1	4	yes	—
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	—	2	2	no	—
88	C₃×C₄²	—	4	1	no	—
89	C₃×C₄²	—	4	1	no	—
90	C₃×C₄²	—	4	2	no	—
91	C₃×C₄²	—	4	2	no	—
92	C₃×C₄²	D₈	1	2	yes	—
93	C₃×C₄²	SD₁₆	1	2	yes	—
94	C₃×C₄²	SD₁₆	1	2	yes	—
95	C₃×C₄²	Q₁₆	1	2	yes	—
96	C₃×C₄²	C₂×Q₈	1	2	yes	—
97	C₃×C₄²	#41	1	2	yes	—
98	C₂×C₃⋊C₈	—	2	4	no	—
99	C₂×C₃⋊C₈	—	4	2	no	—
100	C₂×C₃⋊C₈	—	4	2	no	—
101	C₂×C₃⋊C₈	—	4	2	no	—
102	C₂×C₃⋊C₈	—	4	2	no	—
103	C₂×C₃⋊C₈	—	4	2	no	—
104	C₂²×Q₈	—	2	1	no	—
105	C₂²×C₈	—	6	2	no	—
106	C₂²×C₈	—	6	2	no	—
107	C₄⋊Q₈	—	2	4	no	—
108	C₄×Q₈	—	4	2	no	—
109	C₄×Q₈	—	4	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
115	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₄²	#67	1	1	yes	characteristic
120	C₂×C₄²	#67	1	1	yes	characteristic
121	C₂×C₄²	#67	1	1	yes	characteristic
122	C₂×C₄²	#67	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
124	C₄⋊C₈	—	6	4	no	—
125	C₄⋊C₈	—	12	2	no	—
126	C₄⋊C₈	—	12	2	no	—
127	C₄×C₈	—	3	4	no	—
128	C₄×C₈	—	6	2	no	—
129	#95	—	2	2	no	—
130	#95	—	2	2	no	—
131	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₃×Q₈	—	4	8	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	#102	1	2	yes	—
150	C₂×C₃×C₄	#102	1	2	yes	—
151	C₂×C₃×C₄	#103	1	2	yes	—
152	C₂×C₃×C₄	#103	1	2	yes	—
153	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
154	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₃×C₄	#124	1	1	yes	characteristic
156	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
158	C₂×C₃×C₄	#131	1	2	yes	—
159	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
163	C₃⋊C₈	—	4	4	no	—
164	C₃⋊C₈	—	8	2	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	#217	1	1	yes	characteristic
171	C₂²×C₄	#217	1	1	yes	characteristic
172	C₂²×C₄	#217	1	1	yes	characteristic
173	C₂²×C₄	#217	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×#67	1	1	yes	characteristic
176	C₂²×C₄	C₂×#67	1	1	yes	characteristic
177	C₂×C₈	—	6	4	no	—
178	C₂×C₈	—	12	2	no	—
179	C₂×C₈	—	12	2	no	—
180	C₂×C₈	—	12	2	no	—
181	C₂×C₈	—	12	2	no	—
182	C₂×C₈	—	12	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	#208	1	4	yes	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	—	4	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	#213	1	2	yes	—
197	C₄²	#214	1	2	yes	—
198	C₄²	#215	1	2	yes	—
199	C₄²	#216	1	2	yes	—
200	C₄²	D₃×Q₈	1	2	yes	—
201	C₄²	#239	1	2	yes	—
202	C₂²×C₃	#391	1	1	yes	characteristic
203	C₂²×C₃	#407	1	1	yes	characteristic
204	C₂²×C₃	#408	1	1	yes	characteristic
205	C₂²×C₃	C₂×#102	1	1	yes	characteristic
206	C₂²×C₃	C₂×#103	1	1	yes	characteristic
207	C₂²×C₃	#421	1	1	yes	characteristic
208	C₂²×C₃	#421	1	1	yes	characteristic
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	4	no	—
215	C₃×C₄	—	8	2	no	—
216	C₃×C₄	#446	1	4	yes	—
217	C₃×C₄	#486	1	1	yes	characteristic
218	C₃×C₄	#487	1	1	yes	characteristic
219	C₃×C₄	#498	1	1	yes	characteristic
220	C₃×C₄	#500	1	1	yes	characteristic
221	C₂³	C₂×#217	1	1	yes	characteristic
222	Q₈	—	4	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#832	1	2	yes	—
240	C₂×C₄	#832	1	2	yes	—
241	C₂×C₄	#835	1	2	yes	—
242	C₂×C₄	#835	1	2	yes	—
243	C₂×C₄	C₂×#208	1	2	yes	—
244	C₂×C₄	C₂×#213	1	1	yes	characteristic
245	C₂×C₄	C₂×#214	1	1	yes	characteristic
246	C₂×C₄	C₂×#215	1	1	yes	characteristic
247	C₂×C₄	C₂×#216	1	1	yes	characteristic
248	C₂×C₄	#944	1	1	yes	characteristic
249	C₂×C₄	Q₈×Dic₃	1	2	yes	—
250	C₂×C₄	#946	1	2	yes	—
251	C₂×C₄	#947	1	1	yes	characteristic
252	C₂×C₄	#948	1	2	yes	—
253	C₈	—	12	4	no	—
254	C₈	—	24	2	no	—
255	C₂×C₃	#1480	1	1	yes	characteristic
256	C₂×C₃	#1481	1	1	yes	characteristic
257	C₂×C₃	#1486	1	1	yes	characteristic
258	C₂×C₃	#1487	1	1	yes	characteristic
259	C₂×C₃	#1985	1	1	yes	characteristic
260	C₂×C₃	#1990	1	1	yes	characteristic
261	C₂×C₃	#1995	1	1	yes	characteristic
262	C₂²	#5207	1	1	yes	characteristic
263	C₂²	C₂×#832	1	1	yes	characteristic
264	C₂²	C₂×#835	1	1	yes	characteristic
265	C₂²	#5229	1	1	yes	characteristic
266	C₂²	#5234	1	1	yes	characteristic
267	C₂²	#5236	1	1	yes	characteristic
268	C₂²	#5236	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	8	2	no	—
276	C₄	#5024	1	4	yes	—
277	C₄	#5078	1	1	yes	characteristic
278	C₄	#5080	1	1	yes	characteristic
279	C₄	#5088	1	1	yes	characteristic
280	C₄	#5093	1	1	yes	characteristic
281	C₃	#9874	1	1	yes	characteristic, complemented, Sylow
282	C₂	#71598	1	1	yes	characteristic
283	C₂	#71622	1	1	yes	characteristic
284	C₂	#71635	1	1	yes	characteristic
285	C₂	#71787	1	1	yes	characteristic
286	C₂	#71800	1	1	yes	characteristic
287	C₂	#71824	1	1	yes	characteristic
288	C₂	#71840	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊Q₈	D₃
hypocenter	C₃	#9874
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
51. C₄×C₃⋊C₈	Q₈	semidirect
281. C₃	#9874	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊Q₈	2
#69394	2
#69388	2
#67619	2
#9874	3