Subgroups — #10705128

Order 768.

342 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#69274	C₂	1	1	yes	maximal, characteristic
3	#69052	C₂	1	1	yes	maximal, characteristic
4	C₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#67768	C₂	1	1	yes	maximal, characteristic, complemented
6	#67558	C₂	1	1	yes	maximal, characteristic, complemented
7	#67506	C₂	1	1	yes	maximal, characteristic, complemented
8	#11637	—	3	1	no	maximal, Sylow
9	C₄×D₃×Q₈	C₄	1	1	yes	characteristic
10	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×C₃⋊C₈	C₄	1	1	yes	characteristic
12	D₃×C₄⋊C₈	—	2	1	no	—
13	D₃×C₄⋊C₈	C₄	1	1	yes	characteristic
14	#4830	—	2	1	no	—
15	#4830	C₄	1	1	yes	characteristic
16	C₄×D₃×C₈	—	2	1	no	—
17	C₄×D₃×C₈	C₂²	1	1	yes	characteristic
18	C₈×Dic₆	—	2	1	no	—
19	C₈×Dic₆	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	(C₂×D₃)⋊C₁₆	—	2	1	no	—
23	(C₂×D₃)⋊C₁₆	—	2	1	no	—
24	Dic₃⋊C₁₆	—	2	1	no	—
25	Dic₃×C₁₆	—	2	1	no	—
26	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
27	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₈×Q₈	—	3	1	no	—
29	#2162	—	3	1	no	—
30	#2162	—	3	1	no	—
31	Q₈⋊C₁₆	—	3	1	no	—
32	Q₈⋊C₁₆	—	3	1	no	—
33	Q₈⋊C₁₆	—	3	1	no	—
34	Q₈⋊C₁₆	D₃	1	1	yes	characteristic, complemented
35	C₂×D₃×Q₈	C₈	1	1	yes	characteristic
36	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
37	Q₈×Dic₃	C₈	1	1	yes	characteristic
38	C₂×D₃×C₈	—	2	1	no	—
39	C₂×D₃×C₈	D₄	1	1	yes	characteristic
40	C₂×D₃×C₈	D₄	1	1	yes	characteristic
41	D₃×C₄⋊C₄	—	2	1	no	—
42	D₃×C₄⋊C₄	C₈	1	1	yes	characteristic
43	#887	—	2	1	no	—
44	#887	C₈	1	1	yes	characteristic
45	C₄²×D₃	—	2	1	no	—
46	C₄²×D₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₆	—	2	1	no	—
48	C₄×Dic₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃×C₁₆	—	2	1	no	—
50	C₂×C₃×C₁₆	—	2	1	no	—
51	C₃×C₄⋊C₈	—	2	1	no	—
52	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
55	Dic₃⋊C₈	—	2	1	no	—
56	Dic₃⋊C₈	—	2	1	no	—
57	Dic₃⋊C₈	—	2	1	no	—
58	C₈×Dic₃	—	2	1	no	—
59	C₈×Dic₃	—	2	1	no	—
60	C₈×Dic₃	D₄	1	1	yes	characteristic
61	C₈×Dic₃	D₄	1	1	yes	characteristic
62	C₂×C₃⋊C₁₆	—	2	1	no	—
63	C₂×C₃⋊C₁₆	—	2	1	no	—
64	C₄⋊(C₃⋊C₈)	—	2	1	no	—
65	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
66	C₄×C₃⋊C₈	—	2	1	no	—
67	C₄×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×Q₈	—	3	1	no	—
69	C₈×Q₈	—	3	1	no	—
70	C₈×Q₈	—	3	1	no	—
71	C₈×Q₈	—	3	1	no	—
72	C₈×Q₈	—	6	1	no	—
73	C₈×Q₈	—	6	1	no	—
74	C₈×Q₈	C₂×D₃	1	1	yes	characteristic
75	C₂×C₄⋊C₈	—	3	1	no	—
76	C₂×C₄⋊C₈	—	6	1	no	—
77	C₂×C₄×C₈	—	3	1	no	—
78	C₂×C₄×C₈	—	6	1	no	—
79	C₄⋊C₁₆	—	3	1	no	—
80	C₄⋊C₁₆	—	6	1	no	—
81	C₄⋊C₁₆	C₂×D₃	1	1	yes	characteristic
82	C₂²⋊C₁₆	—	6	1	no	—
83	C₂²⋊C₁₆	—	6	1	no	—
84	C₄×C₁₆	—	3	1	no	—
85	C₄×C₁₆	—	6	1	no	—
86	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
87	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
88	D₃×Q₈	—	2	2	no	—
89	D₃×Q₈	C₁₆	1	4	yes	complemented
90	C₂×C₄×D₃	—	2	1	no	—
91	C₂×C₄×D₃	—	2	1	no	—
92	C₂×C₄×D₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄×D₃	M₁₆	1	1	yes	characteristic
95	C₂×Dic₆	—	2	1	no	—
96	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃×C₄⋊C₄	—	2	1	no	—
101	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₄⋊Dic₃	—	2	1	no	—
105	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
106	Dic₃⋊C₄	—	2	1	no	—
107	Dic₃⋊C₄	—	2	1	no	—
108	Dic₃⋊C₄	—	2	1	no	—
109	C₄×Dic₃	—	2	1	no	—
110	C₄×Dic₃	—	2	1	no	—
111	C₄×Dic₃	—	2	1	no	—
112	C₄×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
113	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₃	M₁₆	1	1	yes	characteristic
115	C₂×C₃⋊C₈	—	2	1	no	—
116	C₂×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
118	D₃×C₈	—	2	1	no	—
119	D₃×C₈	—	2	1	no	—
120	D₃×C₈	—	4	1	no	—
121	D₃×C₈	—	4	2	no	—
122	C₃×C₁₆	—	4	1	no	—
123	C₃×C₁₆	—	4	1	no	—
124	C₃⋊C₁₆	—	4	1	no	—
125	C₃⋊C₁₆	—	4	1	no	—
126	C₂²×Q₈	—	3	1	no	—
127	C₂²×C₈	—	3	1	no	—
128	C₂²×C₈	—	3	1	no	—
129	C₂²×C₈	—	6	1	no	—
130	C₄×Q₈	—	3	1	no	—
131	C₄×Q₈	—	3	1	no	—
132	C₄×Q₈	—	3	1	no	—
133	C₄×Q₈	—	6	1	no	—
134	C₄×Q₈	—	6	1	no	—
135	C₄×Q₈	C₄×D₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	3	1	no	—
137	C₂×C₄⋊C₄	—	6	1	no	—
138	C₂×C₄²	—	3	1	no	—
139	C₂×C₄²	—	6	1	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	—	2	1	no	—
142	C₂×C₁₆	—	6	1	no	—
143	C₂×C₁₆	—	6	1	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	—	3	1	no	—
146	C₄⋊C₈	—	6	1	no	—
147	C₄⋊C₈	—	6	1	no	—
148	C₄⋊C₈	—	6	1	no	—
149	C₄⋊C₈	—	6	1	no	—
150	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	3	1	no	—
153	C₄×C₈	—	3	1	no	—
154	C₄×C₈	—	3	1	no	—
155	C₄×C₈	—	6	1	no	—
156	C₄×C₈	—	6	1	no	—
157	C₄×C₈	—	6	1	no	—
158	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
159	C₂²×D₃	C₂²⋊C₈	1	1	yes	characteristic
160	C₃×Q₈	—	2	1	no	—
161	C₃×Q₈	C₂×C₁₆	1	2	yes	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
167	C₂×Dic₃	—	2	1	no	—
168	C₂×Dic₃	—	2	1	no	—
169	C₂×Dic₃	C₂²⋊C₈	1	1	yes	characteristic
170	C₂×Dic₃	C₂²⋊C₈	1	1	yes	characteristic
171	C₂×Dic₃	C₂²⋊C₈	1	1	yes	characteristic
172	C₄×D₃	—	2	4	no	—
173	C₄×D₃	—	2	1	no	—
174	C₄×D₃	—	2	1	no	—
175	C₄×D₃	—	4	1	no	—
176	C₄×D₃	—	4	2	no	—
177	C₄×D₃	C₂×C₁₆	1	2	yes	—
178	C₄×D₃	M₃₂	1	2	yes	—
179	Dic₆	—	2	1	no	—
180	Dic₆	—	2	4	no	—
181	Dic₆	C₂×C₁₆	1	2	yes	—
182	C₃×C₈	—	2	1	no	—
183	C₃×C₈	—	4	1	no	—
184	C₃×C₈	#137	1	1	yes	characteristic
185	C₃×C₈	#137	1	1	yes	characteristic
186	C₃⋊C₈	—	2	1	no	—
187	C₃⋊C₈	—	2	1	no	—
188	C₃⋊C₈	—	4	1	no	—
189	C₂×Q₈	—	3	4	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×Q₈	—	6	1	no	—
192	C₂×Q₈	—	6	2	no	—
193	C₂×Q₈	D₃×C₈	1	1	yes	characteristic
194	C₂²×C₄	—	3	1	no	—
195	C₂²×C₄	—	3	1	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	6	1	no	—
198	C₂²×C₄	—	6	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	3	1	no	—
201	C₂×C₈	—	3	1	no	—
202	C₂×C₈	—	6	1	no	—
203	C₂×C₈	—	6	1	no	—
204	C₂×C₈	—	6	1	no	—
205	C₂×C₈	—	12	1	no	—
206	C₂×C₈	—	12	2	no	—
207	C₂×C₈	D₃×D₄	1	1	yes	characteristic
208	C₂×C₈	D₃×D₄	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	3	1	no	—
211	C₄⋊C₄	—	6	1	no	—
212	C₄⋊C₄	—	6	1	no	—
213	C₄⋊C₄	—	6	1	no	—
214	C₄⋊C₄	—	6	1	no	—
215	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
216	C₄²	—	2	1	no	—
217	C₄²	—	3	1	no	—
218	C₄²	—	3	1	no	—
219	C₄²	—	3	1	no	—
220	C₄²	—	6	1	no	—
221	C₄²	—	6	1	no	—
222	C₄²	—	6	1	no	—
223	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
224	C₁₆	—	4	1	no	—
225	C₁₆	—	4	1	no	—
226	C₁₆	—	12	1	no	—
227	C₁₆	—	12	1	no	—
228	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
229	C₂×D₃	—	2	1	no	—
230	C₂×D₃	—	2	1	no	—
231	C₂×D₃	C₂²⋊C₁₆	1	2	yes	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	#419	1	1	yes	characteristic
236	C₃×C₄	#421	1	1	yes	characteristic
237	C₃×C₄	C₂²×C₁₆	1	1	yes	characteristic
238	C₃×C₄	C₂×M₃₂	1	1	yes	characteristic
239	Dic₃	—	2	2	no	—
240	Dic₃	—	2	1	no	—
241	Dic₃	—	2	1	no	—
242	Dic₃	—	4	1	no	—
243	Dic₃	C₂²⋊C₁₆	1	2	yes	—
244	C₂³	—	3	1	no	—
245	Q₈	—	2	1	no	—
246	Q₈	—	3	2	no	—
247	Q₈	—	6	4	no	—
248	Q₈	—	6	1	no	—
249	Q₈	D₃×C₁₆	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	3	2	no	—
254	C₂×C₄	—	3	2	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	6	4	no	—
258	C₂×C₄	—	6	1	no	—
259	C₂×C₄	—	6	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	12	1	no	—
263	C₂×C₄	—	12	2	no	—
264	C₂×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
266	C₂×C₄	D₃×M₁₆	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	4	1	no	—
269	C₈	—	6	1	no	—
270	C₈	—	6	1	no	—
271	C₈	—	12	1	no	—
272	C₈	#915	1	1	yes	characteristic
273	C₈	#918	1	1	yes	characteristic
274	C₂×C₃	#1479	1	1	yes	characteristic
275	C₂×C₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
276	C₂×C₃	#2117	1	1	yes	characteristic
277	D₃	—	2	2	no	—
278	C₂²	—	3	2	no	—
279	C₂²	—	6	1	no	—
280	C₂²	—	6	1	no	—
281	C₂²	D₃×C₂²⋊C₈	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	3	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	1	no	—
288	C₄	—	6	2	no	—
289	C₄	—	12	1	no	—
290	C₄	#4802	1	1	yes	characteristic
291	C₄	#4821	1	1	yes	characteristic
292	C₄	C₂×D₃×C₁₆	1	1	yes	characteristic
293	C₄	D₃×M₃₂	1	1	yes	characteristic
294	C₃	#11637	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	6	2	no	—
296	C₂	#68231	1	1	yes	characteristic
297	C₂	D₃×C₂²⋊C₁₆	1	1	yes	characteristic
298	C₂	#69125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₃×D₄
commutator	C₃×C₄	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×Q₈⋊C₁₆	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₃	C₂²⋊C₈
cosocle	C₃×C₄×C₈	C₂³
hypercenter	Q₈⋊C₁₆	D₃
hypocenter	C₃	#11637
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×Q₈⋊C₁₆	C₂	semidirect
5. #67768	C₂	semidirect
6. #67558	C₂	semidirect
7. #67506	C₂	semidirect
34. Q₈⋊C₁₆	D₃	semidirect
89. D₃×Q₈	C₁₆	semidirect
294. C₃	#11637	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₈×Q₈	2
#69274	2
#69052	2
C₃×Q₈⋊C₁₆	2
#67768	2
#67558	2
#67506	2
#11637	3