Subgroups — #10948831

Order 768.

486 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#2163	C₂	1	1	yes	maximal, characteristic
3	#69745	C₂	1	1	yes	maximal, characteristic
4	#69282	C₂	1	4	yes	maximal, complemented
5	#69093	C₂	1	2	yes	maximal
6	#69092	C₂	1	2	yes	maximal
7	#69068	C₂	1	4	yes	maximal, complemented
8	#33497	—	3	1	no	maximal, Sylow
9	C₂×C₄×Dic₆	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₃⋊C₈	C₄	1	1	yes	characteristic
12	C₂×Dic₃⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
16	C₈×Dic₆	C₄	1	2	yes	—
17	C₈×Dic₆	C₄	1	2	yes	—
18	C₈×Dic₆	C₂²	1	2	yes	—
19	C₈×Dic₆	C₂²	1	2	yes	—
20	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₃×#346	C₂²	1	2	yes	—
23	#4549	C₂²	1	2	yes	—
24	#4502	C₂²	1	4	yes	—
25	Dic₃⋊C₁₆	C₂²	1	4	yes	—
26	Dic₃⋊C₁₆	C₂²	1	4	yes	—
27	Dic₃×C₁₆	C₂²	1	4	yes	—
28	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
29	#4461	C₂²	1	2	yes	—
30	C₂×C₈×Q₈	—	3	1	no	—
31	#2183	—	3	4	no	—
32	#2183	—	3	4	no	—
33	#2163	—	3	1	no	—
34	#2163	—	3	2	no	—
35	#2163	D₃	1	1	yes	characteristic
36	#2162	—	3	2	no	—
37	C₂²×Dic₆	C₈	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₃	C₈	1	1	yes	characteristic
42	C₂×Dic₃⋊C₄	C₈	1	1	yes	characteristic
43	C₂×Dic₃⋊C₄	C₈	1	1	yes	characteristic
44	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₆	C₈	1	2	yes	—
49	C₄×Dic₆	C₈	1	2	yes	—
50	C₄×Dic₆	C₂×C₄	1	2	yes	—
51	C₄×Dic₆	C₂×C₄	1	2	yes	—
52	C₂×C₃×C₁₆	C₂³	1	4	yes	—
53	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
57	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
58	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
59	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
60	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
61	C₈×Dic₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₃	C₂³	1	2	yes	—
64	C₈×Dic₃	C₂³	1	2	yes	—
65	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
66	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
67	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
68	C₂×C₄×Q₈	—	3	1	no	—
69	C₈×Q₈	—	3	2	no	—
70	C₈×Q₈	—	3	2	no	—
71	C₈×Q₈	—	3	2	no	—
72	C₈×Q₈	—	3	2	no	—
73	C₂×C₄⋊C₈	—	3	1	no	—
74	C₂×C₄⋊C₈	—	3	1	no	—
75	C₂×C₄⋊C₈	—	3	1	no	—
76	C₂×C₄×C₈	—	3	1	no	—
77	C₂×C₄×C₈	—	3	1	no	—
78	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
79	C₄⋊C₁₆	—	3	4	no	—
80	C₄⋊C₁₆	—	3	4	no	—
81	C₄⋊C₁₆	—	3	2	no	—
82	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
83	C₂²⋊C₁₆	—	3	2	no	—
84	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
85	#346	—	3	4	no	—
86	#346	—	3	2	no	—
87	#346	C₂×D₃	1	2	yes	—
88	C₄×C₁₆	—	3	4	no	—
89	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
93	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
94	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
95	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
96	C₂×Dic₆	—	2	4	no	—
97	C₂×Dic₆	C₂×C₈	1	2	yes	—
98	C₂×Dic₆	C₂×C₈	1	2	yes	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	4	1	no	—
101	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
105	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
106	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
107	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
108	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
109	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
110	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
111	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
112	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
113	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
114	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
115	C₄×Dic₃	C₂×C₈	1	2	yes	—
116	C₄×Dic₃	C₂×C₈	1	2	yes	—
117	C₄×Dic₃	C₂²×C₄	1	2	yes	—
118	C₄×Dic₃	C₂²×C₄	1	2	yes	—
119	C₂×C₃⋊C₈	—	4	1	no	—
120	C₂×C₃⋊C₈	—	4	1	no	—
121	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
123	C₃×C₁₆	—	2	4	no	—
124	C₃⋊C₁₆	—	2	4	no	—
125	C₂²×Q₈	—	3	1	no	—
126	C₂²×C₈	—	3	1	no	—
127	C₂²×C₈	—	3	1	no	—
128	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
130	C₄×Q₈	—	3	2	no	—
131	C₄×Q₈	—	3	2	no	—
132	C₄×Q₈	—	3	2	no	—
133	C₄×Q₈	—	3	2	no	—
134	C₂×C₄⋊C₄	—	3	1	no	—
135	C₂×C₄⋊C₄	—	3	1	no	—
136	C₂×C₄⋊C₄	—	3	1	no	—
137	C₂×C₄²	—	3	1	no	—
138	C₂×C₄²	—	3	1	no	—
139	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
140	C₂×C₁₆	—	3	4	no	—
141	C₂×C₁₆	C₂²×D₃	1	4	yes	—
142	C₄⋊C₈	—	3	2	no	—
143	C₄⋊C₈	—	3	2	no	—
144	C₄⋊C₈	—	3	2	no	—
145	C₄⋊C₈	—	3	2	no	—
146	C₄⋊C₈	—	3	2	no	—
147	C₄⋊C₈	—	3	2	no	—
148	C₄×C₈	—	3	2	no	—
149	C₄×C₈	—	3	2	no	—
150	C₄×C₈	—	3	2	no	—
151	C₄×C₈	—	3	2	no	—
152	C₄×C₈	C₄×D₃	1	1	yes	characteristic
153	C₄×C₈	C₄×D₃	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
156	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
167	C₂×Dic₃	—	2	2	no	—
168	C₂×Dic₃	—	4	1	no	—
169	C₂×Dic₃	—	4	1	no	—
170	C₂×Dic₃	M₃₂	1	4	yes	—
171	C₂×Dic₃	C₂²×C₈	1	2	yes	—
172	C₂×Dic₃	C₂²×C₈	1	2	yes	—
173	C₂×Dic₃	C₂²×C₈	1	2	yes	—
174	C₂×Dic₃	C₂²×C₈	1	2	yes	—
175	Dic₆	—	2	4	no	—
176	Dic₆	—	2	4	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	2	1	no	—
179	C₃×C₈	—	2	1	no	—
180	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
181	C₃⋊C₈	—	2	2	no	—
182	C₃⋊C₈	—	2	2	no	—
183	C₂×Q₈	—	3	2	no	—
184	C₂×Q₈	—	3	2	no	—
185	C₂×Q₈	—	6	4	no	—
186	C₂²×C₄	—	3	1	no	—
187	C₂²×C₄	—	3	1	no	—
188	C₂²×C₄	—	3	1	no	—
189	C₂²×C₄	—	3	1	no	—
190	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
191	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	3	2	no	—
195	C₂×C₈	—	3	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	12	1	no	—
198	C₂×C₈	—	12	1	no	—
199	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	2	no	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	—	3	2	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	—	3	2	no	—
209	C₄²	—	3	2	no	—
210	C₄²	—	3	2	no	—
211	C₄²	—	3	2	no	—
212	C₄²	—	3	2	no	—
213	C₄²	D₃×C₈	1	1	yes	characteristic
214	C₄²	D₃×C₈	1	1	yes	characteristic
215	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
216	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
217	C₁₆	—	2	4	no	—
218	C₁₆	—	6	4	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	C₂×M₃₂	1	2	yes	—
222	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	#519	1	1	yes	characteristic
229	C₃×C₄	#568	1	1	yes	characteristic
230	Dic₃	—	2	2	no	—
231	Dic₃	—	2	2	no	—
232	Dic₃	—	2	2	no	—
233	Dic₃	C₂×M₃₂	1	4	yes	—
234	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
235	Q₈	—	6	4	no	—
236	Q₈	—	6	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	3	4	no	—
241	C₂×C₄	—	3	2	no	—
242	C₂×C₄	—	3	2	no	—
243	C₂×C₄	—	3	2	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	—	12	1	no	—
248	C₂×C₄	—	12	1	no	—
249	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
250	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
251	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
252	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
253	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	6	2	no	—
259	C₈	—	6	2	no	—
260	C₈	#1010	1	2	yes	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	#2982	1	1	yes	characteristic
263	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
264	C₂×C₃	#3407	1	1	yes	characteristic
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	D₃×M₃₂	1	2	yes	—
268	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	3	4	no	—
275	C₄	—	6	2	no	—
276	C₄	—	6	2	no	—
277	C₄	—	6	2	no	—
278	C₄	#5516	1	1	yes	characteristic
279	C₄	#5745	1	1	yes	characteristic
280	C₃	#33497	1	1	yes	characteristic, complemented, Sylow
281	C₂	—	2	2	no	—
282	C₂	#79263	1	1	yes	characteristic
283	C₂	C₂×D₃×M₃₂	1	1	yes	characteristic
284	C₂	#82007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2163	C₂
prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2163	D₃
hypocenter	C₃	#33497
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
4. #69282	C₂	semidirect
7. #69068	C₂	semidirect
280. C₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₆	2
C₃×#2163	2
#69745	2
#69282	2
#69093	2
#69092	2
#69068	2
#33497	3