Subgroups — #10948834

Order 768.

494 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×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#2162	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	#69048	C₂	1	2	yes	maximal, complemented
8	C₁₆×Dic₆	C₂	1	2	yes	maximal, complemented
9	#33498	—	3	1	no	maximal, Sylow
10	C₂×C₄×Dic₆	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×Dic₃⋊C₈	C₄	1	2	yes	—
13	C₂×C₈×Dic₃	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
15	C₈×Dic₆	C₄	1	2	yes	—
16	C₈×Dic₆	C₄	1	1	yes	characteristic
17	C₈×Dic₆	C₄	1	1	yes	characteristic
18	C₈×Dic₆	C₂²	1	2	yes	—
19	C₈×Dic₆	C₂²	1	1	yes	characteristic
20	C₈×Dic₆	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
22	C₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₁₆	C₂²	1	2	yes	—
25	#4549	C₂²	1	2	yes	—
26	#4502	C₂²	1	4	yes	—
27	Dic₃⋊C₁₆	C₂²	1	4	yes	—
28	Dic₃⋊C₁₆	C₂²	1	4	yes	—
29	Dic₃×C₁₆	C₂²	1	4	yes	—
30	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
31	#4461	C₂²	1	2	yes	—
32	C₂×C₈×Q₈	—	3	1	no	—
33	#2183	—	3	4	no	—
34	#2183	—	3	2	no	—
35	Q₈×C₁₆	—	3	2	no	—
36	#2163	—	3	1	no	—
37	#2163	—	3	2	no	—
38	#2162	—	3	2	no	—
39	#2162	D₃	1	1	yes	characteristic
40	C₂²×Dic₆	C₈	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₃	C₈	1	1	yes	characteristic
45	C₂×Dic₃⋊C₄	C₈	1	2	yes	—
46	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
47	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
48	C₄×Dic₆	C₈	1	1	yes	characteristic
49	C₄×Dic₆	C₈	1	2	yes	—
50	C₄×Dic₆	C₈	1	1	yes	characteristic
51	C₄×Dic₆	C₂×C₄	1	2	yes	—
52	C₄×Dic₆	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₆	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₁₆	C₂³	1	2	yes	—
55	C₂×C₃×C₁₆	C₂³	1	2	yes	—
56	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
60	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
61	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
62	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
63	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
64	C₈×Dic₃	C₂×C₄	1	2	yes	—
65	C₈×Dic₃	C₂×C₄	1	2	yes	—
66	C₈×Dic₃	C₂³	1	2	yes	—
67	C₈×Dic₃	C₂³	1	2	yes	—
68	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
69	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
70	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
71	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×Q₈	—	3	1	no	—
73	C₈×Q₈	—	3	2	no	—
74	C₈×Q₈	—	3	1	no	—
75	C₈×Q₈	—	3	1	no	—
76	C₈×Q₈	—	3	1	no	—
77	C₈×Q₈	—	3	2	no	—
78	C₈×Q₈	—	3	1	no	—
79	C₂×C₄⋊C₈	—	3	2	no	—
80	C₂×C₄⋊C₈	—	3	1	no	—
81	C₂×C₄×C₈	—	3	2	no	—
82	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
83	C₄⋊C₁₆	—	3	4	no	—
84	C₄⋊C₁₆	—	3	4	no	—
85	C₄⋊C₁₆	—	3	2	no	—
86	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
87	C₂²⋊C₁₆	—	3	2	no	—
88	C₂²⋊C₁₆	C₂×D₃	1	1	yes	characteristic
89	C₂²⋊C₁₆	C₂×D₃	1	1	yes	characteristic
90	#346	—	3	4	no	—
91	#346	—	3	2	no	—
92	C₄×C₁₆	—	3	4	no	—
93	C₄×C₁₆	C₂×D₃	1	2	yes	—
94	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₂²×Dic₃	C₂×C₈	1	2	yes	—
98	C₂²×Dic₃	C₂×C₈	1	2	yes	—
99	C₂×Dic₆	—	2	4	no	—
100	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
101	C₂×Dic₆	C₂×C₈	1	2	yes	—
102	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	4	1	no	—
105	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
109	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
110	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
111	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
114	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
115	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
116	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
117	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
118	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
119	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
120	C₄×Dic₃	C₂×C₈	1	2	yes	—
121	C₄×Dic₃	C₂×C₈	1	2	yes	—
122	C₄×Dic₃	C₂²×C₄	1	2	yes	—
123	C₄×Dic₃	C₂²×C₄	1	2	yes	—
124	C₂×C₃⋊C₈	—	4	2	no	—
125	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
127	C₃×C₁₆	—	2	2	no	—
128	C₃×C₁₆	—	2	2	no	—
129	C₃⋊C₁₆	—	2	4	no	—
130	C₂²×Q₈	—	3	1	no	—
131	C₂²×C₈	—	3	2	no	—
132	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
134	C₄×Q₈	—	3	1	no	—
135	C₄×Q₈	—	3	2	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×Q₈	—	3	2	no	—
139	C₄×Q₈	—	3	1	no	—
140	C₂×C₄⋊C₄	—	3	2	no	—
141	C₂×C₄⋊C₄	—	3	1	no	—
142	C₂×C₄²	—	3	2	no	—
143	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
144	C₂×C₁₆	—	3	4	no	—
145	C₂×C₁₆	C₂²×D₃	1	2	yes	—
146	C₂×C₁₆	C₂²×D₃	1	2	yes	—
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	1	no	—
152	C₄⋊C₈	—	3	2	no	—
153	C₄⋊C₈	—	3	1	no	—
154	C₄×C₈	—	3	2	no	—
155	C₄×C₈	—	3	2	no	—
156	C₄×C₈	—	3	2	no	—
157	C₄×C₈	—	3	2	no	—
158	C₄×C₈	C₄×D₃	1	1	yes	characteristic
159	C₄×C₈	C₄×D₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
162	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
173	C₂×Dic₃	—	2	4	no	—
174	C₂×Dic₃	—	2	4	no	—
175	C₂×Dic₃	C₂²×C₈	1	2	yes	—
176	C₂×Dic₃	C₂²×C₈	1	2	yes	—
177	C₂×Dic₃	C₂²×C₈	1	2	yes	—
178	C₂×Dic₃	C₂²×C₈	1	2	yes	—
179	Dic₆	—	2	2	no	—
180	Dic₆	—	2	4	no	—
181	Dic₆	M₃₂	1	4	yes	—
182	C₃×C₈	—	2	1	no	—
183	C₃×C₈	—	2	1	no	—
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
186	C₃⋊C₈	—	2	2	no	—
187	C₃⋊C₈	—	2	2	no	—
188	C₂×Q₈	—	3	1	no	—
189	C₂×Q₈	—	3	2	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×Q₈	—	6	4	no	—
192	C₂²×C₄	—	3	2	no	—
193	C₂²×C₄	—	3	2	no	—
194	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
195	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	3	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	12	2	no	—
202	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
206	C₄⋊C₄	—	3	2	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	—	3	2	no	—
209	C₄⋊C₄	—	3	2	no	—
210	C₄⋊C₄	—	3	2	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	1	no	—
213	C₄²	—	3	2	no	—
214	C₄²	—	3	2	no	—
215	C₄²	—	3	2	no	—
216	C₄²	—	3	2	no	—
217	C₄²	D₃×C₈	1	1	yes	characteristic
218	C₄²	D₃×C₈	1	1	yes	characteristic
219	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
220	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	—	6	4	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	#504	1	2	yes	—
227	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	C₂×M₃₂	1	2	yes	—
233	C₃×C₄	#519	1	1	yes	characteristic
234	C₃×C₄	#568	1	1	yes	characteristic
235	Dic₃	—	2	2	no	—
236	Dic₃	—	2	2	no	—
237	Dic₃	—	2	2	no	—
238	Dic₃	C₂×M₃₂	1	4	yes	—
239	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
240	Q₈	—	3	4	no	—
241	Q₈	—	6	4	no	—
242	Q₈	—	6	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	3	2	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	6	4	no	—
252	C₂×C₄	—	6	4	no	—
253	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
254	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
255	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
257	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	6	2	no	—
263	C₈	—	6	2	no	—
264	C₈	#1010	1	2	yes	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	#2982	1	1	yes	characteristic
267	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
268	C₂×C₃	C₂×#504	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	#4901	1	2	yes	—
272	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	3	4	no	—
278	C₄	—	6	2	no	—
279	C₄	—	6	2	no	—
280	C₄	—	6	2	no	—
281	C₄	D₃×M₃₂	1	2	yes	—
282	C₄	#5516	1	1	yes	characteristic
283	C₄	#5745	1	1	yes	characteristic
284	C₃	#33498	1	1	yes	characteristic, complemented, Sylow
285	C₂	—	2	2	no	—
286	C₂	#79263	1	1	yes	characteristic
287	C₂	C₂×#4901	1	1	yes	characteristic
288	C₂	C₂×D₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2162	C₂
prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2162	D₃
hypocenter	C₃	#33498
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
4. #69282	C₂	semidirect
7. #69048	C₂	semidirect
8. C₁₆×Dic₆	C₂	semidirect
284. C₃	#33498	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×#2162	2
#69745	2
#69282	2
#69093	2
#69092	2
#69048	2
C₁₆×Dic₆	2
#33498	3