Subgroups — #10948833

Order 768.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 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	C₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	#70949	C₂	1	1	yes	maximal, characteristic, complemented
4	#69282	C₂	1	2	yes	maximal, complemented
5	#69280	C₂	1	1	yes	maximal, characteristic, complemented
6	#69275	C₂	1	1	yes	maximal, characteristic
7	#69274	C₂	1	2	yes	maximal
8	#69073	C₂	1	2	yes	maximal
9	#69068	C₂	1	2	yes	maximal, complemented
10	#69052	C₂	1	1	yes	maximal, characteristic
11	C₁₆×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#33498	—	3	1	no	maximal, Sylow
13	C₄×D₃×Q₈	C₄	1	1	yes	characteristic
14	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
15	Q₈×C₃⋊C₈	C₄	1	1	yes	characteristic
16	D₃×C₄⋊C₈	C₄	1	1	yes	characteristic
17	D₃×C₄⋊C₈	C₄	1	2	yes	—
18	#4830	C₄	1	2	yes	—
19	#4830	C₄	1	1	yes	characteristic
20	C₄×D₃×C₈	C₂²	1	2	yes	—
21	C₄×D₃×C₈	C₂²	1	1	yes	characteristic
22	C₈×Dic₆	C₂²	1	1	yes	characteristic
23	C₈×Dic₆	C₂²	1	2	yes	—
24	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
25	C₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₃×#346	C₂²	1	2	yes	—
27	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
28	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
29	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
31	#4502	C₂²	1	2	yes	—
32	Dic₃⋊C₁₆	C₂²	1	2	yes	—
33	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
34	Dic₃⋊C₁₆	C₂²	1	2	yes	—
35	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
36	Dic₃×C₁₆	C₂²	1	2	yes	—
37	Dic₃×C₁₆	C₂²	1	2	yes	—
38	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
39	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
40	#4461	C₂²	1	2	yes	—
41	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₂×C₈×Q₈	—	3	1	no	—
43	#2183	—	3	1	no	—
44	#2183	—	3	2	no	—
45	#2183	—	3	2	no	—
46	#2183	D₃	1	1	yes	characteristic, complemented
47	Q₈×C₁₆	—	3	1	no	—
48	Q₈×C₁₆	—	3	1	no	—
49	#2163	—	3	1	no	—
50	#2163	—	3	2	no	—
51	#2162	—	3	2	no	—
52	#2162	—	3	1	no	—
53	C₂×D₃×Q₈	C₈	1	1	yes	characteristic
54	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
55	Q₈×Dic₃	C₈	1	1	yes	characteristic
56	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
57	C₂×D₃×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
59	D₃×C₄⋊C₄	C₈	1	1	yes	characteristic
60	D₃×C₄⋊C₄	C₈	1	2	yes	—
61	#887	C₈	1	2	yes	—
62	#887	C₈	1	1	yes	characteristic
63	C₄²×D₃	C₂×C₄	1	2	yes	—
64	C₄²×D₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₆	C₂×C₄	1	2	yes	—
66	C₄×Dic₆	C₂×C₄	1	1	yes	characteristic
67	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₁₆	C₂³	1	2	yes	—
69	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
71	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂³	1	2	yes	—
74	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
75	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
76	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
77	C₈×Dic₃	C₂×C₄	1	2	yes	—
78	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₃	C₂³	1	2	yes	—
80	C₈×Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
83	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
84	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
85	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
86	C₄×C₃⋊C₈	C₂×C₄	1	2	yes	—
87	C₄×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄×Q₈	—	3	1	no	—
89	C₈×Q₈	—	3	1	no	—
90	C₈×Q₈	—	3	2	no	—
91	C₈×Q₈	—	3	1	no	—
92	C₈×Q₈	—	3	1	no	—
93	C₈×Q₈	—	3	2	no	—
94	C₈×Q₈	C₂×D₃	1	1	yes	characteristic
95	C₂×C₄⋊C₈	—	3	2	no	—
96	C₂×C₄⋊C₈	—	3	1	no	—
97	C₂×C₄×C₈	—	3	2	no	—
98	C₂×C₄×C₈	—	3	1	no	—
99	C₄⋊C₁₆	—	3	1	no	—
100	C₄⋊C₁₆	—	3	1	no	—
101	C₄⋊C₁₆	—	3	2	no	—
102	C₄⋊C₁₆	—	3	2	no	—
103	C₄⋊C₁₆	—	3	2	no	—
104	C₄⋊C₁₆	—	3	1	no	—
105	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
106	C₄⋊C₁₆	C₂×D₃	1	1	yes	characteristic
107	C₂²⋊C₁₆	—	3	1	no	—
108	C₂²⋊C₁₆	—	3	2	no	—
109	C₂²⋊C₁₆	—	3	1	no	—
110	#346	—	3	2	no	—
111	#346	—	3	2	no	—
112	#346	C₂×D₃	1	2	yes	—
113	C₄×C₁₆	—	3	2	no	—
114	C₄×C₁₆	—	3	2	no	—
115	C₄×C₁₆	—	3	1	no	—
116	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
117	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
118	D₃×Q₈	—	2	4	no	—
119	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
120	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
121	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
122	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
123	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₆	C₂×C₈	1	2	yes	—
125	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
128	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
130	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
131	C₃×C₄²	C₂²×C₄	1	2	yes	—
132	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
133	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
134	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
135	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
136	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
137	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
138	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
139	C₄×Dic₃	C₂×C₈	1	2	yes	—
140	C₄×Dic₃	C₂×C₈	1	2	yes	—
141	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
142	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
143	C₄×Dic₃	C₂²×C₄	1	2	yes	—
144	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
146	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
147	D₃×C₈	—	2	1	no	—
148	D₃×C₈	—	2	1	no	—
149	D₃×C₈	—	4	2	no	—
150	D₃×C₈	—	4	1	no	—
151	C₃×C₁₆	—	2	1	no	—
152	C₃×C₁₆	—	2	1	no	—
153	C₃×C₁₆	—	2	2	no	—
154	C₃⋊C₁₆	—	2	1	no	—
155	C₃⋊C₁₆	—	2	2	no	—
156	C₃⋊C₁₆	—	2	1	no	—
157	C₂²×Q₈	—	3	1	no	—
158	C₂²×C₈	—	3	1	no	—
159	C₂²×C₈	—	3	2	no	—
160	C₂²×C₈	—	3	1	no	—
161	C₄×Q₈	—	3	1	no	—
162	C₄×Q₈	—	3	1	no	—
163	C₄×Q₈	—	3	2	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×Q₈	—	3	2	no	—
166	C₄×Q₈	C₄×D₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	3	2	no	—
168	C₂×C₄⋊C₄	—	3	1	no	—
169	C₂×C₄²	—	3	2	no	—
170	C₂×C₄²	—	3	1	no	—
171	C₂×C₁₆	—	3	1	no	—
172	C₂×C₁₆	—	3	2	no	—
173	C₂×C₁₆	—	3	1	no	—
174	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂²×D₃	1	2	yes	—
176	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
177	C₄⋊C₈	—	3	2	no	—
178	C₄⋊C₈	—	3	2	no	—
179	C₄⋊C₈	—	3	2	no	—
180	C₄⋊C₈	—	3	1	no	—
181	C₄⋊C₈	—	3	2	no	—
182	C₄⋊C₈	C₄×D₃	1	2	yes	—
183	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
184	C₄×C₈	—	3	2	no	—
185	C₄×C₈	—	3	1	no	—
186	C₄×C₈	—	3	2	no	—
187	C₄×C₈	—	3	1	no	—
188	C₄×C₈	—	3	2	no	—
189	C₄×C₈	—	3	1	no	—
190	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₃	1	2	yes	—
192	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
193	C₃×Q₈	—	2	2	no	—
194	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
196	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
197	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
199	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
200	C₂×Dic₃	C₂²×C₈	1	2	yes	—
201	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
202	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
203	C₂×Dic₃	C₂²×C₈	1	2	yes	—
204	C₄×D₃	—	2	4	no	—
205	C₄×D₃	—	2	2	no	—
206	C₄×D₃	—	2	1	no	—
207	C₄×D₃	—	2	1	no	—
208	C₄×D₃	—	2	2	no	—
209	C₄×D₃	—	2	4	no	—
210	Dic₆	—	2	4	no	—
211	Dic₆	M₃₂	1	4	yes	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	2₋¹⁺⁴	1	1	yes	characteristic
215	C₃×C₈	2₋¹⁺⁴	1	1	yes	characteristic
216	C₃⋊C₈	—	2	1	no	—
217	C₃⋊C₈	—	2	2	no	—
218	C₃⋊C₈	—	2	1	no	—
219	C₂×Q₈	—	3	1	no	—
220	C₂×Q₈	—	3	2	no	—
221	C₂×Q₈	—	6	4	no	—
222	C₂×Q₈	D₃×C₈	1	1	yes	characteristic
223	C₂²×C₄	—	3	2	no	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	—	3	2	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	3	1	no	—
228	C₂×C₈	—	3	1	no	—
229	C₂×C₈	—	3	2	no	—
230	C₂×C₈	—	3	1	no	—
231	C₂×C₈	—	6	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	—	12	1	no	—
234	C₂×C₈	—	12	2	no	—
235	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
237	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
238	C₄⋊C₄	—	3	2	no	—
239	C₄⋊C₄	—	3	2	no	—
240	C₄⋊C₄	—	3	2	no	—
241	C₄⋊C₄	—	3	2	no	—
242	C₄⋊C₄	—	3	1	no	—
243	C₄⋊C₄	D₃×C₈	1	2	yes	—
244	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
245	C₄²	—	3	2	no	—
246	C₄²	—	3	2	no	—
247	C₄²	—	3	1	no	—
248	C₄²	—	3	1	no	—
249	C₄²	—	3	2	no	—
250	C₄²	—	3	1	no	—
251	C₄²	C₂×C₄×D₃	1	2	yes	—
252	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
253	C₁₆	—	2	1	no	—
254	C₁₆	—	2	1	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	6	1	no	—
257	C₁₆	—	6	2	no	—
258	C₁₆	—	6	1	no	—
259	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
260	C₂×D₃	—	2	1	no	—
261	C₂×D₃	—	2	1	no	—
262	C₂×D₃	#504	1	2	yes	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	C₂×M₃₂	1	2	yes	—
267	C₃×C₄	#519	1	1	yes	characteristic
268	C₃×C₄	#568	1	1	yes	characteristic
269	Dic₃	—	2	1	no	—
270	Dic₃	—	2	2	no	—
271	Dic₃	—	2	1	no	—
272	Dic₃	—	2	1	no	—
273	Dic₃	C₂×M₃₂	1	4	yes	—
274	C₂³	—	3	1	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	3	4	no	—
277	Q₈	—	6	4	no	—
278	C₂×C₄	—	3	2	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	3	2	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	3	1	no	—
283	C₂×C₄	—	6	4	no	—
284	C₂×C₄	—	6	4	no	—
285	C₂×C₄	—	6	2	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	—	6	1	no	—
288	C₂×C₄	—	6	1	no	—
289	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
290	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
291	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
292	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	2	no	—
296	C₈	—	6	2	no	—
297	C₈	—	6	1	no	—
298	C₈	—	6	1	no	—
299	C₈	#1007	1	1	yes	characteristic
300	C₈	#1010	1	1	yes	characteristic
301	C₂×C₃	#2982	1	1	yes	characteristic
302	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
303	C₂×C₃	C₂×#504	1	1	yes	characteristic
304	D₃	—	2	2	no	—
305	C₂²	—	3	2	no	—
306	C₂²	—	6	1	no	—
307	C₂²	—	6	1	no	—
308	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	3	4	no	—
313	C₄	—	6	1	no	—
314	C₄	—	6	1	no	—
315	C₄	—	6	2	no	—
316	C₄	—	6	1	no	—
317	C₄	D₃×M₃₂	1	2	yes	—
318	C₄	#5572	1	1	yes	characteristic
319	C₄	#5750	1	1	yes	characteristic
320	C₃	#33498	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	6	2	no	—
322	C₂	#79920	1	1	yes	characteristic
323	C₂	C₂×D₃×M₃₂	1	1	yes	characteristic
324	C₂	D₃×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2183	C₂
prime radical	C₂²×D₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2183	D₃
hypocenter	C₃	#33498
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. C₃×#2183	C₂	semidirect
3. #70949	C₂	semidirect
4. #69282	C₂	semidirect
5. #69280	C₂	semidirect
9. #69068	C₂	semidirect
11. C₁₆×Dic₆	C₂	semidirect
46. #2183	D₃	semidirect
320. C₃	#33498	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
C₃×#2183	2
#70949	2
#69282	2
#69280	2
#69275	2
#69274	2
#69073	2
#69068	2
#69052	2
C₁₆×Dic₆	2
#33498	3