Subgroups — #10948838

Order 768.

452 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79917	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₁₆	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	#69281	C₂	1	1	yes	maximal, characteristic, complemented
6	#69276	C₂	1	2	yes	maximal
7	#69275	C₂	1	1	yes	maximal, characteristic
8	#69053	C₂	1	2	yes	maximal
9	#69052	C₂	1	1	yes	maximal, characteristic
10	#69048	C₂	1	2	yes	maximal, complemented
11	C₁₆×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#33499	—	3	1	no	maximal, Sylow
13	#5578	C₄	1	1	yes	characteristic
14	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
15	#5024	C₄	1	1	yes	characteristic
16	#4839	C₄	1	2	yes	—
17	#4837	C₄	1	1	yes	characteristic
18	#4831	C₄	1	1	yes	characteristic
19	#4828	C₄	1	2	yes	—
20	#4689	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	1	yes	characteristic
25	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
27	C₃×C₄×C₁₆	C₂²	1	2	yes	—
28	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
29	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
30	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
31	#4502	C₂²	1	2	yes	—
32	#4502	C₂²	1	2	yes	—
33	Dic₃⋊C₁₆	C₂²	1	2	yes	—
34	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
35	Dic₃⋊C₁₆	C₂²	1	2	yes	—
36	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
37	Dic₃×C₁₆	C₂²	1	2	yes	—
38	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
40	#4461	C₂²	1	2	yes	—
41	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
42	#2960	—	3	1	no	—
43	#2183	—	3	1	no	—
44	#2183	—	3	1	no	—
45	#2183	—	3	2	no	—
46	#2183	—	3	2	no	—
47	Q₈×C₁₆	—	3	1	no	—
48	Q₈×C₁₆	D₃	1	1	yes	characteristic, complemented
49	#2164	—	3	2	no	—
50	#2164	—	3	2	no	—
51	#2163	—	3	1	no	—
52	#2162	—	3	1	no	—
53	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
54	#946	C₈	1	1	yes	characteristic
55	#944	C₈	1	1	yes	characteristic
56	C₂×ΓL(1,𝔽₂₅)	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	#897	C₈	1	1	yes	characteristic
60	#896	C₈	1	2	yes	—
61	#889	C₈	1	2	yes	—
62	#888	C₈	1	1	yes	characteristic
63	#872	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	#815	C₂×C₄	1	2	yes	—
75	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
76	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
77	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
78	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₃	C₂³	1	1	yes	characteristic
80	C₈×Dic₃	C₂³	1	2	yes	—
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	#803	C₂×C₄	1	2	yes	—
87	C₄×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
88	#533	—	3	1	no	—
89	#446	—	3	1	no	—
90	#446	—	3	2	no	—
91	#446	—	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	#424	—	3	2	no	—
96	#423	—	3	1	no	—
97	#405	—	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	1	no	—
104	C₄⋊C₁₆	—	3	2	no	—
105	C₄⋊C₁₆	C₂×D₃	1	1	yes	characteristic
106	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
107	C₂²⋊C₁₆	—	3	1	no	—
108	C₂²⋊C₁₆	—	3	1	no	—
109	C₂²⋊C₁₆	—	3	2	no	—
110	#346	—	3	2	no	—
111	#346	—	3	2	no	—
112	#346	—	3	2	no	—
113	C₄×C₁₆	—	3	2	no	—
114	C₄×C₁₆	—	3	1	no	—
115	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
116	C₄×C₁₆	C₂×D₃	1	2	yes	—
117	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
118	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
119	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
120	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
122	C₂×Dic₆	C₂×C₈	1	2	yes	—
123	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
126	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
127	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
128	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
129	C₃×C₄²	C₂²×C₄	1	2	yes	—
130	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
131	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
132	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
133	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
134	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
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	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
139	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
140	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
141	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
142	C₄×Dic₃	C₂²×C₄	1	2	yes	—
143	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
146	ΓL(1,𝔽₂₅)	—	4	2	no	—
147	D₃×C₈	—	2	1	no	—
148	D₃×C₈	—	2	1	no	—
149	D₃×C₈	—	4	1	no	—
150	C₃×C₁₆	—	2	1	no	—
151	C₃×C₁₆	—	2	1	no	—
152	C₃×C₁₆	—	2	2	no	—
153	C₃⋊C₁₆	—	2	1	no	—
154	C₃⋊C₁₆	—	2	2	no	—
155	C₃⋊C₁₆	—	2	1	no	—
156	C₂×M₁₆	—	3	2	no	—
157	C₂²×C₈	—	3	1	no	—
158	C₂²×C₈	—	3	1	no	—
159	C₄⋊Q₈	—	3	1	no	—
160	C₄⋊Q₈	—	3	1	no	—
161	#125	—	3	2	no	—
162	C₂²⋊Q₈	—	3	1	no	—
163	C₂²⋊Q₈	—	3	2	no	—
164	C₂²⋊Q₈	—	3	1	no	—
165	C₄×Q₈	—	3	2	no	—
166	C₄×Q₈	—	3	1	no	—
167	C₄×Q₈	C₄×D₃	1	1	yes	characteristic
168	#117	—	3	2	no	—
169	C₂×C₄²	—	3	1	no	—
170	C₂×C₁₆	—	3	1	no	—
171	C₂×C₁₆	—	3	2	no	—
172	C₂×C₁₆	—	3	1	no	—
173	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
174	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂²×D₃	1	2	yes	—
176	C₄⋊C₈	—	3	2	no	—
177	C₄⋊C₈	—	3	2	no	—
178	C₄⋊C₈	—	3	2	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₄⋊C₈	—	3	2	no	—
181	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
182	C₄⋊C₈	C₄×D₃	1	2	yes	—
183	#97	—	3	2	no	—
184	#97	—	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	1	no	—
189	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₃	1	2	yes	—
191	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
192	C₃×Q₈	—	2	2	no	—
193	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
195	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
196	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
198	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
199	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
200	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
201	C₂×Dic₃	C₂²×C₈	1	2	yes	—
202	C₂×Dic₃	C₂²×C₈	1	2	yes	—
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	2	2	no	—
206	C₄×D₃	—	4	1	no	—
207	Dic₆	—	2	2	no	—
208	Dic₆	—	4	2	no	—
209	C₃×C₈	—	2	1	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	2₋¹⁺⁴	1	1	yes	characteristic
212	C₃×C₈	2₋¹⁺⁴	1	1	yes	characteristic
213	C₃⋊C₈	—	2	1	no	—
214	C₃⋊C₈	—	2	1	no	—
215	C₃⋊C₈	—	2	2	no	—
216	C₂×Q₈	—	3	1	no	—
217	C₂×Q₈	—	3	2	no	—
218	C₂×Q₈	D₃×C₈	1	1	yes	characteristic
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	3	1	no	—
221	C₂²×C₄	—	3	1	no	—
222	M₁₆	—	12	2	no	—
223	C₂×C₈	—	3	1	no	—
224	C₂×C₈	—	3	1	no	—
225	C₂×C₈	—	3	2	no	—
226	C₂×C₈	—	6	1	no	—
227	C₂×C₈	—	6	1	no	—
228	C₂×C₈	—	12	1	no	—
229	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
231	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
232	C₄⋊C₄	—	3	2	no	—
233	C₄⋊C₄	—	3	2	no	—
234	C₄⋊C₄	—	3	2	no	—
235	C₄⋊C₄	—	3	2	no	—
236	C₄⋊C₄	—	3	2	no	—
237	C₄⋊C₄	—	3	2	no	—
238	C₄⋊C₄	—	3	1	no	—
239	C₄⋊C₄	D₃×C₈	1	2	yes	—
240	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
241	C₂²⋊C₄	—	3	2	no	—
242	C₂²⋊C₄	—	3	2	no	—
243	C₄²	—	3	1	no	—
244	C₄²	—	3	1	no	—
245	C₄²	—	3	1	no	—
246	C₄²	—	3	2	no	—
247	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₃	1	2	yes	—
249	C₁₆	—	2	1	no	—
250	C₁₆	—	2	1	no	—
251	C₁₆	—	2	2	no	—
252	C₁₆	—	6	1	no	—
253	C₁₆	—	6	2	no	—
254	C₁₆	—	6	1	no	—
255	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	—	2	1	no	—
258	C₂×D₃	—	2	1	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	#504	1	2	yes	—
263	C₃×C₄	#519	1	1	yes	characteristic
264	C₃×C₄	#568	1	1	yes	characteristic
265	Dic₃	—	2	2	no	—
266	Dic₃	—	2	1	no	—
267	Dic₃	—	2	1	no	—
268	Dic₃	—	2	2	no	—
269	Dic₃	—	2	1	no	—
270	C₂³	—	3	1	no	—
271	Q₈	—	2	2	no	—
272	Q₈	—	6	2	no	—
273	Q₈	—	12	2	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	2	no	—
278	C₂×C₄	—	3	2	no	—
279	C₂×C₄	—	6	2	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	12	1	no	—
283	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
284	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
285	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
286	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	2	2	no	—
290	C₈	—	6	1	no	—
291	C₈	—	6	2	no	—
292	C₈	—	6	1	no	—
293	C₈	#1007	1	1	yes	characteristic
294	C₈	#1010	1	1	yes	characteristic
295	C₂×C₃	#2982	1	1	yes	characteristic
296	C₂×C₃	C₂×#504	1	1	yes	characteristic
297	C₂×C₃	#3407	1	1	yes	characteristic
298	D₃	—	4	1	no	—
299	C₂²	—	6	1	no	—
300	C₂²	—	6	1	no	—
301	C₂²	—	6	1	no	—
302	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	6	1	no	—
307	C₄	—	6	1	no	—
308	C₄	—	6	2	no	—
309	C₄	—	6	1	no	—
310	C₄	—	6	2	no	—
311	C₄	#4901	1	2	yes	—
312	C₄	#5572	1	1	yes	characteristic
313	C₄	#5750	1	1	yes	characteristic
314	C₃	#33499	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	12	1	no	—
316	C₂	#79920	1	1	yes	characteristic
317	C₂	C₂×#4901	1	1	yes	characteristic
318	C₂	#82012	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×Q₈×C₁₆	C₂
prime radical	C₂²×D₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	Q₈×C₁₆	D₃
hypocenter	C₃	#33499
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₁₆	C₂	semidirect
3. #70949	C₂	semidirect
4. #69282	C₂	semidirect
5. #69281	C₂	semidirect
10. #69048	C₂	semidirect
11. C₁₆×Dic₆	C₂	semidirect
48. Q₈×C₁₆	D₃	semidirect
314. C₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#79917	2
C₃×Q₈×C₁₆	2
#70949	2
#69282	2
#69281	2
#69276	2
#69275	2
#69053	2
#69052	2
#69048	2
C₁₆×Dic₆	2
#33499	3