Subgroups — #10948751

Order 768.

516 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78448	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#71433	C₂	1	2	yes	maximal, complemented
4	#71430	C₂	1	2	yes	maximal, complemented
5	#69744	C₂	1	1	yes	maximal, characteristic, complemented
6	#69074	C₂	1	2	yes	maximal, complemented
7	#69072	C₂	1	2	yes	maximal, complemented
8	#69055	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#69048	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₆×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#33483	—	3	1	no	maximal, Sylow
13	#5479	C₄	1	1	yes	characteristic
14	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	#5111	C₄	1	2	yes	—
18	C₈×#67	C₂²	1	2	yes	—
19	#4926	C₄	1	1	yes	characteristic
20	C₂×#798	C₂²	1	2	yes	—
21	C₂×D₃×C₁₆	C₂²	1	2	yes	—
22	#4690	C₄	1	2	yes	—
23	#4689	C₂²	1	2	yes	—
24	#4688	C₄	1	1	yes	characteristic
25	C₈×D₁₂	C₂²	1	1	yes	characteristic
26	#4679	C₄	1	1	yes	characteristic
27	C₈×Dic₆	C₂²	1	1	yes	characteristic
28	C₃×#346	C₂²	1	1	yes	characteristic
29	C₃×#346	C₂²	1	1	yes	characteristic
30	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#4549	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
35	#4502	C₂²	1	2	yes	—
36	Dic₃⋊C₁₆	C₂²	1	2	yes	—
37	Dic₃⋊C₁₆	C₂²	1	2	yes	—
38	Dic₃×C₁₆	C₂²	1	2	yes	—
39	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
40	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
42	#2966	—	3	1	no	—
43	#2183	—	3	1	no	—
44	Q₈×C₁₆	—	3	1	no	—
45	#2169	—	3	2	no	—
46	#2169	—	3	1	no	—
47	#2168	—	3	2	no	—
48	D₄×C₁₆	—	3	1	no	—
49	#2164	—	3	2	no	—
50	#2162	—	3	1	no	—
51	#2108	—	3	2	no	—
52	C₄×M₃₂	D₃	1	1	yes	characteristic, complemented
53	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
56	#930	C₈	1	1	yes	characteristic
57	#929	C₈	1	2	yes	—
58	C₄×#67	C₂×C₄	1	2	yes	—
59	#924	C₈	1	1	yes	characteristic
60	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
61	C₂×D₃×C₈	C₂³	1	2	yes	—
62	#876	C₈	1	2	yes	—
63	#875	C₈	1	1	yes	characteristic
64	C₄×D₁₂	C₂×C₄	1	1	yes	characteristic
65	#872	C₂×C₄	1	2	yes	—
66	#870	C₈	1	1	yes	characteristic
67	C₄×Dic₆	C₂×C₄	1	1	yes	characteristic
68	C₃×M₃₂	—	2	2	no	—
69	C₃×M₃₂	—	2	2	no	—
70	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
77	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
78	#830	C₂×C₄	1	2	yes	—
79	(C₂×D₃)⋊C₈	C₂×C₄	1	2	yes	—
80	(C₂×D₃)⋊C₈	C₂×C₄	1	2	yes	—
81	#815	C₂×C₄	1	2	yes	—
82	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
83	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
84	C₈×Dic₃	C₂³	1	2	yes	—
85	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
86	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
87	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
88	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
89	#803	C₂×C₄	1	2	yes	—
90	#798	—	2	4	no	—
91	D₃×C₁₆	—	2	4	no	—
92	#529	—	3	1	no	—
93	C₂×M₃₂	—	3	2	no	—
94	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
95	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
96	C₂²×C₁₆	—	3	2	no	—
97	#446	—	3	1	no	—
98	C₈×Q₈	—	3	1	no	—
99	#436	—	3	1	no	—
100	#435	—	3	2	no	—
101	C₈×D₄	—	3	2	no	—
102	C₈×D₄	—	3	1	no	—
103	#433	—	3	2	no	—
104	#432	—	3	1	no	—
105	#405	—	3	2	no	—
106	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
107	C₄⋊C₁₆	—	3	2	no	—
108	C₄⋊C₁₆	—	3	2	no	—
109	C₄⋊C₁₆	—	3	1	no	—
110	C₄⋊C₁₆	—	3	1	no	—
111	C₂²⋊C₁₆	—	3	2	no	—
112	C₂²⋊C₁₆	—	3	2	no	—
113	C₂²⋊C₁₆	—	3	2	no	—
114	#346	—	3	2	no	—
115	#346	C₂×D₃	1	1	yes	characteristic
116	#346	C₂×D₃	1	1	yes	characteristic
117	C₄×C₁₆	—	3	2	no	—
118	C₄×C₁₆	—	3	2	no	—
119	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
120	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
124	C₂×#67	C₂×C₈	1	2	yes	—
125	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
126	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
127	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
135	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
136	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
137	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
138	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
139	#217	C₂×C₈	1	2	yes	—
140	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
141	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
142	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
143	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
144	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
145	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
146	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
147	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
148	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
149	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
150	C₄×Dic₃	C₂²×C₄	1	2	yes	—
151	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
152	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
153	ΓL(1,𝔽₂₅)	—	2	4	no	—
154	D₃×C₈	—	2	2	no	—
155	D₃×C₈	—	2	2	no	—
156	C₃×C₁₆	—	2	1	no	—
157	C₃×C₁₆	—	2	1	no	—
158	C₃×C₁₆	#41	1	2	yes	—
159	C₃×C₁₆	#41	1	2	yes	—
160	C₃⋊C₁₆	—	2	2	no	—
161	C₃⋊C₁₆	—	2	2	no	—
162	C₂×M₁₆	—	3	2	no	—
163	C₂²×C₈	—	3	2	no	—
164	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
166	#126	—	3	2	no	—
167	#125	—	3	1	no	—
168	#124	—	3	1	no	—
169	#123	—	3	2	no	—
170	C₂²⋊Q₈	—	3	1	no	—
171	#121	—	3	1	no	—
172	C₄×Q₈	—	3	1	no	—
173	C₄×D₄	—	3	2	no	—
174	C₄×D₄	—	3	1	no	—
175	#117	—	3	2	no	—
176	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
177	M₃₂	—	2	2	no	—
178	M₃₂	—	2	2	no	—
179	M₃₂	—	6	4	no	—
180	C₂×C₁₆	—	3	2	no	—
181	C₂×C₁₆	—	3	2	no	—
182	C₂×C₁₆	—	6	4	no	—
183	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
185	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
187	C₄⋊C₈	—	3	2	no	—
188	C₄⋊C₈	—	3	2	no	—
189	C₄⋊C₈	—	3	1	no	—
190	C₄⋊C₈	—	3	1	no	—
191	C₂²⋊C₈	—	3	2	no	—
192	C₂²⋊C₈	—	3	2	no	—
193	C₂²⋊C₈	—	3	2	no	—
194	#97	—	3	2	no	—
195	#97	—	3	2	no	—
196	C₄×C₈	—	3	2	no	—
197	C₄×C₈	C₄×D₃	1	1	yes	characteristic
198	C₄×C₈	C₄×D₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
200	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
201	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
202	C₂²×D₃	C₂²×C₈	1	2	yes	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
213	#67	—	4	2	no	—
214	C₂×Dic₃	C₂²×C₈	1	2	yes	—
215	C₂×Dic₃	C₂²×C₈	1	2	yes	—
216	C₂×Dic₃	C₂²×C₈	1	2	yes	—
217	D₁₂	—	2	2	no	—
218	C₄×D₃	—	2	2	no	—
219	C₄×D₃	—	2	2	no	—
220	Dic₆	—	2	2	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	#131	1	2	yes	—
224	C₃×C₈	C₂×#41	1	1	yes	characteristic
225	C₃×C₈	C₂×#41	1	1	yes	characteristic
226	C₃⋊C₈	—	2	2	no	—
227	C₃⋊C₈	—	2	2	no	—
228	C₂×Q₈	—	3	1	no	—
229	C₂×D₄	—	3	2	no	—
230	C₂×D₄	—	3	1	no	—
231	C₂²×C₄	—	3	2	no	—
232	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
233	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
235	M₁₆	—	6	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	3	2	no	—
240	C₂×C₈	—	3	2	no	—
241	C₂×C₈	—	6	2	no	—
242	C₂×C₈	—	6	2	no	—
243	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
246	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
247	C₄⋊C₄	—	3	2	no	—
248	C₄⋊C₄	—	3	2	no	—
249	C₄⋊C₄	—	3	1	no	—
250	C₄⋊C₄	—	3	2	no	—
251	C₄⋊C₄	—	3	1	no	—
252	C₄⋊C₄	—	3	2	no	—
253	C₂²⋊C₄	—	3	2	no	—
254	C₂²⋊C₄	—	3	2	no	—
255	C₂²⋊C₄	—	3	2	no	—
256	C₂²⋊C₄	—	3	2	no	—
257	C₂²⋊C₄	—	3	2	no	—
258	C₄²	—	3	2	no	—
259	C₄²	D₃×C₈	1	1	yes	characteristic
260	C₄²	D₃×C₈	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
263	C₁₆	—	2	1	no	—
264	C₁₆	—	2	1	no	—
265	C₁₆	—	6	2	no	—
266	C₁₆	—	6	2	no	—
267	C₁₆	#235	1	2	yes	—
268	C₁₆	#235	1	2	yes	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
273	C₂×D₃	—	2	2	no	—
274	C₂×D₃	—	2	2	no	—
275	C₂×D₃	—	2	2	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	#504	1	2	yes	—
281	C₃×C₄	C₄×#41	1	1	yes	characteristic
282	C₃×C₄	C₂×#131	1	1	yes	characteristic
283	Dic₃	—	2	2	no	—
284	Dic₃	—	2	2	no	—
285	Dic₃	—	2	2	no	—
286	C₂³	—	3	2	no	—
287	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
288	Q₈	—	6	2	no	—
289	D₄	—	6	2	no	—
290	D₄	—	12	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	3	2	no	—
295	C₂×C₄	—	3	2	no	—
296	C₂×C₄	—	3	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	6	2	no	—
300	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
303	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
304	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	6	2	no	—
309	C₈	—	6	2	no	—
310	C₈	#901	1	2	yes	—
311	C₈	C₂×#235	1	1	yes	characteristic
312	C₈	C₂×#235	1	1	yes	characteristic
313	C₂×C₃	—	4	1	no	—
314	C₂×C₃	C₈×#41	1	1	yes	characteristic
315	C₂×C₃	C₂×#504	1	1	yes	characteristic
316	C₂×C₃	#3407	1	1	yes	characteristic
317	D₃	—	4	2	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	6	2	no	—
322	C₂²	—	6	2	no	—
323	C₂²	—	6	2	no	—
324	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	6	2	no	—
330	C₄	—	6	2	no	—
331	C₄	—	6	2	no	—
332	C₄	#4907	1	2	yes	—
333	C₄	C₄×#235	1	1	yes	characteristic
334	C₄	C₂×#901	1	1	yes	characteristic
335	C₃	#33483	1	1	yes	characteristic, complemented, Sylow
336	C₂	—	4	1	no	—
337	C₂	—	12	2	no	—
338	C₂	C₈×#235	1	1	yes	characteristic
339	C₂	C₂×#4907	1	1	yes	characteristic
340	C₂	#82007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×C₄×M₃₂	C₂
prime radical	#930	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₄×M₃₂	D₃
hypocenter	C₃	#33483
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×M₃₂	C₂	semidirect
3. #71433	C₂	semidirect
4. #71430	C₂	semidirect
5. #69744	C₂	semidirect
6. #69074	C₂	semidirect
7. #69072	C₂	semidirect
8. #69055	C₂	semidirect
9. C₁₆×D₁₂	C₂	semidirect
10. #69048	C₂	semidirect
11. C₁₆×Dic₆	C₂	semidirect
52. C₄×M₃₂	D₃	semidirect
335. C₃	#33483	semidirect

Maximal subgroups

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

subgroup	index
#78448	2
C₃×C₄×M₃₂	2
#71433	2
#71430	2
#69744	2
#69074	2
#69072	2
#69055	2
C₁₆×D₁₂	2
#69048	2
C₁₆×Dic₆	2
#33483	3