Subgroups — #10686284

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78458	C₂	1	1	yes	maximal, characteristic
2	C₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	#70431	C₂	1	1	yes	maximal, characteristic
4	#69724	C₂	1	1	yes	maximal, characteristic, complemented
5	#69469	C₂	1	1	yes	maximal, characteristic
6	#67707	C₂	1	1	yes	maximal, characteristic, complemented
7	#67620	C₂	1	1	yes	maximal, characteristic, complemented
8	#8255	—	3	1	no	maximal, Sylow
9	C₂×#901	—	2	1	no	—
10	#5477	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×#97	—	2	1	no	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#5110	—	2	1	no	—
16	#4925	C₂²	1	1	yes	characteristic
17	#4925	C₂²	1	1	yes	characteristic
18	#4924	C₂²	1	1	yes	characteristic
19	#4688	—	2	1	no	—
20	#4688	C₄	1	2	yes	complemented
21	C₈×D₁₂	—	2	1	no	—
22	C₈×D₁₂	C₄	1	2	yes	complemented
23	C₃×#336	—	2	1	no	—
24	#4485	C₄	1	2	yes	complemented
25	#4482	C₄	1	2	yes	complemented
26	#4455	—	2	1	no	—
27	C₈×C₃⋊C₈	—	2	1	no	—
28	#4451	C₄	1	4	yes	—
29	#2931	—	3	1	no	—
30	#1770	D₃	1	1	yes	characteristic, complemented
31	#1540	—	3	1	no	—
32	#1502	—	3	1	no	—
33	#1450	—	3	1	no	—
34	#1303	—	3	1	no	—
35	#1301	—	3	1	no	—
36	C₂×#235	—	2	1	no	—
37	C₂²×C₃×C₈	—	2	1	no	—
38	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₃×C₄²	—	2	1	no	—
42	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
43	#930	—	2	1	no	—
44	#930	C₂×C₄	1	2	yes	—
45	C₂×#208	—	2	1	no	—
46	#901	—	4	2	no	—
47	#901	—	4	2	no	—
48	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
49	C₂×D₃×C₈	—	2	2	no	—
50	#875	—	2	1	no	—
51	#874	C₂×C₄	1	1	yes	characteristic
52	C₄×D₁₂	—	2	1	no	—
53	C₄×D₁₂	C₂×C₄	1	2	yes	—
54	#869	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	1	no	—
56	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₃×#97	—	4	2	no	—
58	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₃×C₄×C₈	D₄	1	1	yes	characteristic
61	C₃×C₄×C₈	D₄	1	1	yes	characteristic
62	C₃×#95	C₂×C₄	1	2	yes	—
63	#830	—	2	1	no	—
64	#830	C₂×C₄	1	2	yes	—
65	(C₂×D₃)⋊C₈	—	2	2	no	—
66	(C₂×D₃)⋊C₈	—	2	2	no	—
67	C₄⋊(C₃⋊C₈)	—	2	1	no	—
68	C₄⋊(C₃⋊C₈)	—	2	1	no	—
69	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
70	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
71	C₄×C₃⋊C₈	—	2	1	no	—
72	C₄×C₃⋊C₈	—	2	1	no	—
73	C₂×#131	—	6	1	no	—
74	#532	—	3	1	no	—
75	#436	—	3	2	no	—
76	#436	—	6	1	no	—
77	C₈×D₄	—	3	2	no	—
78	C₈×D₄	—	6	1	no	—
79	#431	—	3	1	no	—
80	#431	—	3	1	no	—
81	#423	—	3	1	no	—
82	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
83	#408	—	6	1	no	—
84	C₂×#97	—	2	1	no	—
85	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
86	C₄×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
87	#336	—	2	1	no	—
88	#332	—	3	2	no	—
89	#331	—	3	2	no	—
90	#327	—	3	4	no	—
91	#322	—	6	1	no	—
92	C₈²	—	6	1	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₃×C₄	#41	1	1	yes	characteristic
97	C₂×#67	—	2	2	no	—
98	#235	—	4	1	no	—
99	#235	—	4	1	no	—
100	#235	—	4	1	no	—
101	#235	—	4	1	no	—
102	C₂×D₁₂	—	2	1	no	—
103	C₂×D₁₂	—	2	1	no	—
104	C₂×D₁₂	C₄²	1	2	yes	—
105	C₂×C₄×D₃	—	2	2	no	—
106	C₂×Dic₆	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	4	1	no	—
111	C₂×C₃×C₈	—	4	1	no	—
112	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₈	D₈	1	2	yes	—
117	C₂×C₃×C₈	SD₁₆	1	2	yes	—
118	C₃×C₄⋊C₄	—	4	2	no	—
119	C₃×C₄²	—	4	1	no	—
120	C₃×C₄²	—	4	1	no	—
121	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
123	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
124	C₃×C₄²	#41	1	1	yes	characteristic
125	(C₂×D₃)⋊C₄	—	2	2	no	—
126	(C₂×D₃)⋊C₄	—	2	2	no	—
127	C₄⋊Dic₃	—	2	1	no	—
128	C₄⋊Dic₃	C₄²	1	2	yes	—
129	#208	—	4	1	no	—
130	#208	—	4	1	no	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	—	2	2	no	—
133	C₂×C₃⋊C₈	—	2	2	no	—
134	ΓL(1,𝔽₂₅)	—	4	4	no	—
135	D₃×C₈	—	4	4	no	—
136	C₂×#41	—	6	1	no	—
137	#131	—	12	2	no	—
138	#131	—	12	2	no	—
139	C₂×M₁₆	—	6	1	no	—
140	C₂×M₁₆	—	6	2	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	6	2	no	—
143	C₂²×C₈	#67	1	1	yes	characteristic
144	C₂²×C₈	#67	1	1	yes	characteristic
145	C₄⋊Q₈	—	3	1	no	—
146	#127	—	3	1	no	—
147	#124	—	6	1	no	—
148	#121	—	3	2	no	—
149	#121	—	6	1	no	—
150	C₄×D₄	—	3	2	no	—
151	C₄×D₄	—	6	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	3	2	no	—
157	C₄⋊C₈	—	3	2	no	—
158	C₄⋊C₈	—	6	1	no	—
159	C₄⋊C₈	—	6	1	no	—
160	C₄⋊C₈	C₄×D₃	1	2	yes	—
161	C₂²⋊C₈	—	3	2	no	—
162	C₂²⋊C₈	—	6	1	no	—
163	C₂²⋊C₈	—	6	2	no	—
164	C₂²⋊C₈	—	6	2	no	—
165	#97	—	4	2	no	—
166	C₄×C₈	—	6	1	no	—
167	C₄×C₈	—	6	1	no	—
168	C₄×C₈	C₄×D₃	1	1	yes	characteristic
169	C₄×C₈	C₄×D₃	1	1	yes	characteristic
170	C₄×C₈	D₁₂	1	1	yes	characteristic
171	C₄×C₈	D₁₂	1	1	yes	characteristic
172	#95	C₄×D₃	1	2	yes	—
173	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
174	C₂²×D₃	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	4	1	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	#117	1	1	yes	characteristic
183	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
188	C₂×C₃×C₄	#135	1	1	yes	characteristic
189	C₂×C₃×C₄	#135	1	1	yes	characteristic
190	#67	—	4	2	no	—
191	#67	—	4	2	no	—
192	C₂×Dic₃	—	2	2	no	—
193	D₁₂	—	4	2	no	—
194	D₁₂	—	4	1	no	—
195	D₁₂	—	4	1	no	—
196	D₁₂	—	4	2	no	—
197	C₄×D₃	—	4	2	no	—
198	C₄×D₃	—	4	2	no	—
199	Dic₆	—	4	1	no	—
200	Dic₆	—	4	1	no	—
201	C₃×C₈	—	2	1	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	4	2	no	—
204	C₃×C₈	#102	1	2	yes	—
205	C₃×C₈	#102	1	2	yes	—
206	C₃⋊C₈	—	4	2	no	—
207	C₃⋊C₈	—	4	2	no	—
208	C₃⋊C₈	—	4	2	no	—
209	#41	—	12	1	no	—
210	#41	—	12	1	no	—
211	#41	—	12	1	no	—
212	#41	—	12	1	no	—
213	C₂×Q₈	—	6	1	no	—
214	C₂×D₄	—	3	2	no	—
215	C₂×D₄	—	6	2	no	—
216	C₂×D₄	—	6	1	no	—
217	C₂×D₄	—	6	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	6	2	no	—
220	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
221	C₂²×C₄	#235	1	1	yes	characteristic
222	C₂²×C₄	C₂×#67	1	1	yes	characteristic
223	M₁₆	—	12	1	no	—
224	M₁₆	—	12	1	no	—
225	M₁₆	—	12	4	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	6	2	no	—
232	C₂×C₈	—	6	2	no	—
233	C₂×C₈	—	6	2	no	—
234	C₂×C₈	—	12	4	no	—
235	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
237	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
238	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₂×C₈	#213	1	2	yes	—
240	C₂×C₈	#215	1	2	yes	—
241	C₄⋊C₄	—	3	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	6	1	no	—
244	C₂²⋊C₄	—	6	2	no	—
245	C₂²⋊C₄	—	6	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
250	C₄²	C₂×D₁₂	1	1	yes	characteristic
251	C₄²	#235	1	1	yes	characteristic
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
255	C₂²×C₃	C₄×D₈	1	1	yes	characteristic
256	C₂²×C₃	C₄×SD₁₆	1	1	yes	characteristic
257	C₂×D₃	—	4	2	no	—
258	C₂×D₃	—	4	2	no	—
259	C₂×D₃	—	4	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	8	2	no	—
264	C₃×C₄	C₂×#102	1	1	yes	characteristic
265	C₃×C₄	#416	1	1	yes	characteristic
266	C₃×C₄	C₄×D₈	1	1	yes	characteristic
267	C₃×C₄	C₄×SD₁₆	1	1	yes	characteristic
268	C₃×C₄	#444	1	2	yes	—
269	Dic₃	—	4	2	no	—
270	C₂³	—	6	2	no	—
271	C₂³	C₄×#67	1	1	yes	characteristic
272	Q₈	—	12	1	no	—
273	Q₈	—	12	1	no	—
274	D₄	—	12	2	no	—
275	D₄	—	12	2	no	—
276	D₄	—	12	2	no	—
277	D₄	—	12	2	no	—
278	D₄	—	12	1	no	—
279	D₄	—	12	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	6	2	no	—
286	C₂×C₄	—	12	2	no	—
287	C₂×C₄	—	12	2	no	—
288	C₂×C₄	C₄²×D₃	1	1	yes	characteristic
289	C₂×C₄	#872	1	1	yes	characteristic
290	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
291	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
292	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₄×#67	1	1	yes	characteristic
294	C₂×C₄	C₂×#213	1	1	yes	characteristic
295	C₂×C₄	C₂×#215	1	1	yes	characteristic
296	C₂×C₄	#950	1	1	yes	characteristic
297	C₂×C₄	#950	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	2	no	—
301	C₈	—	12	2	no	—
302	C₈	—	12	2	no	—
303	C₈	—	12	2	no	—
304	C₈	#809	1	2	yes	—
305	C₈	#809	1	2	yes	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	#1757	1	1	yes	characteristic
309	C₂×C₃	C₄×#102	1	1	yes	characteristic
310	C₂×C₃	#1765	1	1	yes	characteristic
311	D₃	—	8	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	12	2	no	—
315	C₂²	—	12	2	no	—
316	C₂²	—	12	2	no	—
317	C₂²	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
318	C₂²	C₄×#213	1	1	yes	characteristic
319	C₂²	C₄×#215	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	12	2	no	—
325	C₄	#4717	1	2	yes	—
326	C₄	C₂×#809	1	1	yes	characteristic
327	C₄	#5002	1	1	yes	characteristic
328	C₄	C₄×#213	1	1	yes	characteristic
329	C₄	C₄×#215	1	1	yes	characteristic
330	C₃	#8255	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	24	2	no	—
334	C₂	C₄×#809	1	1	yes	characteristic
335	C₂	#69839	1	1	yes	characteristic
336	C₂	#69898	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1770	C₂
socle	C₂²×C₃	C₄×C₂²⋊C₄
prime radical	#5477	C₂²
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1770	D₃
hypocenter	C₃	#8255
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. C₃×#1770	C₂	semidirect
4. #69724	C₂	semidirect
6. #67707	C₂	semidirect
7. #67620	C₂	semidirect
20. #4688	C₄	semidirect
22. C₈×D₁₂	C₄	semidirect
24. #4485	C₄	semidirect
25. #4482	C₄	semidirect
30. #1770	D₃	semidirect
330. C₃	#8255	semidirect

Maximal subgroups

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

subgroup	index
#78458	2
C₃×#1770	2
#70431	2
#69724	2
#69469	2
#67707	2
#67620	2
#8255	3