Subgroups — #10712936

Order 768.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#71437	C₂	1	1	yes	maximal, characteristic
2	C₂×#4508	C₂	1	1	yes	maximal, characteristic
3	#71399	C₂	1	1	yes	maximal, characteristic, complemented
4	#69898	C₂	1	1	yes	maximal, characteristic
5	#69885	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	#67814	C₂	1	1	yes	maximal, characteristic, complemented
8	#13123	—	3	1	no	maximal, Sylow
9	C₂×#901	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×#405	C₂²	1	1	yes	characteristic
13	#5132	C₄	1	1	yes	characteristic
14	#5101	C₂²	1	1	yes	characteristic
15	C₂×#812	C₂²	1	1	yes	characteristic
16	C₂×#812	C₂²	1	1	yes	characteristic
17	#5006	C₄	1	1	yes	characteristic
18	#4559	C₄	1	1	yes	characteristic
19	#4533	C₄	1	1	yes	characteristic
20	#4508	—	2	1	no	—
21	#4508	—	2	1	no	—
22	#4508	—	2	2	no	—
23	#4502	—	2	1	no	—
24	Dic₃×C₁₆	—	2	1	no	—
25	#4463	—	2	2	no	—
26	#2153	—	3	1	no	—
27	#2117	—	3	1	no	—
28	C₂×#350	—	3	1	no	—
29	#2108	—	3	1	no	—
30	#1757	—	3	1	no	—
31	#1379	D₃	1	1	yes	characteristic, complemented
32	#1378	—	3	1	no	—
33	C₂×#235	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
36	C₃×#117	C₂×C₄	1	1	yes	characteristic
37	#926	C₂×C₄	1	1	yes	characteristic
38	C₂×#208	C₂×C₄	1	1	yes	characteristic
39	C₂×#208	C₂×C₄	1	1	yes	characteristic
40	#901	—	2	1	no	—
41	#901	—	2	1	no	—
42	#901	—	2	1	no	—
43	#901	—	2	1	no	—
44	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
45	C₂×D₃×C₈	—	2	1	no	—
46	C₃×M₃₂	—	2	1	no	—
47	C₃×M₃₂	—	2	1	no	—
48	C₂×C₃×C₁₆	—	2	1	no	—
49	C₂×C₃×C₁₆	—	2	1	no	—
50	C₂×C₃×C₁₆	—	2	1	no	—
51	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
52	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
53	C₃×#97	—	2	1	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	#824	C₈	1	2	yes	—
56	#823	C₈	1	2	yes	—
57	#815	—	2	1	no	—
58	C₈×Dic₃	—	2	1	no	—
59	#812	—	2	1	no	—
60	#812	—	2	1	no	—
61	#812	—	2	1	no	—
62	#812	D₄	1	2	yes	—
63	C₂×C₃⋊C₁₆	—	2	1	no	—
64	C₂×C₃⋊C₁₆	—	2	1	no	—
65	#806	—	2	2	no	—
66	C₂×#131	—	3	1	no	—
67	C₂×M₃₂	—	3	1	no	—
68	C₂×M₃₂	—	3	1	no	—
69	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
71	#413	—	3	1	no	—
72	#410	—	3	1	no	—
73	#405	—	3	1	no	—
74	#405	C₂×D₃	1	1	yes	characteristic
75	#364	—	6	2	no	—
76	#350	—	6	2	no	—
77	#350	—	6	1	no	—
78	#350	—	6	1	no	—
79	#346	—	6	1	no	—
80	C₄×C₁₆	—	6	1	no	—
81	#344	—	3	1	no	—
82	#344	—	3	1	no	—
83	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×#67	—	2	1	no	—
85	#235	—	2	1	no	—
86	#235	—	2	1	no	—
87	#235	—	2	1	no	—
88	#235	—	2	1	no	—
89	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×D₃	—	2	1	no	—
91	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
92	C₃×M₁₆	—	2	1	no	—
93	C₃×M₁₆	C₂×C₈	1	2	yes	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₈	#41	1	1	yes	characteristic
98	C₂×C₃×C₈	#41	1	1	yes	characteristic
99	C₂×C₃×C₈	#41	1	1	yes	characteristic
100	C₂×C₃×C₈	#41	1	1	yes	characteristic
101	C₃×C₄⋊C₄	—	2	1	no	—
102	C₃×C₂²⋊C₄	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	#217	M₁₆	1	2	yes	—
105	C₄⋊Dic₃	—	2	1	no	—
106	C₄×Dic₃	—	2	1	no	—
107	#208	—	2	1	no	—
108	#208	—	2	1	no	—
109	#208	—	2	1	no	—
110	#208	C₂×C₈	1	2	yes	—
111	C₂×C₃⋊C₈	—	2	1	no	—
112	C₂×C₃⋊C₈	—	2	1	no	—
113	ΓL(1,𝔽₂₅)	—	4	1	no	—
114	ΓL(1,𝔽₂₅)	—	4	1	no	—
115	D₃×C₈	—	4	1	no	—
116	D₃×C₈	—	4	1	no	—
117	C₃×C₁₆	—	2	1	no	—
118	C₃×C₁₆	—	2	1	no	—
119	C₃×C₁₆	—	4	1	no	—
120	C₃⋊C₁₆	—	2	2	no	—
121	C₃⋊C₁₆	—	4	1	no	—
122	C₂×#41	—	3	1	no	—
123	#131	—	6	1	no	—
124	#131	—	6	1	no	—
125	#131	—	6	1	no	—
126	#131	—	6	1	no	—
127	C₂×M₁₆	—	3	1	no	—
128	C₂×M₁₆	—	3	1	no	—
129	C₂×M₁₆	—	6	1	no	—
130	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
131	C₂²×C₈	—	6	1	no	—
132	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
133	#117	—	3	1	no	—
134	#117	C₄×D₃	1	1	yes	characteristic
135	M₃₂	—	2	1	no	—
136	M₃₂	—	2	1	no	—
137	M₃₂	—	3	2	no	—
138	M₃₂	—	6	1	no	—
139	M₃₂	—	6	1	no	—
140	M₃₂	—	6	1	no	—
141	C₂×C₁₆	—	2	1	no	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	6	1	no	—
145	C₂×C₁₆	—	6	1	no	—
146	C₂×C₁₆	D₁₂	1	1	yes	characteristic
147	C₂×C₁₆	D₁₂	1	1	yes	characteristic
148	#101	—	3	2	no	—
149	#100	—	3	2	no	—
150	#99	—	6	2	no	—
151	#97	—	2	1	no	—
152	#97	—	6	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	6	1	no	—
155	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
156	C₂²×D₃	—	2	1	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	#117	1	1	yes	characteristic
159	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₃×C₄	#131	1	1	yes	characteristic
163	C₂×C₃×C₄	#131	1	1	yes	characteristic
164	#67	—	4	1	no	—
165	#67	—	4	1	no	—
166	C₂×Dic₃	—	2	1	no	—
167	C₂×Dic₃	—	2	2	no	—
168	D₁₂	—	2	1	no	—
169	D₁₂	—	2	1	no	—
170	C₄×D₃	—	4	1	no	—
171	C₄×D₃	—	4	1	no	—
172	Dic₆	—	2	1	no	—
173	Dic₆	—	2	1	no	—
174	C₃×C₈	—	2	2	no	—
175	C₃×C₈	#121	1	1	yes	characteristic
176	C₃×C₈	#123	1	1	yes	characteristic
177	C₃×C₈	#124	1	1	yes	characteristic
178	C₃×C₈	#126	1	1	yes	characteristic
179	C₃⋊C₈	—	2	2	no	—
180	C₃⋊C₈	—	4	1	no	—
181	#41	—	6	1	no	—
182	#41	—	6	1	no	—
183	#41	—	6	1	no	—
184	#41	—	6	1	no	—
185	C₂×Q₈	—	3	1	no	—
186	C₂×D₄	—	3	1	no	—
187	C₂×D₄	—	6	1	no	—
188	C₂²×C₄	—	6	1	no	—
189	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
190	M₁₆	—	2	1	no	—
191	M₁₆	—	3	2	no	—
192	M₁₆	—	6	1	no	—
193	M₁₆	—	6	1	no	—
194	M₁₆	—	6	1	no	—
195	M₁₆	—	12	1	no	—
196	M₁₆	—	12	1	no	—
197	M₁₆	D₃×C₈	1	2	yes	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	6	1	no	—
200	C₂×C₈	—	6	1	no	—
201	C₂×C₈	—	12	1	no	—
202	C₂×C₈	—	12	1	no	—
203	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
204	C₂×C₈	#235	1	1	yes	characteristic
205	C₂×C₈	D₃×D₄	1	1	yes	characteristic
206	C₂×C₈	#237	1	1	yes	characteristic
207	C₂×C₈	#237	1	1	yes	characteristic
208	C₂×C₈	#239	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	6	1	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	3	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	6	1	no	—
215	C₁₆	—	2	1	no	—
216	C₁₆	—	2	1	no	—
217	C₁₆	—	4	1	no	—
218	C₁₆	—	6	2	no	—
219	C₁₆	—	12	1	no	—
220	C₂²×C₃	—	4	1	no	—
221	C₂²×C₃	#431	1	1	yes	characteristic
222	C₂²×C₃	#435	1	1	yes	characteristic
223	C₂²×C₃	#436	1	1	yes	characteristic
224	C₂×D₃	—	4	1	no	—
225	C₂×D₃	—	8	1	no	—
226	C₃×C₄	—	8	1	no	—
227	C₃×C₄	#388	1	1	yes	characteristic
228	C₃×C₄	#433	1	1	yes	characteristic
229	C₃×C₄	C₈×D₄	1	1	yes	characteristic
230	C₃×C₄	C₈×D₄	1	1	yes	characteristic
231	Dic₃	—	4	2	no	—
232	Dic₃	—	4	1	no	—
233	C₂³	—	6	1	no	—
234	C₂³	D₃×M₁₆	1	1	yes	characteristic
235	Q₈	—	6	1	no	—
236	Q₈	—	6	1	no	—
237	D₄	—	6	1	no	—
238	D₄	—	6	1	no	—
239	D₄	—	12	1	no	—
240	D₄	—	12	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	—	6	1	no	—
244	C₂×C₄	—	12	1	no	—
245	C₂×C₄	—	12	1	no	—
246	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
247	C₂×C₄	#67⋊C₄	1	1	yes	characteristic
248	C₂×C₄	#892	1	1	yes	characteristic
249	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
250	C₂×C₄	#901	1	1	yes	characteristic
251	C₂×C₄	#907	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	6	2	no	—
254	C₈	—	12	1	no	—
255	C₈	#885	1	1	yes	characteristic
256	C₈	#886	1	1	yes	characteristic
257	C₈	#895	1	1	yes	characteristic
258	C₈	#898	1	1	yes	characteristic
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	#1923	1	1	yes	characteristic
263	D₃	—	8	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	12	1	no	—
266	C₂²	—	24	1	no	—
267	C₂²	#4706	1	1	yes	characteristic
268	C₂²	#4729	1	1	yes	characteristic
269	C₂²	#4833	1	1	yes	characteristic
270	C₄	—	8	1	no	—
271	C₄	—	12	2	no	—
272	C₄	—	12	1	no	—
273	C₄	#4670	1	1	yes	characteristic
274	C₄	C₈×D₁₂	1	1	yes	characteristic
275	C₄	#67⋊C₈	1	1	yes	characteristic
276	C₄	#4840	1	1	yes	characteristic
277	C₃	#13123	1	1	yes	characteristic, complemented, Sylow
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	24	1	no	—
282	C₂	#68615	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#895
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1379	C₂
socle	C₂×C₃	#1923
prime radical	C₂×#235	C₂×C₄
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1379	D₃
hypocenter	C₃	#13123
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
3. #71399	C₂	semidirect
5. #69885	C₂	semidirect
6. C₃×#1379	C₂	semidirect
7. #67814	C₂	semidirect
31. #1379	D₃	semidirect
277. C₃	#13123	semidirect

Maximal subgroups

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

subgroup	index
#71437	2
C₂×#4508	2
#71399	2
#69898	2
#69885	2
C₃×#1379	2
#67814	2
#13123	3