Subgroups — #10686191

Order 768.

452 conjugacy classes of subgroups.

The subgroup lattice has 296 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×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#70773	C₂	1	1	yes	maximal, characteristic
4	#69859	C₂	1	1	yes	maximal, characteristic
5	C₈×#208	C₂	1	1	yes	maximal, characteristic, complemented
6	#67743	C₂	1	1	yes	maximal, characteristic, complemented
7	#67669	C₂	1	1	yes	maximal, characteristic, complemented
8	#8241	—	3	1	no	maximal, Sylow
9	C₂×#901	C₄	1	2	yes	—
10	#5477	C₂²	1	1	yes	characteristic
11	C₃×#431	C₂²	1	1	yes	characteristic
12	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	#5110	C₄	1	2	yes	—
15	#4925	C₂²	1	1	yes	characteristic
16	#4924	C₂²	1	1	yes	characteristic
17	C₄×#208	C₂²	1	1	yes	characteristic
18	#4688	—	2	1	no	—
19	#4688	—	2	1	no	—
20	C₈×D₁₂	—	2	1	no	—
21	C₈×D₁₂	—	2	1	no	—
22	C₃×#322	—	2	1	no	—
23	C₃×C₈²	—	2	1	no	—
24	#4485	C₄	1	2	yes	—
25	#4481	C₄	1	2	yes	—
26	#4464	C₄	1	4	yes	—
27	#4454	—	2	1	no	—
28	C₈×C₃⋊C₈	—	2	1	no	—
29	#2931	—	3	1	no	—
30	#1543	—	3	1	no	—
31	#1525	—	3	1	no	—
32	C₈×M₁₆	—	3	1	no	—
33	C₈×M₁₆	D₃	1	1	yes	characteristic, complemented
34	#1299	—	3	1	no	—
35	#1299	—	3	1	no	—
36	C₂×#235	C₂×C₄	1	2	yes	—
37	C₂×C₃×M₁₆	—	2	1	no	—
38	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
41	#930	—	2	1	no	—
42	#930	—	2	1	no	—
43	C₂×#208	—	2	1	no	—
44	C₂×#208	C₂×C₄	1	2	yes	—
45	#901	—	4	4	no	—
46	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
47	C₂×D₃×C₈	—	2	2	no	—
48	#875	C₂×C₄	1	2	yes	—
49	#874	C₂×C₄	1	1	yes	characteristic
50	C₄×D₁₂	—	2	1	no	—
51	C₄×D₁₂	—	2	1	no	—
52	#869	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₃×C₂²⋊C₈	—	2	1	no	—
55	C₃×#97	C₂×C₄	1	2	yes	—
56	C₃×C₄×C₈	—	2	1	no	—
57	C₃×C₄×C₈	—	2	1	no	—
58	C₃×C₄×C₈	D₄	1	1	yes	characteristic
59	C₃×C₄×C₈	D₄	1	1	yes	characteristic
60	C₃×C₄×C₈	D₄	1	1	yes	characteristic
61	C₃×C₄×C₈	D₄	1	1	yes	characteristic
62	#830	—	2	1	no	—
63	(C₂×D₃)⋊C₈	—	2	2	no	—
64	(C₂×D₃)⋊C₈	—	2	2	no	—
65	C₄⋊(C₃⋊C₈)	—	2	1	no	—
66	C₄⋊(C₃⋊C₈)	—	2	1	no	—
67	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
68	#803	C₂×C₄	1	2	yes	—
69	C₄×C₃⋊C₈	—	2	1	no	—
70	C₄×C₃⋊C₈	—	2	1	no	—
71	C₂×#131	—	3	2	no	—
72	#532	—	3	1	no	—
73	#436	—	6	1	no	—
74	#436	—	6	1	no	—
75	C₈×D₄	—	6	1	no	—
76	C₈×D₄	—	6	1	no	—
77	#431	—	3	1	no	—
78	#431	C₂×D₃	1	1	yes	characteristic
79	#423	—	3	1	no	—
80	#408	—	3	2	no	—
81	C₄×M₁₆	—	3	1	no	—
82	C₄×M₁₆	C₂×D₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
84	#333	—	3	2	no	—
85	#331	—	3	2	no	—
86	#329	—	3	4	no	—
87	#322	—	2	1	no	—
88	#322	—	6	1	no	—
89	C₈²	—	2	1	no	—
90	C₈²	—	6	1	no	—
91	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	#41	1	1	yes	characteristic
93	C₂²×C₃×C₄	#41	1	1	yes	characteristic
94	C₂×#67	—	2	2	no	—
95	#235	—	4	2	no	—
96	#235	—	4	2	no	—
97	C₂×D₁₂	—	2	1	no	—
98	C₂×D₁₂	—	2	1	no	—
99	C₂×D₁₂	C₄²	1	2	yes	—
100	C₂×C₄×D₃	—	2	2	no	—
101	C₂×Dic₆	C₄²	1	2	yes	—
102	C₃×M₁₆	—	2	4	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	4	1	no	—
107	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
114	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
115	(C₂×D₃)⋊C₄	—	2	2	no	—
116	(C₂×D₃)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₃	—	2	1	no	—
118	C₄⋊Dic₃	—	2	1	no	—
119	#208	—	2	4	no	—
120	#208	—	4	2	no	—
121	C₂×C₃⋊C₈	—	2	2	no	—
122	C₂×C₃⋊C₈	—	2	2	no	—
123	C₂×C₃⋊C₈	—	2	2	no	—
124	ΓL(1,𝔽₂₅)	—	4	4	no	—
125	D₃×C₈	—	4	4	no	—
126	C₂×#41	—	3	2	no	—
127	#131	—	12	4	no	—
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	3	2	no	—
130	C₂×M₁₆	—	6	1	no	—
131	C₂×M₁₆	—	6	2	no	—
132	C₂²×C₈	—	6	2	no	—
133	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
134	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	3	1	no	—
136	#127	—	3	1	no	—
137	#124	—	3	2	no	—
138	#121	—	6	1	no	—
139	#121	—	6	1	no	—
140	C₄×D₄	—	6	1	no	—
141	C₄×D₄	—	6	1	no	—
142	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
143	C₄⋊C₈	—	3	2	no	—
144	C₄⋊C₈	—	6	1	no	—
145	C₄⋊C₈	—	6	1	no	—
146	C₄⋊C₈	C₄×D₃	1	2	yes	—
147	C₂²⋊C₈	—	2	1	no	—
148	C₂²⋊C₈	—	6	1	no	—
149	C₂²⋊C₈	—	6	2	no	—
150	C₂²⋊C₈	—	6	2	no	—
151	#97	—	3	2	no	—
152	#97	C₄×D₃	1	2	yes	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	6	1	no	—
156	C₄×C₈	—	6	1	no	—
157	C₄×C₈	D₁₂	1	1	yes	characteristic
158	C₄×C₈	D₁₂	1	1	yes	characteristic
159	C₄×C₈	#67	1	1	yes	characteristic
160	C₄×C₈	#67	1	1	yes	characteristic
161	C₂³×C₃	#117	1	1	yes	characteristic
162	C₂²×D₃	—	2	2	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
173	#67	—	4	4	no	—
174	C₂×Dic₃	—	2	2	no	—
175	D₁₂	—	4	2	no	—
176	D₁₂	—	4	2	no	—
177	D₁₂	—	4	2	no	—
178	C₄×D₃	—	4	2	no	—
179	C₄×D₃	—	4	2	no	—
180	Dic₆	—	4	2	no	—
181	C₃×C₈	—	2	4	no	—
182	C₃×C₈	—	2	1	no	—
183	C₃×C₈	—	2	1	no	—
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	—	4	2	no	—
186	C₃×C₈	#100	1	2	yes	—
187	C₃⋊C₈	—	2	4	no	—
188	C₃⋊C₈	—	4	2	no	—
189	C₃⋊C₈	—	4	2	no	—
190	#41	—	12	2	no	—
191	#41	—	12	2	no	—
192	C₂×Q₈	—	3	2	no	—
193	C₂×D₄	—	3	2	no	—
194	C₂×D₄	—	6	2	no	—
195	C₂×D₄	—	6	1	no	—
196	C₂×D₄	—	6	1	no	—
197	C₂²×C₄	—	6	2	no	—
198	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂²×C₄	#235	1	1	yes	characteristic
200	C₂²×C₄	#235	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	6	4	no	—
203	M₁₆	—	12	4	no	—
204	M₁₆	—	12	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	6	2	no	—
210	C₂×C₈	—	6	2	no	—
211	C₂×C₈	—	6	2	no	—
212	C₂×C₈	—	12	4	no	—
213	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
215	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
216	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
217	C₄⋊C₄	—	6	1	no	—
218	C₄⋊C₄	—	6	1	no	—
219	C₂²⋊C₄	—	6	2	no	—
220	C₂²⋊C₄	—	6	2	no	—
221	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
222	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
223	C₄²	C₂×D₁₂	1	1	yes	characteristic
224	C₄²	C₂×#67	1	1	yes	characteristic
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×D₃	—	4	2	no	—
230	C₂×D₃	—	4	2	no	—
231	C₂×D₃	—	4	2	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	C₂×#100	1	1	yes	characteristic
236	C₃×C₄	#413	1	1	yes	characteristic
237	C₃×C₄	#443	1	2	yes	—
238	C₃×C₄	#443	1	2	yes	—
239	Dic₃	—	4	2	no	—
240	C₂³	—	6	2	no	—
241	C₂³	#872	1	1	yes	characteristic
242	Q₈	—	12	2	no	—
243	D₄	—	12	2	no	—
244	D₄	—	12	4	no	—
245	D₄	—	12	2	no	—
246	D₄	—	12	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	6	2	no	—
252	C₂×C₄	—	12	2	no	—
253	C₂×C₄	—	12	2	no	—
254	C₂×C₄	C₄²×D₃	1	1	yes	characteristic
255	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
256	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
257	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₄×#67	1	1	yes	characteristic
259	C₂×C₄	C₄×#67	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	4	2	no	—
265	C₈	—	6	4	no	—
266	C₈	—	12	2	no	—
267	C₈	—	12	2	no	—
268	C₈	#823	1	2	yes	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	C₄×#100	1	1	yes	characteristic
271	C₂×C₃	#1764	1	1	yes	characteristic
272	C₂×C₃	#1764	1	1	yes	characteristic
273	D₃	—	8	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	12	2	no	—
278	C₂²	—	12	2	no	—
279	C₂²	—	12	2	no	—
280	C₂²	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	12	2	no	—
285	C₄	#4700	1	2	yes	—
286	C₄	C₂×#823	1	1	yes	characteristic
287	C₄	#5132	1	1	yes	characteristic
288	C₄	#5144	1	2	yes	—
289	C₃	#8241	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	4	1	no	—
291	C₂	—	24	2	no	—
292	C₂	#69789	1	1	yes	characteristic
293	C₂	C₄×#823	1	1	yes	characteristic
294	C₂	#69887	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₈×M₁₆	C₂
socle	C₂²×C₃	C₄×C₂²⋊C₄
prime radical	#5477	C₂²
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₈×M₁₆	D₃
hypocenter	C₃	#8241
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×M₁₆	C₂	semidirect
5. C₈×#208	C₂	semidirect
6. #67743	C₂	semidirect
7. #67669	C₂	semidirect
33. C₈×M₁₆	D₃	semidirect
289. C₃	#8241	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×C₈×M₁₆	2
#70773	2
#69859	2
C₈×#208	2
#67743	2
#67669	2
#8241	3