Subgroups — #10694119

Order 768.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#71624	C₂	1	2	yes	maximal
3	#71621	C₂	1	2	yes	maximal
4	#69729	C₂	1	1	yes	maximal, characteristic
5	#69724	C₂	1	1	yes	maximal, characteristic
6	#9669	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
9	C₃×C₄×Q₁₆	—	2	1	no	—
10	C₃×C₄×Q₁₆	—	2	1	no	—
11	C₃×C₄×Q₁₆	—	2	1	no	—
12	C₃×C₄×Q₁₆	—	2	1	no	—
13	C₂×C₃×#107	C₄	1	1	yes	characteristic
14	C₂×C₃×#103	C₄	1	2	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#4925	C₂²	1	2	yes	—
17	#4925	C₂²	1	1	yes	characteristic
18	#4925	C₂²	1	1	yes	characteristic
19	#4535	—	2	4	no	—
20	#4528	—	2	4	no	—
21	#4458	—	2	1	no	—
22	#4457	—	2	1	no	—
23	#4455	—	2	1	no	—
24	C₈×C₃⋊C₈	—	2	1	no	—
25	C₂×C₄×Q₁₆	D₃	1	1	yes	characteristic
26	#1564	—	3	1	no	—
27	#1479	—	3	2	no	—
28	#1461	—	3	2	no	—
29	#1450	—	3	1	no	—
30	C₂²×C₃×Q₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×Q₁₆	—	2	1	no	—
32	C₂×C₃×Q₁₆	—	2	1	no	—
33	C₂×C₃×Q₁₆	—	2	2	no	—
34	C₂×C₃×Q₁₆	C₈	1	4	yes	complemented
35	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Q₈	—	2	2	no	—
38	C₃×C₄×Q₈	—	2	2	no	—
39	C₃×C₄×Q₈	—	2	2	no	—
40	C₃×C₄×Q₈	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₄²	—	2	2	no	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₃×#107	—	2	1	no	—
46	C₃×#107	—	2	1	no	—
47	C₃×#103	—	2	2	no	—
48	C₃×#103	—	2	2	no	—
49	C₃×C₄×C₈	D₄	1	1	yes	characteristic
50	C₃×C₄×C₈	D₄	1	1	yes	characteristic
51	C₃×C₄×C₈	D₄	1	1	yes	characteristic
52	C₃×C₄×C₈	D₄	1	1	yes	characteristic
53	#830	—	2	1	no	—
54	#830	—	2	2	no	—
55	#830	D₄	1	2	yes	—
56	C₄⋊(C₃⋊C₈)	—	2	1	no	—
57	C₄⋊(C₃⋊C₈)	—	2	2	no	—
58	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
59	C₄×C₃⋊C₈	—	2	1	no	—
60	C₄×C₃⋊C₈	—	2	1	no	—
61	C₄×C₃⋊C₈	—	2	2	no	—
62	C₂²×Q₁₆	Dic₃	1	1	yes	characteristic
63	C₂×C₄×Q₈	C₂×D₃	1	2	yes	—
64	C₄×Q₁₆	—	2	1	no	—
65	C₄×Q₁₆	—	2	1	no	—
66	C₄×Q₁₆	—	2	1	no	—
67	C₄×Q₁₆	—	2	1	no	—
68	#431	—	3	1	no	—
69	#431	—	3	1	no	—
70	#431	—	3	2	no	—
71	C₂×#107	Dic₃	1	1	yes	characteristic
72	C₂×#103	Dic₃	1	2	yes	—
73	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	#335	—	6	1	no	—
75	#334	—	6	1	no	—
76	Q₈⋊C₈	—	6	4	no	—
77	#324	—	6	4	no	—
78	#322	—	6	1	no	—
79	C₈²	—	6	1	no	—
80	C₂×C₃×Q₈	—	2	2	no	—
81	C₂×C₃×Q₈	—	2	2	no	—
82	C₂×C₃×Q₈	—	2	2	no	—
83	C₂×C₃×Q₈	—	2	2	no	—
84	C₂×C₃×Q₈	—	2	2	no	—
85	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₃×C₄	#41	1	1	yes	characteristic
91	C₃×Q₁₆	—	2	4	no	—
92	C₃×Q₁₆	—	2	4	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
100	C₂×C₃×C₈	M₁₆	1	2	yes	—
101	C₃×C₄⋊C₄	—	2	2	no	—
102	C₃×C₄⋊C₄	—	2	2	no	—
103	C₃×C₄⋊C₄	—	4	2	no	—
104	C₃×C₄⋊C₄	—	4	2	no	—
105	C₃×C₄²	—	4	2	no	—
106	C₃×C₄²	—	4	2	no	—
107	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
109	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
110	C₃×C₄²	#41	1	1	yes	characteristic
111	C₂×C₃⋊C₈	—	2	2	no	—
112	C₂×C₃⋊C₈	—	2	2	no	—
113	C₂×C₃⋊C₈	—	4	2	no	—
114	C₂²×Q₈	C₂×Dic₃	1	2	yes	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	1	no	—
117	C₂×Q₁₆	—	2	1	no	—
118	C₂×Q₁₆	C₃⋊C₈	1	4	yes	complemented
119	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
121	C₄×Q₈	—	2	2	no	—
122	C₄×Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
129	#107	—	2	1	no	—
130	#107	—	2	1	no	—
131	C₄⋊C₈	—	3	2	no	—
132	C₄⋊C₈	—	6	1	no	—
133	C₄⋊C₈	—	6	2	no	—
134	#103	—	2	2	no	—
135	#103	—	2	2	no	—
136	C₂²⋊C₈	—	3	2	no	—
137	C₂²⋊C₈	—	6	1	no	—
138	C₂²⋊C₈	—	6	2	no	—
139	C₄×C₈	—	6	2	no	—
140	C₄×C₈	—	6	1	no	—
141	C₄×C₈	—	6	1	no	—
142	C₄×C₈	#67	1	1	yes	characteristic
143	C₄×C₈	#67	1	1	yes	characteristic
144	C₄×C₈	#67	1	1	yes	characteristic
145	C₄×C₈	#67	1	1	yes	characteristic
146	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
147	C₃×Q₈	—	2	4	no	—
148	C₃×Q₈	—	2	4	no	—
149	C₃×Q₈	—	4	2	no	—
150	C₃×Q₈	—	4	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	4	1	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	8	2	no	—
158	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	#121	1	1	yes	characteristic
161	C₂×C₃×C₄	#121	1	1	yes	characteristic
162	C₂×C₃×C₄	#124	1	1	yes	characteristic
163	C₂×C₃×C₄	#127	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
166	C₂×C₃×C₄	#131	1	1	yes	characteristic
167	C₂×C₃×C₄	#131	1	1	yes	characteristic
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	C₂²⋊C₈	1	2	yes	—
171	C₃×C₈	C₂²⋊C₈	1	2	yes	—
172	C₃⋊C₈	—	4	2	no	—
173	C₃⋊C₈	—	4	2	no	—
174	C₃⋊C₈	—	8	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
184	C₂²×C₄	#237	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
186	Q₁₆	—	2	4	no	—
187	Q₁₆	—	2	4	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	6	2	no	—
191	C₂×C₈	—	6	2	no	—
192	C₂×C₈	—	12	2	no	—
193	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
194	C₂×C₈	#208	1	2	yes	—
195	C₂×C₈	#217	1	1	yes	characteristic
196	C₂×C₈	#217	1	1	yes	characteristic
197	C₂×C₈	#217	1	1	yes	characteristic
198	C₂×C₈	#217	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	D₃×D₄	1	1	yes	characteristic
206	C₄²	#237	1	1	yes	characteristic
207	C₄²	C₂×#67	1	1	yes	characteristic
208	C₄²	C₂×#67	1	1	yes	characteristic
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	#390	1	1	yes	characteristic
212	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
213	C₂²×C₃	#436	1	1	yes	characteristic
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	4	4	no	—
218	C₃×C₄	—	8	2	no	—
219	C₃×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
220	C₃×C₄	#408	1	1	yes	characteristic
221	C₃×C₄	C₈×D₄	1	1	yes	characteristic
222	C₃×C₄	#436	1	1	yes	characteristic
223	C₃×C₄	#473	1	2	yes	—
224	C₂³	D₄×Dic₃	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
237	C₂×C₄	C₂×#208	1	1	yes	characteristic
238	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
239	C₂×C₄	#936	1	1	yes	characteristic
240	C₂×C₄	#938	1	1	yes	characteristic
241	C₂×C₄	#938	1	1	yes	characteristic
242	C₂×C₄	#940	1	1	yes	characteristic
243	C₂×C₄	#948	1	1	yes	characteristic
244	C₂×C₄	#948	1	1	yes	characteristic
245	C₂×C₄	C₂×#217	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	12	2	no	—
249	C₈	—	12	2	no	—
250	C₈	—	24	2	no	—
251	C₈	#830	1	2	yes	—
252	C₈	#830	1	2	yes	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	#1954	1	1	yes	characteristic
256	C₂×C₃	#1976	1	1	yes	characteristic
257	C₂×C₃	#1979	1	1	yes	characteristic
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	D₄×C₃⋊C₈	1	1	yes	characteristic
261	C₂²	#5012	1	1	yes	characteristic
262	C₂²	#5221	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	4	no	—
267	C₄	—	8	2	no	—
268	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
269	C₄	#5012	1	1	yes	characteristic
270	C₄	#5192	1	2	yes	—
271	C₄	C₂×#830	1	1	yes	characteristic
272	C₄	#5234	1	1	yes	characteristic
273	C₃	#9669	1	1	yes	characteristic, complemented, Sylow
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#71614	1	1	yes	characteristic
277	C₂	#72069	1	1	yes	characteristic
278	C₂	#72086	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#938
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₄×Q₁₆	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₄×Q₁₆	D₃
hypocenter	C₃	#9669
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
34. C₂×C₃×Q₁₆	C₈	semidirect
118. C₂×Q₁₆	C₃⋊C₈	semidirect
273. C₃	#9669	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×Q₁₆	2
#71624	2
#71621	2
#69729	2
#69724	2
#9669	3