Subgroups — #10694046

Order 768.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#71634	C₂	1	2	yes	maximal
3	C₂×#4535	C₂	1	2	yes	maximal
4	C₂×#4458	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#9656	—	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	2	no	—
10	C₃×C₄×Q₁₆	—	2	2	no	—
11	C₂×C₃×#107	C₄	1	1	yes	characteristic
12	C₂×C₃×#103	C₄	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	#4535	—	2	4	no	—
18	#4524	—	2	4	no	—
19	#4458	—	2	2	no	—
20	C₈×C₃⋊C₈	—	2	2	no	—
21	C₂×C₄×Q₁₆	D₃	1	1	yes	characteristic
22	#1984	—	3	2	no	—
23	C₂×#335	—	3	1	no	—
24	C₂×Q₈⋊C₈	—	3	2	no	—
25	C₂×C₈²	—	3	1	no	—
26	C₂²×C₃×Q₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×Q₁₆	—	2	2	no	—
28	C₂×C₃×Q₁₆	—	2	2	no	—
29	C₂×C₃×Q₁₆	C₈	1	4	yes	complemented
30	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×Q₈	—	2	2	no	—
33	C₃×C₄×Q₈	—	2	2	no	—
34	C₃×C₄×Q₈	—	2	2	no	—
35	C₃×C₄×Q₈	—	2	2	no	—
36	C₂×C₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄²	—	2	2	no	—
39	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
40	C₂²×C₃⋊C₈	—	2	1	no	—
41	C₂²×C₃⋊C₈	—	2	2	no	—
42	C₂²×C₃⋊C₈	D₄	1	2	yes	—
43	C₃×#107	—	2	2	no	—
44	C₃×#103	—	2	4	no	—
45	C₃×C₄×C₈	D₄	1	2	yes	—
46	C₃×C₄×C₈	D₄	1	2	yes	—
47	C₄⋊(C₃⋊C₈)	—	2	2	no	—
48	C₄⋊(C₃⋊C₈)	—	2	2	no	—
49	C₄×C₃⋊C₈	—	2	1	no	—
50	C₄×C₃⋊C₈	—	2	1	no	—
51	C₄×C₃⋊C₈	—	2	1	no	—
52	C₄×C₃⋊C₈	D₄	1	2	yes	—
53	C₂²×Q₁₆	Dic₃	1	1	yes	characteristic
54	C₂×C₄×Q₈	C₂×D₃	1	2	yes	—
55	C₄×Q₁₆	—	2	2	no	—
56	C₄×Q₁₆	—	2	2	no	—
57	C₂×#107	Dic₃	1	1	yes	characteristic
58	C₂×C₄⋊C₈	—	3	2	no	—
59	C₂×#103	Dic₃	1	2	yes	—
60	C₂×C₄×C₈	—	3	1	no	—
61	C₂×C₄×C₈	—	3	1	no	—
62	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	#336	—	6	4	no	—
64	#335	—	6	2	no	—
65	Q₈⋊C₈	—	6	4	no	—
66	C₈²	—	6	2	no	—
67	C₂×C₃×Q₈	—	2	2	no	—
68	C₂×C₃×Q₈	—	2	2	no	—
69	C₂×C₃×Q₈	—	2	2	no	—
70	C₂×C₃×Q₈	—	2	2	no	—
71	C₂×C₃×Q₈	—	2	2	no	—
72	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
73	C₂²×C₃×C₄	—	2	2	no	—
74	C₂²×C₃×C₄	—	2	2	no	—
75	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	#41	1	1	yes	characteristic
78	C₃×Q₁₆	—	2	8	no	—
79	C₂×C₃×C₈	—	2	1	no	—
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
84	C₂×C₃×C₈	M₁₆	1	2	yes	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	—	2	2	no	—
87	C₃×C₄⋊C₄	—	4	2	no	—
88	C₃×C₄⋊C₄	—	4	2	no	—
89	C₃×C₄²	—	4	2	no	—
90	C₃×C₄²	—	4	2	no	—
91	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₃×C₄²	#41	1	1	yes	characteristic
95	C₂×C₃⋊C₈	—	2	4	no	—
96	C₂×C₃⋊C₈	—	2	2	no	—
97	C₂×C₃⋊C₈	—	2	2	no	—
98	C₂×C₃⋊C₈	—	2	2	no	—
99	C₂×C₃⋊C₈	—	4	2	no	—
100	C₂×C₃⋊C₈	—	4	4	no	—
101	C₂×C₃⋊C₈	Q₁₆	1	4	yes	complemented
102	C₂²×Q₈	C₂×Dic₃	1	2	yes	—
103	C₂×Q₁₆	—	2	2	no	—
104	C₂×Q₁₆	—	2	2	no	—
105	C₂×Q₁₆	C₃⋊C₈	1	4	yes	complemented
106	C₂²×C₈	—	3	2	no	—
107	C₂²×C₈	—	6	1	no	—
108	C₂²×C₈	—	6	2	no	—
109	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
111	C₄×Q₈	—	2	2	no	—
112	C₄×Q₈	—	2	2	no	—
113	C₄×Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
117	C₂×C₄²	—	2	2	no	—
118	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
119	#107	—	2	2	no	—
120	C₄⋊C₈	—	6	2	no	—
121	C₄⋊C₈	—	6	2	no	—
122	#103	—	2	4	no	—
123	C₄×C₈	—	3	2	no	—
124	C₄×C₈	—	6	1	no	—
125	C₄×C₈	—	6	1	no	—
126	C₄×C₈	—	6	1	no	—
127	C₄×C₈	#67	1	2	yes	—
128	C₄×C₈	#67	1	2	yes	—
129	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
130	C₃×Q₈	—	2	4	no	—
131	C₃×Q₈	—	2	4	no	—
132	C₃×Q₈	—	4	2	no	—
133	C₃×Q₈	—	4	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	4	no	—
140	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	#121	1	1	yes	characteristic
143	C₂×C₃×C₄	#121	1	1	yes	characteristic
144	C₂×C₃×C₄	#124	1	1	yes	characteristic
145	C₂×C₃×C₄	#127	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
148	C₂×C₃×C₄	#131	1	1	yes	characteristic
149	C₂×C₃×C₄	#131	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
151	C₂×C₃×C₄	#135	1	2	yes	—
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	C₂²⋊C₈	1	4	yes	—
154	C₃⋊C₈	—	2	4	no	—
155	C₃⋊C₈	—	2	4	no	—
156	C₃⋊C₈	—	8	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
166	C₂²×C₄	#237	1	1	yes	characteristic
167	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
168	Q₁₆	—	2	8	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	3	4	no	—
172	C₂×C₈	—	6	4	no	—
173	C₂×C₈	—	6	2	no	—
174	C₂×C₈	—	6	2	no	—
175	C₂×C₈	—	6	2	no	—
176	C₂×C₈	—	12	4	no	—
177	C₂×C₈	—	12	2	no	—
178	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
179	C₂×C₈	#208	1	2	yes	—
180	C₂×C₈	#217	1	2	yes	—
181	C₂×C₈	#217	1	2	yes	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	D₃×D₄	1	1	yes	characteristic
189	C₄²	#237	1	1	yes	characteristic
190	C₄²	C₂×#67	1	1	yes	characteristic
191	C₄²	C₂×#67	1	1	yes	characteristic
192	C₂²×C₃	#390	1	1	yes	characteristic
193	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
194	C₂²×C₃	#436	1	1	yes	characteristic
195	C₂²×C₃	C₄×Q₁₆	1	2	yes	—
196	C₂²×C₃	#443	1	2	yes	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	4	4	no	—
200	C₃×C₄	—	8	2	no	—
201	C₃×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
202	C₃×C₄	#408	1	1	yes	characteristic
203	C₃×C₄	C₈×D₄	1	1	yes	characteristic
204	C₃×C₄	#436	1	1	yes	characteristic
205	C₃×C₄	#467	1	2	yes	—
206	C₃×C₄	#494	1	1	yes	characteristic
207	C₃×C₄	#495	1	1	yes	characteristic
208	C₂³	D₄×Dic₃	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	2	4	no	—
211	Q₈	—	4	2	no	—
212	Q₈	—	4	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	D₃×Q₁₆	1	2	yes	—
220	C₂×C₄	#919	1	2	yes	—
221	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
222	C₂×C₄	C₂×#208	1	1	yes	characteristic
223	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
224	C₂×C₄	#936	1	1	yes	characteristic
225	C₂×C₄	#938	1	1	yes	characteristic
226	C₂×C₄	#938	1	1	yes	characteristic
227	C₂×C₄	#940	1	1	yes	characteristic
228	C₂×C₄	#948	1	1	yes	characteristic
229	C₂×C₄	#948	1	1	yes	characteristic
230	C₂×C₄	C₂×#217	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	6	4	no	—
233	C₈	—	6	4	no	—
234	C₈	—	24	2	no	—
235	C₈	#830	1	4	yes	—
236	C₂×C₃	C₈×Q₁₆	1	2	yes	—
237	C₂×C₃	#1591	1	2	yes	—
238	C₂×C₃	#1954	1	1	yes	characteristic
239	C₂×C₃	#1969	1	1	yes	characteristic
240	C₂×C₃	#1974	1	1	yes	characteristic
241	C₂²	D₄×C₃⋊C₈	1	1	yes	characteristic
242	C₂²	#5012	1	1	yes	characteristic
243	C₂²	Dic₃×Q₁₆	1	2	yes	—
244	C₂²	#5196	1	2	yes	—
245	C₂²	#5221	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	4	no	—
249	C₄	—	8	2	no	—
250	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
251	C₄	#5012	1	1	yes	characteristic
252	C₄	#5183	1	1	yes	characteristic
253	C₄	#5188	1	2	yes	—
254	C₄	#5191	1	1	yes	characteristic
255	C₄	C₂×#830	1	1	yes	characteristic
256	C₄	#5234	1	1	yes	characteristic
257	C₃	#9656	1	1	yes	characteristic, complemented, Sylow
258	C₂	Q₁₆×C₃⋊C₈	1	2	yes	—
259	C₂	#70905	1	2	yes	—
260	C₂	#71614	1	1	yes	characteristic
261	C₂	#72066	1	1	yes	characteristic
262	C₂	#72084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₄×Q₁₆	C₂
socle, prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₄×Q₁₆	D₃
hypocenter	C₃	#9656
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
29. C₂×C₃×Q₁₆	C₈	semidirect
101. C₂×C₃⋊C₈	Q₁₆	semidirect
105. C₂×Q₁₆	C₃⋊C₈	semidirect
257. C₃	#9656	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
#71634	2
C₂×#4535	2
C₂×#4458	2
C₂×C₈×C₃⋊C₈	2
#9656	3