Subgroups — #10694056

Order 768.

486 conjugacy classes of subgroups.

The subgroup lattice has 258 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×#4457	C₂	1	1	yes	maximal, characteristic
5	C₂×#4455	C₂	1	1	yes	maximal, characteristic
6	#9658	—	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	#4457	—	2	2	no	—
20	#4455	—	2	2	no	—
21	C₂×C₄×Q₁₆	D₃	1	1	yes	characteristic
22	#1984	—	3	2	no	—
23	C₂×#334	—	3	1	no	—
24	C₂×Q₈⋊C₈	—	3	2	no	—
25	C₂×#322	—	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	#334	—	6	2	no	—
65	Q₈⋊C₈	—	6	4	no	—
66	#322	—	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	2	no	—
96	C₂×C₃⋊C₈	—	2	2	no	—
97	C₂×C₃⋊C₈	—	2	4	no	—
98	C₂×C₃⋊C₈	—	4	2	no	—
99	C₂×C₃⋊C₈	—	4	2	no	—
100	C₂×C₃⋊C₈	—	4	4	no	—
101	C₂²×Q₈	C₂×Dic₃	1	2	yes	—
102	C₂×Q₁₆	—	2	2	no	—
103	C₂×Q₁₆	—	2	2	no	—
104	C₂×Q₁₆	C₃⋊C₈	1	4	yes	complemented
105	C₂²×C₈	—	3	2	no	—
106	C₂²×C₈	—	6	1	no	—
107	C₂²×C₈	—	6	2	no	—
108	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₄×Q₈	—	2	2	no	—
111	C₄×Q₈	—	2	2	no	—
112	C₄×Q₈	—	2	2	no	—
113	C₄×Q₈	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
118	#107	—	2	2	no	—
119	C₄⋊C₈	—	6	2	no	—
120	C₄⋊C₈	—	6	2	no	—
121	#103	—	2	4	no	—
122	C₄×C₈	—	3	2	no	—
123	C₄×C₈	—	6	1	no	—
124	C₄×C₈	—	6	1	no	—
125	C₄×C₈	—	6	1	no	—
126	C₄×C₈	#67	1	2	yes	—
127	C₄×C₈	#67	1	2	yes	—
128	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
129	C₃×Q₈	—	2	4	no	—
130	C₃×Q₈	—	2	4	no	—
131	C₃×Q₈	—	4	2	no	—
132	C₃×Q₈	—	4	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	#121	1	1	yes	characteristic
142	C₂×C₃×C₄	#121	1	1	yes	characteristic
143	C₂×C₃×C₄	#124	1	1	yes	characteristic
144	C₂×C₃×C₄	#127	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₃×C₄	#131	1	1	yes	characteristic
148	C₂×C₃×C₄	#131	1	1	yes	characteristic
149	C₂×C₃×C₄	#137	1	4	yes	—
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	C₂²⋊C₈	1	4	yes	—
152	C₃⋊C₈	—	4	2	no	—
153	C₃⋊C₈	—	4	2	no	—
154	C₃⋊C₈	—	8	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
164	C₂²×C₄	#237	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
166	Q₁₆	—	2	8	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	6	4	no	—
170	C₂×C₈	—	6	2	no	—
171	C₂×C₈	—	6	2	no	—
172	C₂×C₈	—	12	2	no	—
173	C₂×C₈	—	12	4	no	—
174	C₂×C₈	—	12	2	no	—
175	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
176	C₂×C₈	#208	1	2	yes	—
177	C₂×C₈	#217	1	2	yes	—
178	C₂×C₈	#217	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	D₃×D₄	1	1	yes	characteristic
186	C₄²	#237	1	1	yes	characteristic
187	C₄²	C₂×#67	1	1	yes	characteristic
188	C₄²	C₂×#67	1	1	yes	characteristic
189	C₂²×C₃	#390	1	1	yes	characteristic
190	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
191	C₂²×C₃	#436	1	1	yes	characteristic
192	C₂²×C₃	#441	1	2	yes	—
193	C₂²×C₃	#444	1	2	yes	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	4	4	no	—
197	C₃×C₄	—	8	2	no	—
198	C₃×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
199	C₃×C₄	#408	1	1	yes	characteristic
200	C₃×C₄	C₈×D₄	1	1	yes	characteristic
201	C₃×C₄	#436	1	1	yes	characteristic
202	C₃×C₄	#470	1	2	yes	—
203	C₃×C₄	#497	1	2	yes	—
204	C₂³	D₄×Dic₃	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	2	4	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	4	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	#918	1	4	yes	—
216	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
217	C₂×C₄	C₂×#208	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
219	C₂×C₄	#936	1	1	yes	characteristic
220	C₂×C₄	#938	1	1	yes	characteristic
221	C₂×C₄	#938	1	1	yes	characteristic
222	C₂×C₄	#940	1	1	yes	characteristic
223	C₂×C₄	#948	1	1	yes	characteristic
224	C₂×C₄	#948	1	1	yes	characteristic
225	C₂×C₄	C₂×#217	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	12	2	no	—
228	C₈	—	12	2	no	—
229	C₈	—	24	2	no	—
230	C₈	#830	1	4	yes	—
231	C₂×C₃	#1579	1	2	yes	—
232	C₂×C₃	#1593	1	2	yes	—
233	C₂×C₃	#1954	1	1	yes	characteristic
234	C₂×C₃	#1971	1	1	yes	characteristic
235	C₂×C₃	#1975	1	1	yes	characteristic
236	C₂²	D₄×C₃⋊C₈	1	1	yes	characteristic
237	C₂²	#5012	1	1	yes	characteristic
238	C₂²	#5184	1	2	yes	—
239	C₂²	#5197	1	2	yes	—
240	C₂²	#5221	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	4	no	—
244	C₄	—	8	2	no	—
245	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
246	C₄	#5012	1	1	yes	characteristic
247	C₄	#5189	1	2	yes	—
248	C₄	#5190	1	2	yes	—
249	C₄	C₂×#830	1	1	yes	characteristic
250	C₄	#5234	1	1	yes	characteristic
251	C₃	#9658	1	1	yes	characteristic, complemented, Sylow
252	C₂	#70902	1	2	yes	—
253	C₂	#70906	1	2	yes	—
254	C₂	#71614	1	1	yes	characteristic
255	C₂	#72067	1	1	yes	characteristic
256	C₂	#72085	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	#9658
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
104. C₂×Q₁₆	C₃⋊C₈	semidirect
251. C₃	#9658	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×#4457	2
C₂×#4455	2
#9658	3