Subgroups — #10694090

Order 768.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#4454	C₂	1	1	yes	maximal, characteristic
6	#9664	—	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	#4454	—	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	4	no	—
28	C₂×C₃×Q₁₆	—	2	2	no	—
29	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄×Q₈	—	2	2	no	—
32	C₃×C₄×Q₈	—	2	2	no	—
33	C₃×C₄×Q₈	—	2	2	no	—
34	C₃×C₄×Q₈	—	2	2	no	—
35	C₂×C₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄²	—	2	2	no	—
38	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
39	C₂²×C₃⋊C₈	—	2	1	no	—
40	C₂²×C₃⋊C₈	—	2	2	no	—
41	C₂²×C₃⋊C₈	D₄	1	2	yes	—
42	C₃×#107	—	2	2	no	—
43	C₃×#103	—	2	4	no	—
44	C₃×C₄×C₈	D₄	1	2	yes	—
45	C₃×C₄×C₈	D₄	1	2	yes	—
46	C₄⋊(C₃⋊C₈)	—	2	2	no	—
47	C₄⋊(C₃⋊C₈)	—	2	2	no	—
48	C₄×C₃⋊C₈	—	2	1	no	—
49	C₄×C₃⋊C₈	—	2	1	no	—
50	C₄×C₃⋊C₈	—	2	1	no	—
51	C₄×C₃⋊C₈	D₄	1	2	yes	—
52	C₂²×Q₁₆	Dic₃	1	1	yes	characteristic
53	C₂×C₄×Q₈	C₂×D₃	1	2	yes	—
54	C₄×Q₁₆	—	2	2	no	—
55	C₄×Q₁₆	—	2	2	no	—
56	C₂×#107	Dic₃	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	3	2	no	—
58	C₂×#103	Dic₃	1	2	yes	—
59	C₂×C₄×C₈	—	3	1	no	—
60	C₂×C₄×C₈	—	3	1	no	—
61	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	#336	—	6	4	no	—
63	#334	—	6	2	no	—
64	Q₈⋊C₈	—	6	4	no	—
65	#322	—	6	2	no	—
66	C₂×C₃×Q₈	—	2	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	M₁₆	1	4	yes	—
72	C₂²×C₃×C₄	—	2	2	no	—
73	C₂²×C₃×C₄	—	2	2	no	—
74	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	#41	1	1	yes	characteristic
77	C₃×Q₁₆	—	2	8	no	—
78	C₂×C₃×C₈	—	2	2	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	—	4	2	no	—
85	C₃×C₄⋊C₄	—	4	2	no	—
86	C₃×C₄²	—	4	2	no	—
87	C₃×C₄²	—	4	2	no	—
88	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄²	#41	1	1	yes	characteristic
92	C₂×C₃⋊C₈	—	2	4	no	—
93	C₂×C₃⋊C₈	—	2	2	no	—
94	C₂×C₃⋊C₈	—	2	2	no	—
95	C₂×C₃⋊C₈	—	2	4	no	—
96	C₂×C₃⋊C₈	—	4	2	no	—
97	C₂×C₃⋊C₈	—	4	4	no	—
98	C₂²×Q₈	C₂×Dic₃	1	2	yes	—
99	C₂×Q₁₆	—	2	4	no	—
100	C₂×Q₁₆	—	2	2	no	—
101	C₂²×C₈	—	3	2	no	—
102	C₂²×C₈	—	6	1	no	—
103	C₂²×C₈	—	6	2	no	—
104	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
106	C₄×Q₈	—	2	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
114	#107	—	2	2	no	—
115	C₄⋊C₈	—	6	2	no	—
116	C₄⋊C₈	—	6	2	no	—
117	#103	—	2	4	no	—
118	C₄×C₈	—	3	2	no	—
119	C₄×C₈	—	6	1	no	—
120	C₄×C₈	—	6	1	no	—
121	C₄×C₈	—	6	1	no	—
122	C₄×C₈	#67	1	2	yes	—
123	C₄×C₈	#67	1	2	yes	—
124	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
125	C₃×Q₈	—	2	4	no	—
126	C₃×Q₈	—	2	4	no	—
127	C₃×Q₈	—	4	2	no	—
128	C₃×Q₈	—	4	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	—	4	2	no	—
133	C₂×C₃×C₄	—	4	2	no	—
134	C₂×C₃×C₄	—	4	4	no	—
135	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	#121	1	1	yes	characteristic
138	C₂×C₃×C₄	#121	1	1	yes	characteristic
139	C₂×C₃×C₄	#124	1	1	yes	characteristic
140	C₂×C₃×C₄	#127	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
142	C₂×C₃×C₄	#131	1	2	yes	—
143	C₂×C₃×C₄	#137	1	4	yes	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	2	2	no	—
146	C₃⋊C₈	—	2	4	no	—
147	C₃⋊C₈	—	4	2	no	—
148	C₃⋊C₈	—	8	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	#208	1	4	yes	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
158	C₂²×C₄	#237	1	1	yes	characteristic
159	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
160	Q₁₆	—	2	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	6	4	no	—
164	C₂×C₈	—	6	2	no	—
165	C₂×C₈	—	6	4	no	—
166	C₂×C₈	—	6	2	no	—
167	C₂×C₈	—	12	4	no	—
168	C₂×C₈	—	12	2	no	—
169	C₂×C₈	#217	1	2	yes	—
170	C₂×C₈	#217	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄²	—	4	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	D₃×D₄	1	1	yes	characteristic
178	C₄²	#237	1	1	yes	characteristic
179	C₄²	C₂×#67	1	1	yes	characteristic
180	C₄²	C₂×#67	1	1	yes	characteristic
181	C₂²×C₃	#390	1	1	yes	characteristic
182	C₂²×C₃	#435	1	2	yes	—
183	C₂²×C₃	#441	1	2	yes	—
184	C₂²×C₃	#443	1	2	yes	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	4	4	no	—
188	C₃×C₄	—	8	2	no	—
189	C₃×C₄	#407	1	1	yes	characteristic
190	C₃×C₄	#408	1	1	yes	characteristic
191	C₃×C₄	#435	1	2	yes	—
192	C₃×C₄	#470	1	2	yes	—
193	C₃×C₄	#497	1	2	yes	—
194	C₂³	D₄×Dic₃	1	1	yes	characteristic
195	Q₈	—	2	4	no	—
196	Q₈	—	2	4	no	—
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	#918	1	4	yes	—
206	C₂×C₄	C₂×#208	1	2	yes	—
207	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
208	C₂×C₄	#936	1	1	yes	characteristic
209	C₂×C₄	#938	1	1	yes	characteristic
210	C₂×C₄	#938	1	1	yes	characteristic
211	C₂×C₄	#940	1	1	yes	characteristic
212	C₂×C₄	#948	1	2	yes	—
213	C₂×C₄	C₂×#217	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	6	4	no	—
217	C₈	—	12	2	no	—
218	C₈	—	24	2	no	—
219	C₂×C₃	#1587	1	4	yes	—
220	C₂×C₃	#1955	1	1	yes	characteristic
221	C₂×C₃	#1971	1	1	yes	characteristic
222	C₂×C₃	#1974	1	1	yes	characteristic
223	C₂²	#5011	1	2	yes	—
224	C₂²	#5184	1	2	yes	—
225	C₂²	#5196	1	2	yes	—
226	C₂²	#5221	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	4	no	—
230	C₄	—	8	2	no	—
231	C₄	#5011	1	2	yes	—
232	C₄	#5189	1	2	yes	—
233	C₄	#5190	1	2	yes	—
234	C₄	#5207	1	1	yes	characteristic
235	C₄	#5234	1	1	yes	characteristic
236	C₃	#9664	1	1	yes	characteristic, complemented, Sylow
237	C₂	#70904	1	4	yes	—
238	C₂	#71615	1	1	yes	characteristic
239	C₂	#72067	1	1	yes	characteristic
240	C₂	#72084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃×D₄
commutator	C₂²×C₃×C₄	C₂²×C₄
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	#9664
LCS	G ▷ C₂²×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
236. C₃	#9664	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×#4454	2
#9664	3