Subgroups — #10694472

Order 768.

474 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	C₂	1	1	yes	maximal, characteristic
2	#71778	C₂	1	2	yes	maximal
3	C₂×#4547	C₂	1	2	yes	maximal
4	#69757	C₂	1	1	yes	maximal, characteristic
5	C₂×#4457	C₂	1	1	yes	maximal, characteristic
6	#9740	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#500	—	2	2	no	—
9	C₃×#500	C₄	1	4	yes	complemented
10	C₂×C₃×#107	—	2	1	no	—
11	C₂×C₃×#107	—	2	1	no	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#835	—	2	4	no	—
14	C₂×#815	—	2	1	no	—
15	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	#4550	—	2	1	no	—
19	#4547	—	2	4	no	—
20	#4457	—	2	2	no	—
21	C₂×#500	D₃	1	1	yes	characteristic
22	#1981	—	3	2	no	—
23	#1770	—	3	1	no	—
24	C₂×#334	—	3	1	no	—
25	C₂×#332	—	3	2	no	—
26	C₂²×C₃×Q₈	—	2	2	no	—
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₃×C₄⋊Q₈	—	2	2	no	—
30	C₃×C₄⋊Q₈	—	2	2	no	—
31	C₃×C₄⋊Q₈	—	2	2	no	—
32	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
33	C₂×C₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₃	—	2	1	no	—
38	C₂²×C₃⋊C₈	—	2	1	no	—
39	C₂²×C₃⋊C₈	—	2	2	no	—
40	C₃×#107	—	2	4	no	—
41	C₃×#107	—	2	4	no	—
42	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₃×C₄×C₈	D₄	1	2	yes	—
44	#835	—	4	8	no	—
45	C₃⋊#95	—	2	1	no	—
46	#815	—	4	2	no	—
47	C₄⋊(C₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₃⋊C₈)	—	2	2	no	—
49	C₄⋊(C₃⋊C₈)	—	2	2	no	—
50	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
51	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
52	#500	—	2	2	no	—
53	#500	Dic₃	1	4	yes	complemented
54	C₂×#107	—	2	1	no	—
55	C₂×#107	—	2	1	no	—
56	C₂×C₄⋊C₈	—	3	1	no	—
57	C₂×C₄⋊C₈	—	3	2	no	—
58	C₂×#103	—	6	4	no	—
59	C₂×#97	—	6	1	no	—
60	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
61	C₄×C₄⋊C₄	—	3	1	no	—
62	#336	—	6	1	no	—
63	#334	—	6	2	no	—
64	#332	—	6	4	no	—
65	C₂×C₃×Q₈	—	2	4	no	—
66	C₂×C₃×Q₈	—	4	2	no	—
67	C₂×C₃×Q₈	—	4	4	no	—
68	C₂×C₃×Q₈	—	4	4	no	—
69	C₂²×C₃×C₄	—	4	2	no	—
70	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	#41	1	1	yes	characteristic
73	C₂²×Dic₃	—	2	2	no	—
74	C₂×C₃×C₈	—	2	1	no	—
75	C₂×C₃×C₈	—	2	1	no	—
76	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₈	SD₁₆	1	2	yes	—
79	C₂×C₃×C₈	Q₁₆	1	2	yes	—
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₄⋊C₄	—	2	4	no	—
82	C₃×C₄⋊C₄	—	4	2	no	—
83	C₃×C₄⋊C₄	—	4	2	no	—
84	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
86	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₃×C₄²	#41	1	1	yes	characteristic
88	C₄⋊Dic₃	—	2	4	no	—
89	C₄×Dic₃	—	4	1	no	—
90	C₄×Dic₃	—	4	1	no	—
91	C₂×C₃⋊C₈	—	2	2	no	—
92	C₂×C₃⋊C₈	—	4	1	no	—
93	C₂×C₃⋊C₈	—	4	1	no	—
94	C₂×C₃⋊C₈	—	4	2	no	—
95	C₂×C₃⋊C₈	—	4	4	no	—
96	C₂²×Q₈	—	2	2	no	—
97	C₂²×C₈	—	6	1	no	—
98	C₂²×C₈	—	6	2	no	—
99	C₂²×C₈	#67	1	1	yes	characteristic
100	C₂²×C₈	#67	1	1	yes	characteristic
101	C₄⋊Q₈	—	2	2	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	—	2	2	no	—
104	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	3	2	no	—
108	C₂×C₄²	—	6	1	no	—
109	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
110	#107	—	2	4	no	—
111	#107	—	2	4	no	—
112	C₄⋊C₈	—	3	2	no	—
113	C₄⋊C₈	—	6	2	no	—
114	C₄⋊C₈	—	6	1	no	—
115	C₄⋊C₈	—	6	2	no	—
116	#103	—	12	8	no	—
117	#97	—	12	2	no	—
118	C₄×C₈	C₂×Dic₃	1	2	yes	—
119	C₄×C₈	#67	1	2	yes	—
120	#95	—	6	1	no	—
121	C₂³×C₃	#121	1	1	yes	characteristic
122	C₃×Q₈	—	4	8	no	—
123	C₃×Q₈	—	8	2	no	—
124	C₃×Q₈	—	8	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	4	4	no	—
127	C₂×C₃×C₄	—	8	4	no	—
128	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	#121	1	1	yes	characteristic
132	C₂×C₃×C₄	#124	1	1	yes	characteristic
133	C₂×C₃×C₄	#127	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
136	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₃×C₄	#137	1	1	yes	characteristic
138	C₂×C₃×C₄	#137	1	1	yes	characteristic
139	C₂×Dic₃	—	4	2	no	—
140	C₂×Dic₃	—	4	2	no	—
141	C₂×Dic₃	—	4	2	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	#103	1	4	yes	complemented
144	C₃⋊C₈	—	4	2	no	—
145	C₃⋊C₈	—	8	2	no	—
146	C₂×Q₈	—	2	4	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	4	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	6	2	no	—
152	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
153	C₂²×C₄	#237	1	1	yes	characteristic
154	C₂²×C₄	C₂×#67	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	6	2	no	—
158	C₂×C₈	—	12	1	no	—
159	C₂×C₈	—	12	1	no	—
160	C₂×C₈	—	12	4	no	—
161	C₂×C₈	—	12	2	no	—
162	C₂×C₈	#215	1	2	yes	—
163	C₂×C₈	#216	1	2	yes	—
164	C₂×C₈	#217	1	2	yes	—
165	C₂×C₈	#217	1	2	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	6	4	no	—
171	C₄²	—	12	1	no	—
172	C₄²	—	12	1	no	—
173	C₄²	D₃×D₄	1	1	yes	characteristic
174	C₄²	#237	1	1	yes	characteristic
175	C₄²	C₂²×Dic₃	1	1	yes	characteristic
176	C₄²	C₂×#67	1	1	yes	characteristic
177	C₂²×C₃	#390	1	1	yes	characteristic
178	C₂²×C₃	#461	1	1	yes	characteristic
179	C₂²×C₃	#462	1	1	yes	characteristic
180	C₂²×C₃	#469	1	2	yes	—
181	C₂²×C₃	#473	1	2	yes	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	8	4	no	—
185	C₃×C₄	C₂×#103	1	1	yes	characteristic
186	C₃×C₄	#419	1	1	yes	characteristic
187	C₃×C₄	#442	1	2	yes	—
188	C₃×C₄	#461	1	1	yes	characteristic
189	C₃×C₄	#462	1	1	yes	characteristic
190	C₃×C₄	#496	1	2	yes	—
191	Dic₃	—	8	2	no	—
192	C₂³	#938	1	1	yes	characteristic
193	Q₈	—	4	8	no	—
194	Q₈	—	8	2	no	—
195	Q₈	—	8	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	—	12	2	no	—
200	C₂×C₄	—	12	2	no	—
201	C₂×C₄	—	12	2	no	—
202	C₂×C₄	#911	1	4	yes	—
203	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
204	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
205	C₂×C₄	#936	1	1	yes	characteristic
206	C₂×C₄	#938	1	1	yes	characteristic
207	C₂×C₄	#940	1	1	yes	characteristic
208	C₂×C₄	C₂×#215	1	1	yes	characteristic
209	C₂×C₄	#942	1	1	yes	characteristic
210	C₂×C₄	#942	1	1	yes	characteristic
211	C₂×C₄	C₂×#216	1	1	yes	characteristic
212	C₂×C₄	C₂×#217	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	12	2	no	—
215	C₈	—	24	2	no	—
216	C₈	#835	1	4	yes	complemented
217	C₂×C₃	#1693	1	2	yes	—
218	C₂×C₃	#1694	1	2	yes	—
219	C₂×C₃	#1972	1	1	yes	characteristic
220	C₂×C₃	#1979	1	1	yes	characteristic
221	C₂×C₃	#1981	1	1	yes	characteristic
222	C₂²	#5083	1	1	yes	characteristic
223	C₂²	#5090	1	1	yes	characteristic
224	C₂²	#5159	1	2	yes	—
225	C₂²	#5192	1	2	yes	—
226	C₂²	#5221	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	8	4	no	—
230	C₄	—	24	2	no	—
231	C₄	#5083	1	1	yes	characteristic
232	C₄	#5090	1	1	yes	characteristic
233	C₄	#5152	1	2	yes	—
234	C₄	#5154	1	2	yes	—
235	C₄	C₂×#835	1	1	yes	characteristic
236	C₄	#5225	1	1	yes	characteristic
237	C₃	#9740	1	1	yes	characteristic, complemented, Sylow
238	C₂	#71373	1	2	yes	—
239	C₂	#71379	1	2	yes	—
240	C₂	#71778	1	1	yes	characteristic
241	C₂	#72039	1	1	yes	characteristic
242	C₂	#72069	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#938
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#500	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#500	D₃
hypocenter	C₃	#9740
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
9. C₃×#500	C₄	semidirect
53. #500	Dic₃	semidirect
143. C₃×C₈	#103	semidirect
216. C₈	#835	semidirect
237. C₃	#9740	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#500	2
#71778	2
C₂×#4547	2
#69757	2
C₂×#4457	2
#9740	3