Subgroups — #10685134

Order 768.

542 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×#431	C₂	1	1	yes	maximal, characteristic
2	#82049	C₂	1	2	yes	maximal
3	#67705	C₂	1	4	yes	maximal, complemented
4	#8000	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₃×#431	—	2	2	no	—
7	C₃×#431	—	2	2	no	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₂²⋊C₈	C₄	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
11	#5207	—	2	2	no	—
12	#4926	—	2	2	no	—
13	#4926	C₄	1	4	yes	—
14	#4925	—	2	2	no	—
15	#4925	C₄	1	4	yes	—
16	#4924	C₂²	1	4	yes	—
17	C₄×#208	—	2	2	no	—
18	#2920	—	3	2	no	—
19	C₂×#431	D₃	1	1	yes	characteristic
20	#1292	—	3	4	no	—
21	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂³×C₃×C₄	Q₈	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	2	yes	—
30	C₂×C₃×C₄²	D₄	1	2	yes	—
31	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂×#208	—	2	4	no	—
34	C₃×C₄⋊C₈	—	2	4	no	—
35	C₃×C₂²⋊C₈	—	2	4	no	—
36	C₃×C₄×C₈	—	2	2	no	—
37	C₃×C₄×C₈	—	2	2	no	—
38	#830	—	2	4	no	—
39	#830	—	2	4	no	—
40	C₄⋊(C₃⋊C₈)	—	2	4	no	—
41	C₄⋊(C₃⋊C₈)	C₂×C₄	1	8	yes	—
42	#803	—	2	4	no	—
43	C₄×C₃⋊C₈	—	2	4	no	—
44	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
45	#432	—	3	4	no	—
46	#432	—	6	2	no	—
47	#431	—	2	2	no	—
48	#431	—	2	2	no	—
49	#431	—	3	4	no	—
50	#431	—	6	2	no	—
51	#423	—	3	4	no	—
52	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
53	#407	—	6	2	no	—
54	C₂×C₂²⋊C₈	Dic₃	1	2	yes	—
55	C₄×M₁₆	—	6	2	no	—
56	C₂×C₄×C₈	Dic₃	1	2	yes	—
57	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	2	4	no	—
59	C₂²×C₃×C₄	—	4	2	no	—
60	C₂²×C₃×C₄	—	4	2	no	—
61	C₂²×C₃×C₄	C₄²	1	2	yes	—
62	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
66	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
67	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₃×C₈	—	2	4	no	—
74	C₂×C₃×C₈	—	4	4	no	—
75	C₃×C₄²	C₄²	1	1	yes	characteristic
76	C₃×C₄²	C₄²	1	1	yes	characteristic
77	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
78	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
79	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
80	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
81	C₃×C₄²	C₄⋊C₄	1	2	yes	—
82	C₃×C₄²	C₄⋊C₄	1	2	yes	—
83	C₃×C₄²	C₂²×C₄	1	2	yes	—
84	C₃×C₄²	C₂×D₄	1	2	yes	—
85	#208	—	4	8	no	—
86	C₂×C₃⋊C₈	—	2	8	no	—
87	C₂³×C₄	Dic₆	1	1	yes	characteristic
88	C₂³×C₄	D₁₂	1	1	yes	characteristic
89	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
90	C₂×M₁₆	—	6	4	no	—
91	C₂²×C₈	C₂×Dic₃	1	4	yes	—
92	C₂×C₄²	Dic₆	1	1	yes	characteristic
93	C₂×C₄²	C₄×D₃	1	2	yes	—
94	C₂×C₄²	C₄×D₃	1	2	yes	—
95	C₂×C₄²	D₁₂	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
97	C₂×C₄²	#67	1	2	yes	—
98	C₂×C₄²	#67	1	2	yes	—
99	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	3	8	no	—
102	C₄⋊C₈	—	6	4	no	—
103	C₂²⋊C₈	—	2	4	no	—
104	C₂²⋊C₈	—	6	4	no	—
105	C₂²⋊C₈	—	6	4	no	—
106	#97	—	6	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	—	6	4	no	—
110	C₂³×C₃	—	2	2	no	—
111	C₂³×C₃	—	4	1	no	—
112	C₂³×C₃	—	4	1	no	—
113	C₂³×C₃	#95	1	2	yes	—
114	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	4	2	no	—
123	C₂×C₃×C₄	—	4	2	no	—
124	C₂×C₃×C₄	—	4	1	no	—
125	C₂×C₃×C₄	—	4	1	no	—
126	C₂×C₃×C₄	#95	1	1	yes	characteristic
127	C₂×C₃×C₄	#95	1	1	yes	characteristic
128	C₂×C₃×C₄	#95	1	1	yes	characteristic
129	C₂×C₃×C₄	#95	1	2	yes	—
130	C₂×C₃×C₄	#95	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₃×C₄	#135	1	4	yes	—
136	C₃×C₈	—	4	4	no	—
137	C₃⋊C₈	—	4	8	no	—
138	C₂⁴	C₄⋊Dic₃	1	1	yes	characteristic
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	C₄×Dic₃	1	2	yes	—
143	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
144	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
148	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
149	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
150	C₂²×C₄	#217	1	2	yes	—
151	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
153	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
154	M₁₆	—	12	8	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	6	8	no	—
158	C₄²	C₄×Dic₃	1	1	yes	characteristic
159	C₄²	C₄×Dic₃	1	1	yes	characteristic
160	C₄²	Dic₃⋊C₄	1	2	yes	—
161	C₄²	Dic₃⋊C₄	1	2	yes	—
162	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
163	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
164	C₄²	#217	1	1	yes	characteristic
165	C₄²	#217	1	1	yes	characteristic
166	C₄²	C₂×C₄×D₃	1	2	yes	—
167	C₄²	C₂×#67	1	2	yes	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	4	4	no	—
174	C₂²×C₃	C₂×#95	1	1	yes	characteristic
175	C₂²×C₃	#427	1	2	yes	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	#338	1	2	yes	—
183	C₃×C₄	#413	1	2	yes	—
184	C₃×C₄	#416	1	4	yes	—
185	C₂³	—	2	2	no	—
186	C₂³	—	4	1	no	—
187	C₂³	—	4	1	no	—
188	C₂³	C₃⋊#95	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
202	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
203	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
204	C₂×C₄	C₃⋊#95	1	2	yes	—
205	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
206	C₂×C₄	#904	1	4	yes	—
207	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
208	C₂×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
210	C₂×C₄	C₂×#217	1	1	yes	characteristic
211	C₈	—	4	4	no	—
212	C₈	—	12	8	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	—	4	2	no	—
215	C₂×C₃	C₂×#338	1	1	yes	characteristic
216	C₂×C₃	#1735	1	1	yes	characteristic
217	C₂×C₃	#1742	1	1	yes	characteristic
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	4	no	—
224	C₂²	#5106	1	2	yes	—
225	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	C₃⋊#338	1	2	yes	—
233	C₄	#5113	1	4	yes	—
234	C₄	#5237	1	2	yes	—
235	C₃	#8000	1	1	yes	characteristic, complemented, Sylow
236	C₂	—	2	2	no	—
237	C₂	—	4	2	no	—
238	C₂	#71587	1	1	yes	characteristic
239	C₂	#71969	1	1	yes	characteristic
240	C₂	C₂×C₃⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₃⋊#95
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#431	C₂
socle	C₂²×C₃	C₂×#95
prime radical	C₂⁴×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#431	D₃
hypocenter	C₃	#8000
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #67705	C₂	semidirect
235. C₃	#8000	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#431	2
#82049	2
#67705	2
#8000	3