Subgroups — #10685137

Order 768.

634 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₂×#4924	C₂	1	2	yes	maximal
3	#67705	C₂	1	4	yes	maximal, complemented
4	#8001	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₂²×#208	—	2	2	no	—
7	C₃×#431	—	2	2	no	—
8	C₃×#431	—	2	2	no	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
12	#4924	—	2	2	no	—
13	#4924	—	2	2	no	—
14	#4924	C₄	1	4	yes	—
15	#4924	C₂²	1	4	yes	—
16	C₂×C₄⋊(C₃⋊C₈)	—	2	2	no	—
17	C₂×C₄⋊(C₃⋊C₈)	C₄	1	4	yes	—
18	C₂×#431	D₃	1	1	yes	characteristic
19	C₂×#423	—	3	2	no	—
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	C₂×#208	—	2	4	no	—
35	C₂×#208	—	4	4	no	—
36	C₂×#208	—	4	4	no	—
37	C₂²×C₃⋊C₈	—	2	4	no	—
38	C₃×C₄⋊C₈	—	2	4	no	—
39	C₃×C₂²⋊C₈	—	2	4	no	—
40	C₃×C₄×C₈	—	2	2	no	—
41	C₃×C₄×C₈	—	2	2	no	—
42	C₄⋊(C₃⋊C₈)	—	2	4	no	—
43	C₄⋊(C₃⋊C₈)	—	2	4	no	—
44	C₄⋊(C₃⋊C₈)	—	2	4	no	—
45	C₄⋊(C₃⋊C₈)	C₂×C₄	1	8	yes	—
46	C₂²×M₁₆	—	6	2	no	—
47	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
48	#431	—	2	2	no	—
49	#431	—	2	2	no	—
50	#423	—	3	4	no	—
51	#423	—	3	4	no	—
52	#423	—	6	2	no	—
53	#423	—	6	2	no	—
54	C₂×C₄⋊C₈	—	3	4	no	—
55	C₂×C₄⋊C₈	—	6	2	no	—
56	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
57	C₂×C₂²⋊C₈	Dic₃	1	2	yes	—
58	C₂×C₄×C₈	Dic₃	1	2	yes	—
59	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	—	4	2	no	—
61	C₂²×C₃×C₄	—	4	2	no	—
62	C₂²×C₃×C₄	C₄²	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	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
67	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
68	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	D₈	1	2	yes	—
72	C₂²×C₃×C₄	SD₁₆	1	4	yes	—
73	C₂²×C₃×C₄	Q₁₆	1	2	yes	—
74	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	—	4	4	no	—
79	C₃×C₄²	C₄²	1	1	yes	characteristic
80	C₃×C₄²	C₄²	1	1	yes	characteristic
81	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
82	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
83	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₄²	C₄⋊C₄	1	2	yes	—
86	C₃×C₄²	C₄⋊C₄	1	2	yes	—
87	C₃×C₄²	C₂²×C₄	1	2	yes	—
88	C₃×C₄²	C₂×D₄	1	2	yes	—
89	#208	—	4	8	no	—
90	#208	—	4	4	no	—
91	#208	—	4	4	no	—
92	C₂×C₃⋊C₈	—	2	8	no	—
93	C₂×C₃⋊C₈	—	4	8	no	—
94	C₂³×C₄	Dic₆	1	1	yes	characteristic
95	C₂³×C₄	D₁₂	1	1	yes	characteristic
96	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
97	C₂×M₁₆	—	6	4	no	—
98	C₂×M₁₆	—	6	4	no	—
99	C₂×M₁₆	—	12	4	no	—
100	C₂×M₁₆	—	12	4	no	—
101	C₂²×C₈	—	6	4	no	—
102	C₂²×C₈	C₂×Dic₃	1	4	yes	—
103	C₂×C₄²	Dic₆	1	1	yes	characteristic
104	C₂×C₄²	C₄×D₃	1	2	yes	—
105	C₂×C₄²	C₄×D₃	1	2	yes	—
106	C₂×C₄²	D₁₂	1	1	yes	characteristic
107	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
108	C₂×C₄²	#67	1	2	yes	—
109	C₂×C₄²	#67	1	2	yes	—
110	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	3	8	no	—
113	C₄⋊C₈	—	6	4	no	—
114	C₄⋊C₈	—	6	4	no	—
115	C₄⋊C₈	—	6	4	no	—
116	C₂²⋊C₈	—	2	4	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₂³×C₃	—	4	1	no	—
120	C₂³×C₃	—	4	1	no	—
121	C₂³×C₃	#95	1	2	yes	—
122	C₂³×C₃	#106	1	2	yes	—
123	C₂³×C₃	#107	1	2	yes	—
124	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	—	4	2	no	—
131	C₂×C₃×C₄	—	4	1	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	#95	1	1	yes	characteristic
134	C₂×C₃×C₄	#95	1	2	yes	—
135	C₂×C₃×C₄	#95	1	1	yes	characteristic
136	C₂×C₃×C₄	#95	1	1	yes	characteristic
137	C₂×C₃×C₄	#95	1	1	yes	characteristic
138	C₂×C₃×C₄	#102	1	4	yes	—
139	C₂×C₃×C₄	#102	1	4	yes	—
140	C₂×C₃×C₄	#103	1	4	yes	—
141	C₂×C₃×C₄	#103	1	4	yes	—
142	C₂×C₃×C₄	#106	1	2	yes	—
143	C₂×C₃×C₄	#107	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
150	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
151	C₃×C₈	—	4	4	no	—
152	C₃⋊C₈	—	4	8	no	—
153	C₂⁴	C₄⋊Dic₃	1	1	yes	characteristic
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	#204	1	4	yes	—
157	C₂²×C₄	D₂₄	1	2	yes	—
158	C₂²×C₄	Dic₁₂	1	2	yes	—
159	C₂²×C₄	C₄×Dic₃	1	2	yes	—
160	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
161	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
162	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
165	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
166	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
167	C₂²×C₄	#217	1	2	yes	—
168	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
170	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
171	M₁₆	—	12	8	no	—
172	M₁₆	—	12	4	no	—
173	M₁₆	—	12	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	6	8	no	—
177	C₂×C₈	—	12	8	no	—
178	C₄²	C₄×Dic₃	1	1	yes	characteristic
179	C₄²	C₄×Dic₃	1	1	yes	characteristic
180	C₄²	Dic₃⋊C₄	1	2	yes	—
181	C₄²	Dic₃⋊C₄	1	2	yes	—
182	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
183	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
184	C₄²	#217	1	1	yes	characteristic
185	C₄²	#217	1	1	yes	characteristic
186	C₄²	C₂×C₄×D₃	1	2	yes	—
187	C₄²	C₂×#67	1	2	yes	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	#340	1	4	yes	—
195	C₂²×C₃	C₂×#95	1	1	yes	characteristic
196	C₂²×C₃	C₂×#106	1	1	yes	characteristic
197	C₂²×C₃	C₂×#107	1	1	yes	characteristic
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	#338	1	2	yes	—
203	C₃×C₄	#340	1	2	yes	—
204	C₃×C₄	#340	1	2	yes	—
205	C₃×C₄	#413	1	2	yes	—
206	C₃×C₄	C₂×#102	1	2	yes	—
207	C₃×C₄	C₂×#103	1	2	yes	—
208	C₂³	—	4	1	no	—
209	C₂³	—	4	1	no	—
210	C₂³	#817	1	2	yes	—
211	C₂³	#818	1	2	yes	—
212	C₂³	C₃⋊#95	1	2	yes	—
213	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	#816	1	4	yes	—
223	C₂×C₄	#816	1	4	yes	—
224	C₂×C₄	#817	1	2	yes	—
225	C₂×C₄	#818	1	2	yes	—
226	C₂×C₄	#821	1	4	yes	—
227	C₂×C₄	#821	1	4	yes	—
228	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
229	C₂×C₄	C₃⋊#95	1	2	yes	—
230	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
231	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
232	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
233	C₂×C₄	C₂×#204	1	2	yes	—
234	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
235	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
237	C₂×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
238	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
239	C₂×C₄	C₂×#217	1	1	yes	characteristic
240	C₈	—	4	4	no	—
241	C₈	—	12	8	no	—
242	C₂×C₃	—	2	4	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	C₂×#338	1	1	yes	characteristic
246	C₂×C₃	C₂×#340	1	1	yes	characteristic
247	C₂×C₃	#1742	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	#4551	1	4	yes	—
255	C₂²	C₂×#817	1	1	yes	characteristic
256	C₂²	C₂×#818	1	1	yes	characteristic
257	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	#4551	1	2	yes	—
263	C₄	#4551	1	2	yes	—
264	C₄	C₃⋊#338	1	2	yes	—
265	C₄	C₂×#816	1	2	yes	—
266	C₄	C₂×#821	1	2	yes	—
267	C₄	#5237	1	2	yes	—
268	C₃	#8001	1	1	yes	characteristic, complemented, Sylow
269	C₂	—	2	4	no	—
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	#71587	1	1	yes	characteristic
273	C₂	C₂×#4551	1	1	yes	characteristic
274	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	#8001
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
268. C₃	#8001	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
C₂×#4924	2
#67705	2
#8001	3