Subgroups — C₂×#67654 #10684643

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
3	#67654	C₂	1	4	yes	maximal, direct product factor
4	C₂×#1277	—	3	1	no	maximal, Sylow
5	C₂³×C₃×C₈	C₄	1	2	yes	—
6	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂³×C₃⋊C₈	—	2	2	no	—
8	C₂×C₃×C₄×C₈	—	2	4	no	—
9	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₃⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊(C₃⋊C₈)	—	2	4	no	—
13	C₂×C₄⋊(C₃⋊C₈)	C₄	1	8	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₈	—	3	2	no	—
16	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
17	#1277	—	3	4	no	—
18	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
20	C₂³×C₃×C₄	Q₈	1	1	yes	characteristic
21	C₂²×C₃×C₈	—	2	4	no	—
22	C₂²×C₃×C₈	—	2	4	no	—
23	C₂²×C₃×C₈	C₈	1	8	yes	complemented
24	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
26	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	D₄	1	4	yes	—
29	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
31	C₂²×C₃⋊C₈	—	2	8	no	—
32	C₂²×C₃⋊C₈	—	2	4	no	—
33	C₂²×C₃⋊C₈	—	4	4	no	—
34	C₂²×C₃⋊C₈	—	4	4	no	—
35	C₃×C₄×C₈	—	2	4	no	—
36	C₃×C₄×C₈	—	2	4	no	—
37	C₄⋊(C₃⋊C₈)	—	2	8	no	—
38	C₄⋊(C₃⋊C₈)	—	2	4	no	—
39	C₄⋊(C₃⋊C₈)	C₂×C₄	1	8	yes	—
40	C₂³×C₈	—	6	2	no	—
41	C₂³×C₈	Dic₃	1	2	yes	—
42	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊C₈	—	3	8	no	—
44	C₂×C₄⋊C₈	—	3	4	no	—
45	C₂×C₄⋊C₈	—	6	2	no	—
46	C₂×C₄⋊C₈	—	6	4	no	—
47	C₂×C₄×C₈	—	2	4	no	—
48	C₂×C₄×C₈	Dic₃	1	2	yes	—
49	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
50	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	—	2	4	no	—
52	C₂²×C₃×C₄	C₄²	1	2	yes	—
53	C₂²×C₃×C₄	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₄	C₄⋊C₄	1	4	yes	—
56	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×C₈	1	4	yes	—
60	C₂²×C₃×C₄	M₁₆	1	4	yes	—
61	C₂²×C₃×C₄	D₈	1	2	yes	—
62	C₂²×C₃×C₄	SD₁₆	1	4	yes	—
63	C₂²×C₃×C₄	Q₁₆	1	2	yes	—
64	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₈	—	2	16	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	—	2	4	no	—
70	C₂×C₃×C₈	C₂×C₈	1	8	yes	complemented
71	C₃×C₄²	C₄²	1	2	yes	—
72	C₃×C₄²	C₂²⋊C₄	1	4	yes	—
73	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
74	C₃×C₄²	C₄⋊C₄	1	4	yes	—
75	C₃×C₄²	C₂²×C₄	1	2	yes	—
76	C₃×C₄²	C₂×D₄	1	2	yes	—
77	C₂×C₃⋊C₈	—	2	8	no	—
78	C₂×C₃⋊C₈	—	4	16	no	—
79	C₂×C₃⋊C₈	—	4	4	no	—
80	C₂×C₃⋊C₈	—	4	4	no	—
81	C₂³×C₄	Dic₆	1	1	yes	characteristic
82	C₂³×C₄	D₁₂	1	1	yes	characteristic
83	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂²×C₈	—	2	4	no	—
85	C₂²×C₈	—	2	4	no	—
86	C₂²×C₈	—	6	8	no	—
87	C₂²×C₈	—	6	4	no	—
88	C₂²×C₈	—	12	4	no	—
89	C₂²×C₈	—	12	4	no	—
90	C₂²×C₈	C₃⋊C₈	1	8	yes	complemented
91	C₂²×C₈	C₂×Dic₃	1	4	yes	—
92	C₂×C₄²	Dic₆	1	1	yes	characteristic
93	C₂×C₄²	C₄×D₃	1	4	yes	—
94	C₂×C₄²	D₁₂	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₄²	#67	1	4	yes	—
97	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
98	C₄⋊C₈	—	3	8	no	—
99	C₄⋊C₈	—	6	8	no	—
100	C₄⋊C₈	—	6	4	no	—
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	2	4	no	—
103	C₂³×C₃	#95	1	2	yes	—
104	C₂³×C₃	C₄⋊C₈	1	4	yes	—
105	C₂³×C₃	#106	1	2	yes	—
106	C₂³×C₃	#107	1	2	yes	—
107	C₂³×C₃	#108	1	4	yes	—
108	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	8	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	#95	1	2	yes	—
113	C₂×C₃×C₄	#95	1	2	yes	—
114	C₂×C₃×C₄	#95	1	2	yes	—
115	C₂×C₃×C₄	C₄×C₈	1	4	yes	—
116	C₂×C₃×C₄	#97	1	4	yes	—
117	C₂×C₃×C₄	C₂²⋊C₈	1	8	yes	—
118	C₂×C₃×C₄	#102	1	8	yes	—
119	C₂×C₃×C₄	#103	1	8	yes	—
120	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
121	C₂×C₃×C₄	#106	1	2	yes	—
122	C₂×C₃×C₄	#107	1	2	yes	—
123	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
128	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
129	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
131	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
132	C₃×C₈	—	2	8	no	—
133	C₃⋊C₈	—	4	8	no	—
134	C₂⁴	C₄⋊Dic₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	#204	1	4	yes	—
137	C₂²×C₄	D₂₄	1	2	yes	—
138	C₂²×C₄	Dic₁₂	1	2	yes	—
139	C₂²×C₄	C₂×C₃⋊C₈	1	4	yes	—
140	C₂²×C₄	#208	1	4	yes	—
141	C₂²×C₄	C₄×Dic₃	1	2	yes	—
142	C₂²×C₄	Dic₃⋊C₄	1	4	yes	—
143	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
146	C₂²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
147	C₂²×C₄	#217	1	2	yes	—
148	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
151	C₂×C₈	—	2	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	6	8	no	—
155	C₂×C₈	—	12	16	no	—
156	C₂×C₈	—	12	4	no	—
157	C₂×C₈	—	12	4	no	—
158	C₂×C₈	C₂×C₃⋊C₈	1	8	yes	complemented
159	C₄²	C₄×Dic₃	1	2	yes	—
160	C₄²	Dic₃⋊C₄	1	4	yes	—
161	C₄²	(C₂×D₃)⋊C₄	1	4	yes	—
162	C₄²	#217	1	2	yes	—
163	C₄²	C₂×C₄×D₃	1	2	yes	—
164	C₄²	C₂×#67	1	2	yes	—
165	C₂²×C₃	#334	1	8	yes	—
166	C₂²×C₃	#335	1	8	yes	—
167	C₂²×C₃	#336	1	4	yes	—
168	C₂²×C₃	#340	1	4	yes	—
169	C₂²×C₃	#341	1	4	yes	—
170	C₂²×C₃	C₂×#95	1	1	yes	characteristic
171	C₂²×C₃	C₂×C₄⋊C₈	1	2	yes	—
172	C₂²×C₃	C₂×#106	1	1	yes	characteristic
173	C₂²×C₃	C₂×#107	1	1	yes	characteristic
174	C₂²×C₃	C₂×#108	1	2	yes	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	#336	1	4	yes	—
178	C₃×C₄	#340	1	4	yes	—
179	C₃×C₄	C₂×C₄×C₈	1	1	yes	characteristic
180	C₃×C₄	C₂×#97	1	1	yes	characteristic
181	C₃×C₄	C₂×C₂²⋊C₈	1	2	yes	—
182	C₃×C₄	C₂×#102	1	2	yes	—
183	C₃×C₄	C₂×#103	1	2	yes	—
184	C₂³	C₄⋊(C₃⋊C₈)	1	4	yes	—
185	C₂³	#817	1	2	yes	—
186	C₂³	#818	1	2	yes	—
187	C₂³	#819	1	4	yes	—
188	C₂³	C₃⋊#95	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	C₄×C₃⋊C₈	1	4	yes	—
194	C₂×C₄	#803	1	4	yes	—
195	C₂×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
196	C₂×C₄	#816	1	8	yes	—
197	C₂×C₄	#817	1	2	yes	—
198	C₂×C₄	#818	1	2	yes	—
199	C₂×C₄	#821	1	8	yes	—
200	C₂×C₄	#830	1	8	yes	—
201	C₂×C₄	C₃⋊#95	1	2	yes	—
202	C₂×C₄	C₃⋊#95	1	2	yes	—
203	C₂×C₄	C₃⋊#95	1	2	yes	—
204	C₂×C₄	C₂×#204	1	2	yes	—
205	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
208	C₂×C₄	C₂×#208	1	2	yes	—
209	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
210	C₂×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
211	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×#217	1	1	yes	characteristic
213	C₈	—	2	8	no	—
214	C₈	—	12	8	no	—
215	C₂×C₃	#1277	1	8	yes	complemented
216	C₂×C₃	C₂×#334	1	2	yes	—
217	C₂×C₃	C₂×#335	1	2	yes	—
218	C₂×C₃	C₂×#336	1	1	yes	characteristic
219	C₂×C₃	C₂×#340	1	1	yes	characteristic
220	C₂×C₃	C₂×#341	1	1	yes	characteristic
221	C₂²	#4457	1	8	yes	—
222	C₂²	#4458	1	8	yes	—
223	C₂²	#4524	1	4	yes	—
224	C₂²	#4551	1	4	yes	—
225	C₂²	#4552	1	4	yes	—
226	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
227	C₂²	C₂×#817	1	1	yes	characteristic
228	C₂²	C₂×#818	1	1	yes	characteristic
229	C₂²	C₂×#819	1	2	yes	—
230	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	#4524	1	4	yes	—
234	C₄	#4551	1	4	yes	—
235	C₄	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
236	C₄	C₂×#803	1	1	yes	characteristic
237	C₄	C₂×#816	1	2	yes	—
238	C₄	C₂×#821	1	2	yes	—
239	C₄	C₂×#830	1	2	yes	—
240	C₃	C₂×#1277	1	1	yes	characteristic, complemented, Sylow
241	C₂	#67654	1	8	yes	direct product factor
242	C₂	C₂×#4457	1	2	yes	—
243	C₂	C₂×#4458	1	2	yes	—
244	C₂	C₂×#4524	1	1	yes	characteristic
245	C₂	C₂×#4551	1	1	yes	characteristic
246	C₂	C₂×#4552	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₁₂
commutator	C₃×C₄	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle, prime radical	C₂⁴×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₂×#1277
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #67654	C₂	direct (N × H)
23. C₂²×C₃×C₈	C₈	semidirect
70. C₂×C₃×C₈	C₂×C₈	semidirect
90. C₂²×C₈	C₃⋊C₈	semidirect
158. C₂×C₈	C₂×C₃⋊C₈	semidirect
215. C₂×C₃	#1277	semidirect
240. C₃	C₂×#1277	semidirect
241. C₂	#67654	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₈	2
C₂²×C₄⋊(C₃⋊C₈)	2
#67654	2
C₂×#1277	3

Subgroups — C2×#67654 #10684643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#67654 #10684643