Subgroups — C₂×#68063 #10782836

Order 768.

814 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×#4699	C₂	1	1	yes	maximal, characteristic
2	C₂×#4681	C₂	1	2	yes	maximal
3	C₂×#4682	C₂	1	1	yes	maximal, characteristic
4	C₂×#4680	C₂	1	2	yes	maximal
5	C₂×C₃×C₈²	C₂	1	1	yes	maximal, characteristic
6	#68063	C₂	1	8	yes	maximal, direct product factor
7	C₂×#1707	—	3	1	no	maximal, Sylow
8	C₂²×Dic₁₂	—	2	2	no	—
9	C₂×#869	C₂²	1	2	yes	—
10	C₂×#869	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×#818	—	2	2	no	—
15	C₂×#817	—	2	4	no	—
16	C₂×#816	—	2	4	no	—
17	#4699	—	2	2	no	—
18	#4699	C₂²	1	4	yes	—
19	#4682	—	2	2	no	—
20	#4682	C₂²	1	4	yes	—
21	#4681	—	2	4	no	—
22	#4681	C₂²	1	8	yes	—
23	#4680	—	2	4	no	—
24	#4680	C₂²	1	8	yes	—
25	C₃×C₈²	C₂²	1	4	yes	—
26	C₂×#500	—	3	1	no	—
27	C₂×#498	—	3	2	no	—
28	C₂×#494	—	3	1	no	—
29	C₂×#487	—	3	2	no	—
30	#1707	—	3	8	no	—
31	C₂×C₈²	D₃	1	1	yes	characteristic
32	C₂²×Dic₆	—	2	2	no	—
33	C₂²×Dic₆	—	2	2	no	—
34	C₂²×C₃×C₈	D₄	1	2	yes	—
35	C₂²×C₃×C₈	D₄	1	2	yes	—
36	C₂²×C₃×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₃	—	2	2	no	—
39	C₂×C₄⋊Dic₃	—	2	2	no	—
40	C₂×C₄⋊Dic₃	—	2	2	no	—
41	C₂×C₄⋊Dic₃	—	2	2	no	—
42	C₂×Dic₁₂	—	2	8	no	—
43	C₂×Dic₁₂	—	4	8	no	—
44	#869	—	2	2	no	—
45	#869	—	2	2	no	—
46	#869	—	2	2	no	—
47	#869	—	2	2	no	—
48	#869	—	2	2	no	—
49	#869	—	2	2	no	—
50	#869	C₂³	1	4	yes	—
51	#869	C₂³	1	4	yes	—
52	C₃×C₄×C₈	D₄	1	2	yes	—
53	C₃×C₄×C₈	D₄	1	2	yes	—
54	C₃×C₄×C₈	D₄	1	2	yes	—
55	C₃×C₄×C₈	C₂³	1	2	yes	—
56	C₃×C₄×C₈	C₂³	1	2	yes	—
57	C₃×C₄×C₈	C₂³	1	2	yes	—
58	#818	—	2	8	no	—
59	#817	—	2	16	no	—
60	#816	—	2	16	no	—
61	C₂²×Q₁₆	—	6	2	no	—
62	C₂×C₄⋊Q₈	—	3	2	no	—
63	C₂×C₄⋊Q₈	—	3	2	no	—
64	#500	—	3	4	no	—
65	#500	—	6	2	no	—
66	#498	—	3	8	no	—
67	#498	—	6	4	no	—
68	#494	—	3	4	no	—
69	#494	—	6	2	no	—
70	#487	—	3	8	no	—
71	#487	—	6	4	no	—
72	C₂×#107	—	6	2	no	—
73	C₂×#106	—	6	4	no	—
74	C₂×#103	—	6	4	no	—
75	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
78	C₈²	C₂×D₃	1	4	yes	—
79	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₃	—	4	2	no	—
83	C₂²×Dic₃	—	4	2	no	—
84	C₂×Dic₆	—	2	4	no	—
85	C₂×Dic₆	—	2	4	no	—
86	C₂×Dic₆	—	4	8	no	—
87	C₂×Dic₆	—	4	8	no	—
88	C₂×Dic₆	—	4	2	no	—
89	C₂×Dic₆	—	4	2	no	—
90	C₂×C₃×C₈	—	2	4	no	—
91	C₂×C₃×C₈	SD₁₆	1	8	yes	—
92	C₂×C₃×C₈	Q₁₆	1	8	yes	complemented
93	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
94	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
95	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
96	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₃×C₄²	C₂⁴	1	1	yes	characteristic
100	C₄⋊Dic₃	—	2	4	no	—
101	C₄⋊Dic₃	—	2	4	no	—
102	C₄⋊Dic₃	—	2	4	no	—
103	C₄⋊Dic₃	—	2	4	no	—
104	C₄⋊Dic₃	—	4	2	no	—
105	C₄⋊Dic₃	—	4	2	no	—
106	C₄⋊Dic₃	—	4	2	no	—
107	C₄⋊Dic₃	—	4	2	no	—
108	Dic₁₂	—	4	16	no	—
109	C₂²×Q₈	—	6	2	no	—
110	C₂²×Q₈	—	6	2	no	—
111	C₂×Q₁₆	—	6	8	no	—
112	C₂×Q₁₆	—	12	8	no	—
113	C₂²×C₈	D₁₂	1	2	yes	—
114	C₂²×C₈	D₁₂	1	2	yes	—
115	C₂²×C₈	D₁₂	1	2	yes	—
116	C₄⋊Q₈	—	3	4	no	—
117	C₄⋊Q₈	—	3	4	no	—
118	C₄⋊Q₈	—	6	2	no	—
119	C₄⋊Q₈	—	6	2	no	—
120	C₄⋊Q₈	—	6	2	no	—
121	C₄⋊Q₈	—	6	2	no	—
122	C₄⋊Q₈	—	6	2	no	—
123	C₄⋊Q₈	—	6	2	no	—
124	C₂×C₄⋊C₄	—	6	2	no	—
125	C₂×C₄⋊C₄	—	6	2	no	—
126	C₂×C₄⋊C₄	—	6	2	no	—
127	C₂×C₄⋊C₄	—	6	2	no	—
128	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
129	#107	—	6	8	no	—
130	#106	—	6	16	no	—
131	#103	—	6	16	no	—
132	C₄×C₈	D₁₂	1	2	yes	—
133	C₄×C₈	D₁₂	1	2	yes	—
134	C₄×C₈	D₁₂	1	2	yes	—
135	C₄×C₈	C₂²×D₃	1	2	yes	—
136	C₄×C₈	C₂²×D₃	1	2	yes	—
137	C₄×C₈	C₂²×D₃	1	2	yes	—
138	C₂³×C₃	#127	1	1	yes	characteristic
139	C₂×C₃×C₄	#127	1	1	yes	characteristic
140	C₂×C₃×C₄	#127	1	1	yes	characteristic
141	C₂×C₃×C₄	#127	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×SD₁₆	1	4	yes	—
143	C₂×C₃×C₄	C₂×Q₁₆	1	4	yes	—
144	C₂×C₃×C₄	#135	1	4	yes	—
145	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₃	—	4	4	no	—
149	C₂×Dic₃	—	4	4	no	—
150	C₂×Dic₃	—	8	4	no	—
151	C₂×Dic₃	—	8	4	no	—
152	Dic₆	—	4	8	no	—
153	Dic₆	—	4	8	no	—
154	Dic₆	—	8	4	no	—
155	Dic₆	—	8	4	no	—
156	C₃×C₈	—	2	4	no	—
157	C₃×C₈	C₂×SD₁₆	1	8	yes	—
158	C₃×C₈	C₂×Q₁₆	1	8	yes	complemented
159	C₂×Q₈	—	6	4	no	—
160	C₂×Q₈	—	6	4	no	—
161	C₂×Q₈	—	12	2	no	—
162	C₂×Q₈	—	12	2	no	—
163	C₂×Q₈	—	12	8	no	—
164	C₂×Q₈	—	12	8	no	—
165	C₂²×C₄	—	12	2	no	—
166	C₂²×C₄	—	12	2	no	—
167	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
170	Q₁₆	—	12	16	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	#204	1	8	yes	—
173	C₂×C₈	Dic₁₂	1	8	yes	complemented
174	C₂×C₈	C₂×D₁₂	1	4	yes	—
175	C₂×C₈	C₂×D₁₂	1	4	yes	—
176	C₂×C₈	C₂×D₁₂	1	4	yes	—
177	C₄⋊C₄	—	6	4	no	—
178	C₄⋊C₄	—	6	4	no	—
179	C₄⋊C₄	—	6	4	no	—
180	C₄⋊C₄	—	6	4	no	—
181	C₄⋊C₄	—	12	2	no	—
182	C₄⋊C₄	—	12	2	no	—
183	C₄⋊C₄	—	12	2	no	—
184	C₄⋊C₄	—	12	2	no	—
185	C₄²	C₂×D₁₂	1	1	yes	characteristic
186	C₄²	C₂×D₁₂	1	1	yes	characteristic
187	C₄²	C₂×D₁₂	1	1	yes	characteristic
188	C₄²	C₂³×D₃	1	1	yes	characteristic
189	C₂²×C₃	#492	1	2	yes	—
190	C₂²×C₃	#494	1	2	yes	—
191	C₂²×C₃	#495	1	2	yes	—
192	C₂²×C₃	C₂×#127	1	1	yes	characteristic
193	C₃×C₄	#492	1	2	yes	—
194	C₃×C₄	#494	1	2	yes	—
195	C₃×C₄	#495	1	2	yes	—
196	C₃×C₄	C₂²×SD₁₆	1	2	yes	—
197	C₃×C₄	C₂²×Q₁₆	1	2	yes	—
198	C₃×C₄	C₂×#135	1	2	yes	—
199	Dic₃	—	8	4	no	—
200	Dic₃	—	8	4	no	—
201	C₂³	#874	1	1	yes	characteristic
202	Q₈	—	12	8	no	—
203	Q₈	—	12	8	no	—
204	Q₈	—	24	4	no	—
205	Q₈	—	24	4	no	—
206	C₂×C₄	—	12	4	no	—
207	C₂×C₄	—	12	4	no	—
208	C₂×C₄	—	24	4	no	—
209	C₂×C₄	—	24	4	no	—
210	C₂×C₄	#874	1	1	yes	characteristic
211	C₂×C₄	#874	1	1	yes	characteristic
212	C₂×C₄	#874	1	1	yes	characteristic
213	C₂×C₄	C₂×#204	1	4	yes	—
214	C₂×C₄	#904	1	4	yes	—
215	C₂×C₄	C₂×Dic₁₂	1	4	yes	—
216	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₈	C₂×#204	1	8	yes	—
221	C₈	C₂×Dic₁₂	1	8	yes	complemented
222	C₂×C₃	#1707	1	4	yes	complemented
223	C₂×C₃	C₂×#492	1	1	yes	characteristic
224	C₂×C₃	C₂×#494	1	1	yes	characteristic
225	C₂×C₃	C₂×#495	1	1	yes	characteristic
226	C₂²	#4693	1	2	yes	—
227	C₂²	#4696	1	2	yes	—
228	C₂²	#4699	1	2	yes	—
229	C₂²	C₂×#874	1	1	yes	characteristic
230	C₄	—	24	4	no	—
231	C₄	—	24	4	no	—
232	C₄	#4693	1	2	yes	—
233	C₄	#4696	1	2	yes	—
234	C₄	#4699	1	2	yes	—
235	C₄	C₂²×#204	1	2	yes	—
236	C₄	C₂×#904	1	2	yes	—
237	C₄	C₂²×Dic₁₂	1	2	yes	—
238	C₃	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
239	C₂	#68063	1	4	yes	direct product factor
240	C₂	C₂×#4693	1	1	yes	characteristic
241	C₂	C₂×#4696	1	1	yes	characteristic
242	C₂	C₂×#4699	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#874
commutator, cosocle	C₃×C₄²	C₂⁴
Frattini	C₄²	C₂³×D₃
Fitting	C₂×C₃×C₈²	C₂
socle, prime radical	C₂³×C₃	#127
hypercenter	C₂×C₈²	D₃
hypocenter	C₃	C₂×#1707
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. #68063	C₂	direct (N × H)
92. C₂×C₃×C₈	Q₁₆	semidirect
158. C₃×C₈	C₂×Q₁₆	semidirect
173. C₂×C₈	Dic₁₂	semidirect
221. C₈	C₂×Dic₁₂	semidirect
222. C₂×C₃	#1707	semidirect
238. C₃	C₂×#1707	semidirect
239. C₂	#68063	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#4699	2
C₂×#4681	2
C₂×#4682	2
C₂×#4680	2
C₂×C₃×C₈²	2
#68063	2
C₂×#1707	3

Subgroups — C2×#68063 #10782836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#68063 #10782836