Subgroups — C₂×#68058 #10782830

Order 768.

1046 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4695	C₂	1	1	yes	maximal, characteristic
2	C₂×#4693	C₂	1	2	yes	maximal, complemented
3	C₂×#4694	C₂	1	2	yes	maximal, complemented
4	C₂×#4682	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	#68058	C₂	1	8	yes	maximal, direct product factor
7	C₂×#1706	—	3	1	no	maximal, Sylow
8	C₂²×D₂₄	—	2	2	no	—
9	C₂²×#204	—	2	4	no	—
10	C₂×#874	C₂²	1	2	yes	—
11	C₂×#869	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×#821	—	2	4	no	—
16	C₂×#818	—	2	2	no	—
17	#4695	—	2	2	no	—
18	#4695	C₂²	1	4	yes	—
19	#4694	—	2	4	no	—
20	#4694	C₂²	1	8	yes	complemented
21	#4693	—	2	4	no	—
22	#4693	C₂²	1	8	yes	complemented
23	#4682	—	2	2	no	—
24	#4682	C₂²	1	4	yes	complemented
25	C₃×C₈²	C₂²	1	4	yes	complemented
26	C₂×#500	—	3	1	no	—
27	C₂×#493	—	3	1	no	—
28	C₂×#492	—	3	2	no	—
29	C₂×#486	—	3	2	no	—
30	#1706	—	3	8	no	—
31	C₂×C₈²	D₃	1	1	yes	characteristic, complemented
32	C₂²×D₁₂	—	2	2	no	—
33	C₂²×D₁₂	—	2	2	no	—
34	C₂²×D₁₂	—	2	2	no	—
35	C₂²×Dic₆	—	2	2	no	—
36	C₂²×C₃×C₈	D₄	1	2	yes	—
37	C₂²×C₃×C₈	D₄	1	2	yes	—
38	C₂²×C₃×C₈	D₄	1	2	yes	—
39	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₃	—	2	2	no	—
41	C₂×C₄⋊Dic₃	—	2	2	no	—
42	C₂×D₂₄	—	2	8	no	—
43	C₂×D₂₄	—	4	8	no	—
44	C₂×#204	—	2	16	no	—
45	C₂×#204	—	4	16	no	—
46	#874	—	2	2	no	—
47	#874	—	2	2	no	—
48	#874	—	2	2	no	—
49	#874	C₂³	1	4	yes	—
50	#869	—	2	2	no	—
51	#869	—	2	2	no	—
52	#869	—	2	2	no	—
53	#869	C₂³	1	4	yes	—
54	C₃×C₄×C₈	D₄	1	2	yes	—
55	C₃×C₄×C₈	D₄	1	2	yes	—
56	C₃×C₄×C₈	D₄	1	2	yes	—
57	C₃×C₄×C₈	C₂³	1	2	yes	—
58	C₃×C₄×C₈	C₂³	1	2	yes	—
59	C₃×C₄×C₈	C₂³	1	2	yes	—
60	#821	—	2	16	no	—
61	#818	—	2	8	no	—
62	C₂²×SD₁₆	—	6	4	no	—
63	C₂²×D₈	—	6	2	no	—
64	C₂×C₄⋊Q₈	—	3	2	no	—
65	C₂×#127	—	3	2	no	—
66	#500	—	3	4	no	—
67	#500	—	6	2	no	—
68	#493	—	3	4	no	—
69	#493	—	6	2	no	—
70	#492	—	3	8	no	—
71	#492	—	6	4	no	—
72	#486	—	3	8	no	—
73	#486	—	6	4	no	—
74	C₂×#107	—	6	2	no	—
75	C₂×#102	—	6	4	no	—
76	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
79	C₈²	C₂×D₃	1	4	yes	complemented
80	C₂³×D₃	—	4	2	no	—
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₃	—	4	2	no	—
85	C₂×D₁₂	—	2	4	no	—
86	C₂×D₁₂	—	2	4	no	—
87	C₂×D₁₂	—	2	4	no	—
88	C₂×D₁₂	—	4	8	no	—
89	C₂×D₁₂	—	4	8	no	—
90	C₂×D₁₂	—	4	8	no	—
91	C₂×D₁₂	—	4	2	no	—
92	C₂×D₁₂	—	4	2	no	—
93	C₂×D₁₂	—	4	2	no	—
94	C₂×Dic₆	—	2	4	no	—
95	C₂×Dic₆	—	4	8	no	—
96	C₂×Dic₆	—	4	2	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	D₈	1	8	yes	complemented
99	C₂×C₃×C₈	SD₁₆	1	8	yes	complemented
100	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
101	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
102	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
103	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
104	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
105	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₃×C₄²	C₂⁴	1	1	yes	characteristic
107	C₄⋊Dic₃	—	2	4	no	—
108	C₄⋊Dic₃	—	2	4	no	—
109	C₄⋊Dic₃	—	4	2	no	—
110	C₄⋊Dic₃	—	4	2	no	—
111	D₂₄	—	4	16	no	—
112	#204	—	4	32	no	—
113	C₂²×Q₈	—	6	2	no	—
114	C₂²×D₄	—	6	2	no	—
115	C₂²×D₄	—	6	2	no	—
116	C₂²×D₄	—	6	2	no	—
117	C₂×SD₁₆	—	6	16	no	—
118	C₂×SD₁₆	—	12	16	no	—
119	C₂×D₈	—	6	8	no	—
120	C₂×D₈	—	12	8	no	—
121	C₂²×C₈	D₁₂	1	2	yes	—
122	C₂²×C₈	D₁₂	1	2	yes	—
123	C₂²×C₈	D₁₂	1	2	yes	—
124	C₄⋊Q₈	—	3	4	no	—
125	C₄⋊Q₈	—	6	2	no	—
126	C₄⋊Q₈	—	6	2	no	—
127	C₄⋊Q₈	—	6	2	no	—
128	#127	—	3	4	no	—
129	#127	—	6	2	no	—
130	#127	—	6	2	no	—
131	#127	—	6	2	no	—
132	C₂×C₄⋊C₄	—	6	2	no	—
133	C₂×C₄⋊C₄	—	6	2	no	—
134	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
135	#107	—	6	8	no	—
136	#102	—	6	16	no	—
137	C₄×C₈	D₁₂	1	2	yes	—
138	C₄×C₈	D₁₂	1	2	yes	—
139	C₄×C₈	D₁₂	1	2	yes	—
140	C₄×C₈	C₂²×D₃	1	2	yes	—
141	C₄×C₈	C₂²×D₃	1	2	yes	—
142	C₄×C₈	C₂²×D₃	1	2	yes	—
143	C₂³×C₃	#127	1	1	yes	characteristic
144	C₂²×D₃	—	4	4	no	—
145	C₂²×D₃	—	8	4	no	—
146	C₂²×D₃	—	8	4	no	—
147	C₂²×D₃	—	8	4	no	—
148	C₂×C₃×C₄	#127	1	1	yes	characteristic
149	C₂×C₃×C₄	#127	1	1	yes	characteristic
150	C₂×C₃×C₄	#127	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
152	C₂×C₃×C₄	C₂×SD₁₆	1	4	yes	—
153	C₂×C₃×C₄	#135	1	4	yes	—
154	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂×Dic₃	—	4	4	no	—
158	C₂×Dic₃	—	8	4	no	—
159	D₁₂	—	4	8	no	—
160	D₁₂	—	4	8	no	—
161	D₁₂	—	4	8	no	—
162	D₁₂	—	8	4	no	—
163	D₁₂	—	8	4	no	—
164	D₁₂	—	8	4	no	—
165	Dic₆	—	4	8	no	—
166	Dic₆	—	8	4	no	—
167	C₃×C₈	—	2	4	no	—
168	C₃×C₈	C₂×D₈	1	8	yes	complemented
169	C₃×C₈	C₂×SD₁₆	1	8	yes	complemented
170	C₂⁴	—	12	2	no	—
171	C₂×Q₈	—	6	4	no	—
172	C₂×Q₈	—	12	2	no	—
173	C₂×Q₈	—	12	8	no	—
174	C₂×D₄	—	6	4	no	—
175	C₂×D₄	—	6	4	no	—
176	C₂×D₄	—	6	4	no	—
177	C₂×D₄	—	12	2	no	—
178	C₂×D₄	—	12	2	no	—
179	C₂×D₄	—	12	2	no	—
180	C₂×D₄	—	12	8	no	—
181	C₂×D₄	—	12	8	no	—
182	C₂×D₄	—	12	8	no	—
183	C₂²×C₄	—	12	2	no	—
184	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
187	SD₁₆	—	12	32	no	—
188	D₈	—	12	16	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	#204	1	8	yes	complemented
191	C₂×C₈	D₂₄	1	8	yes	complemented
192	C₂×C₈	C₂×D₁₂	1	4	yes	—
193	C₂×C₈	C₂×D₁₂	1	4	yes	—
194	C₂×C₈	C₂×D₁₂	1	4	yes	—
195	C₄⋊C₄	—	6	4	no	—
196	C₄⋊C₄	—	6	4	no	—
197	C₄⋊C₄	—	12	2	no	—
198	C₄⋊C₄	—	12	2	no	—
199	C₄²	C₂×D₁₂	1	1	yes	characteristic
200	C₄²	C₂×D₁₂	1	1	yes	characteristic
201	C₄²	C₂×D₁₂	1	1	yes	characteristic
202	C₄²	C₂³×D₃	1	1	yes	characteristic
203	C₂²×C₃	#492	1	2	yes	—
204	C₂²×C₃	#493	1	2	yes	—
205	C₂²×C₃	#495	1	2	yes	—
206	C₂²×C₃	C₂×#127	1	1	yes	characteristic
207	C₂×D₃	—	8	4	no	—
208	C₂×D₃	—	8	4	no	—
209	C₂×D₃	—	8	4	no	—
210	C₂×D₃	—	16	8	no	—
211	C₃×C₄	#492	1	2	yes	—
212	C₃×C₄	#493	1	2	yes	—
213	C₃×C₄	#495	1	2	yes	—
214	C₃×C₄	C₂²×D₈	1	2	yes	—
215	C₃×C₄	C₂²×SD₁₆	1	2	yes	—
216	C₃×C₄	C₂×#135	1	2	yes	—
217	Dic₃	—	8	4	no	—
218	C₂³	—	12	4	no	—
219	C₂³	—	24	4	no	—
220	C₂³	—	24	4	no	—
221	C₂³	—	24	4	no	—
222	C₂³	#874	1	1	yes	characteristic
223	Q₈	—	12	8	no	—
224	Q₈	—	24	4	no	—
225	D₄	—	12	8	no	—
226	D₄	—	12	8	no	—
227	D₄	—	12	8	no	—
228	D₄	—	24	4	no	—
229	D₄	—	24	4	no	—
230	D₄	—	24	4	no	—
231	C₂×C₄	—	12	4	no	—
232	C₂×C₄	—	24	4	no	—
233	C₂×C₄	#874	1	1	yes	characteristic
234	C₂×C₄	#874	1	1	yes	characteristic
235	C₂×C₄	#874	1	1	yes	characteristic
236	C₂×C₄	C₂×#204	1	4	yes	—
237	C₂×C₄	C₂×D₂₄	1	4	yes	—
238	C₂×C₄	#904	1	4	yes	—
239	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	C₂×#204	1	8	yes	complemented
244	C₈	C₂×D₂₄	1	8	yes	complemented
245	C₂×C₃	#1706	1	4	yes	complemented
246	C₂×C₃	C₂×#492	1	1	yes	characteristic
247	C₂×C₃	C₂×#493	1	1	yes	characteristic
248	C₂×C₃	C₂×#495	1	1	yes	characteristic
249	D₃	—	16	4	no	—
250	C₂²	—	24	4	no	—
251	C₂²	—	24	4	no	—
252	C₂²	—	24	4	no	—
253	C₂²	—	48	8	no	—
254	C₂²	#4693	1	2	yes	—
255	C₂²	#4695	1	2	yes	—
256	C₂²	#4696	1	2	yes	—
257	C₂²	C₂×#874	1	1	yes	characteristic
258	C₄	—	24	4	no	—
259	C₄	#4693	1	2	yes	—
260	C₄	#4695	1	2	yes	—
261	C₄	#4696	1	2	yes	—
262	C₄	C₂²×#204	1	2	yes	—
263	C₄	C₂²×D₂₄	1	2	yes	—
264	C₄	C₂×#904	1	2	yes	—
265	C₃	C₂×#1706	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	48	4	no	—
267	C₂	#68058	1	4	yes	direct product factor
268	C₂	C₂×#4693	1	1	yes	characteristic
269	C₂	C₂×#4695	1	1	yes	characteristic
270	C₂	C₂×#4696	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	C₂³×C₃	#127
prime radical	C₂×#4695	C₂
hypercenter	C₂×C₈²	D₃
hypocenter	C₃	C₂×#1706
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
2. C₂×#4693	C₂	semidirect
3. C₂×#4694	C₂	semidirect
4. C₂×#4682	C₂	semidirect
5. C₂×C₃×C₈²	C₂	semidirect
6. #68058	C₂	direct (N × H)
20. #4694	C₂²	semidirect
22. #4693	C₂²	semidirect
24. #4682	C₂²	semidirect
25. C₃×C₈²	C₂²	semidirect
31. C₂×C₈²	D₃	semidirect
79. C₈²	C₂×D₃	semidirect
98. C₂×C₃×C₈	D₈	semidirect
99. C₂×C₃×C₈	SD₁₆	semidirect
168. C₃×C₈	C₂×D₈	semidirect
169. C₃×C₈	C₂×SD₁₆	semidirect
190. C₂×C₈	#204	semidirect
191. C₂×C₈	D₂₄	semidirect
243. C₈	C₂×#204	semidirect
244. C₈	C₂×D₂₄	semidirect
245. C₂×C₃	#1706	semidirect
265. C₃	C₂×#1706	semidirect
267. C₂	#68058	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×#4695	2
C₂×#4693	2
C₂×#4694	2
C₂×#4682	2
C₂×C₃×C₈²	2
#68058	2
C₂×#1706	3

Subgroups — C2×#68058 #10782830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#68058 #10782830