Subgroups — C₂×#71359 #10782831

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×C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#5085	C₂	1	2	yes	maximal, complemented
3	C₂×#5088	C₂	1	2	yes	maximal, complemented
4	C₂×#4695	C₂	1	1	yes	maximal, characteristic
5	#71359	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#1706	—	3	1	no	maximal, Sylow
8	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂²×#215	—	2	4	no	—
10	C₂²×D₂₄	—	2	2	no	—
11	C₂×#874	C₂²	1	2	yes	—
12	C₃×#500	—	2	2	no	—
13	C₃×#500	C₂²	1	4	yes	complemented
14	C₂×C₃×#107	—	2	2	no	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#5088	—	2	4	no	—
17	#5088	C₂²	1	8	yes	complemented
18	#5085	—	2	4	no	—
19	#5085	C₂²	1	8	yes	complemented
20	C₂×#809	—	2	4	no	—
21	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	#4695	—	2	2	no	—
24	#4695	C₂²	1	4	yes	—
25	C₈×C₃⋊C₈	C₂²	1	4	yes	complemented
26	C₂×#500	D₃	1	1	yes	characteristic, complemented
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	—	3	1	no	—
32	C₂²×C₃×Q₈	—	2	2	no	—
33	C₂²×D₁₂	—	2	2	no	—
34	C₂²×D₁₂	—	2	2	no	—
35	C₂²×D₁₂	—	2	2	no	—
36	C₂²×C₃×C₈	D₄	1	2	yes	—
37	C₃×C₄⋊Q₈	—	2	2	no	—
38	C₃×C₄⋊Q₈	—	2	2	no	—
39	C₃×C₄⋊Q₈	—	2	2	no	—
40	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₂×C₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
44	C₂×#215	—	2	16	no	—
45	C₂×#215	—	4	16	no	—
46	C₂²×C₃⋊C₈	D₄	1	2	yes	—
47	C₂²×C₃⋊C₈	D₄	1	2	yes	—
48	C₂×D₂₄	—	2	8	no	—
49	C₂×D₂₄	—	4	8	no	—
50	#874	—	2	2	no	—
51	#874	—	2	2	no	—
52	#874	—	2	2	no	—
53	#874	C₂³	1	4	yes	—
54	C₃×#107	—	2	8	no	—
55	C₃×C₄×C₈	D₄	1	2	yes	—
56	C₃×C₄×C₈	C₂³	1	2	yes	—
57	#809	—	2	16	no	—
58	C₄×C₃⋊C₈	D₄	1	2	yes	—
59	C₄×C₃⋊C₈	D₄	1	2	yes	—
60	C₄×C₃⋊C₈	C₂³	1	2	yes	—
61	C₄×C₃⋊C₈	C₂³	1	2	yes	—
62	C₂²×SD₁₆	—	6	4	no	—
63	C₂²×D₈	—	6	2	no	—
64	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
65	C₂×#127	—	3	2	no	—
66	#500	—	2	2	no	—
67	#500	C₂×D₃	1	4	yes	complemented
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	—	2	2	no	—
75	C₂×#102	—	6	4	no	—
76	C₂×C₄×C₈	—	3	1	no	—
77	C₂×C₄×C₈	—	3	1	no	—
78	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
79	C₈²	—	3	4	no	—
80	C₂³×D₃	—	4	2	no	—
81	C₂×C₃×Q₈	—	2	4	no	—
82	C₂×C₃×Q₈	—	4	2	no	—
83	C₂×C₃×Q₈	—	4	8	no	—
84	C₂²×C₃×C₄	—	4	2	no	—
85	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂×D₁₂	—	2	4	no	—
89	C₂×D₁₂	—	2	4	no	—
90	C₂×D₁₂	—	2	4	no	—
91	C₂×D₁₂	—	4	8	no	—
92	C₂×D₁₂	—	4	8	no	—
93	C₂×D₁₂	—	4	8	no	—
94	C₂×D₁₂	—	4	2	no	—
95	C₂×D₁₂	—	4	2	no	—
96	C₂×D₁₂	—	4	2	no	—
97	C₂×C₃×C₈	SD₁₆	1	8	yes	complemented
98	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
99	C₃×C₄⋊C₄	—	2	4	no	—
100	C₃×C₄⋊C₄	—	2	4	no	—
101	C₃×C₄⋊C₄	—	4	2	no	—
102	C₃×C₄⋊C₄	—	4	2	no	—
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	#215	—	4	32	no	—
108	C₂×C₃⋊C₈	—	2	4	no	—
109	C₂×C₃⋊C₈	D₈	1	8	yes	complemented
110	C₂×C₃⋊C₈	C₂×D₄	1	4	yes	—
111	C₂×C₃⋊C₈	C₂×D₄	1	4	yes	—
112	D₂₄	—	4	16	no	—
113	C₂²×Q₈	—	2	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	—	3	2	no	—
122	C₂²×C₈	—	3	2	no	—
123	C₂²×C₈	#67	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂²×D₃	1	4	yes	—
128	#127	—	3	4	no	—
129	#127	—	6	2	no	—
130	#127	—	6	2	no	—
131	#127	—	6	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
135	#107	—	2	8	no	—
136	#102	—	6	16	no	—
137	C₄×C₈	—	3	2	no	—
138	C₄×C₈	—	3	2	no	—
139	C₄×C₈	—	3	2	no	—
140	C₄×C₈	—	3	2	no	—
141	C₄×C₈	#67	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	C₃×Q₈	—	4	8	no	—
149	C₃×Q₈	—	8	4	no	—
150	C₂×C₃×C₄	—	4	4	no	—
151	C₂×C₃×C₄	—	8	4	no	—
152	C₂×C₃×C₄	#127	1	1	yes	characteristic
153	C₂×C₃×C₄	#127	1	1	yes	characteristic
154	C₂×C₃×C₄	#127	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
156	C₂×C₃×C₄	C₂×SD₁₆	1	4	yes	—
157	C₂×C₃×C₄	#135	1	4	yes	—
158	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
161	D₁₂	—	4	8	no	—
162	D₁₂	—	4	8	no	—
163	D₁₂	—	4	8	no	—
164	D₁₂	—	8	4	no	—
165	D₁₂	—	8	4	no	—
166	D₁₂	—	8	4	no	—
167	C₃×C₈	C₂×SD₁₆	1	8	yes	complemented
168	C₃⋊C₈	—	2	4	no	—
169	C₃⋊C₈	C₂×D₈	1	8	yes	complemented
170	C₂⁴	—	12	2	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	8	no	—
174	C₂×D₄	—	6	4	no	—
175	C₂×D₄	—	6	4	no	—
176	C₂×D₄	—	6	4	no	—
177	C₂×D₄	—	12	8	no	—
178	C₂×D₄	—	12	8	no	—
179	C₂×D₄	—	12	2	no	—
180	C₂×D₄	—	12	2	no	—
181	C₂×D₄	—	12	2	no	—
182	C₂×D₄	—	12	8	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
185	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×#67	1	1	yes	characteristic
187	SD₁₆	—	12	32	no	—
188	D₈	—	12	16	no	—
189	C₂×C₈	—	3	8	no	—
190	C₂×C₈	—	3	4	no	—
191	C₂×C₈	—	3	4	no	—
192	C₂×C₈	—	6	4	no	—
193	C₂×C₈	#215	1	8	yes	complemented
194	C₂×C₈	C₂×#67	1	4	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄²	D₃×D₄	1	1	yes	characteristic
200	C₄²	D₃×D₄	1	1	yes	characteristic
201	C₄²	C₂×#67	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	—	8	4	no	—
212	C₃×C₄	#492	1	2	yes	—
213	C₃×C₄	#493	1	2	yes	—
214	C₃×C₄	#495	1	2	yes	—
215	C₃×C₄	C₂²×D₈	1	2	yes	—
216	C₃×C₄	C₂²×SD₁₆	1	2	yes	—
217	C₃×C₄	C₂×#135	1	2	yes	—
218	C₂³	—	12	4	no	—
219	C₂³	—	24	4	no	—
220	C₂³	—	24	4	no	—
221	C₂³	—	24	4	no	—
222	C₂³	#940	1	1	yes	characteristic
223	Q₈	—	4	8	no	—
224	Q₈	—	8	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	—	4	4	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	D₃×D₈	1	4	yes	—
234	C₂×C₄	#919	1	4	yes	—
235	C₂×C₄	#940	1	1	yes	characteristic
236	C₂×C₄	#940	1	1	yes	characteristic
237	C₂×C₄	#940	1	1	yes	characteristic
238	C₂×C₄	C₂×#215	1	4	yes	—
239	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
240	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂²×#67	1	1	yes	characteristic
242	C₈	—	3	8	no	—
243	C₈	—	6	4	no	—
244	C₈	C₂×#215	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	#5085	1	2	yes	—
255	C₂²	#5151	1	2	yes	—
256	C₂²	#5191	1	2	yes	—
257	C₂²	C₂×#940	1	1	yes	characteristic
258	C₄	—	8	4	no	—
259	C₄	#5085	1	2	yes	—
260	C₄	#5151	1	2	yes	—
261	C₄	#5191	1	2	yes	—
262	C₄	C₂×D₃×D₈	1	2	yes	—
263	C₄	C₂×#919	1	2	yes	—
264	C₄	C₂²×#215	1	2	yes	—
265	C₃	C₂×#1706	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	48	4	no	—
267	C₂	#71359	1	4	yes	direct product factor
268	C₂	C₂×#5085	1	1	yes	characteristic
269	C₂	C₂×#5151	1	1	yes	characteristic
270	C₂	C₂×#5191	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#940
commutator, cosocle	C₃×C₄²	C₂⁴
Frattini	C₄²	C₂³×D₃
Fitting	C₂×C₃×#500	C₂
socle	C₂³×C₃	#127
prime radical	C₂×#4695	C₂
hypercenter	C₂×#500	D₃
hypocenter	C₃	C₂×#1706
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#500	C₂	semidirect
2. C₂×#5085	C₂	semidirect
3. C₂×#5088	C₂	semidirect
5. #71359	C₂	direct (N × H)
6. C₂×C₈×C₃⋊C₈	C₂	semidirect
13. C₃×#500	C₂²	semidirect
17. #5088	C₂²	semidirect
19. #5085	C₂²	semidirect
25. C₈×C₃⋊C₈	C₂²	semidirect
26. C₂×#500	D₃	semidirect
67. #500	C₂×D₃	semidirect
97. C₂×C₃×C₈	SD₁₆	semidirect
109. C₂×C₃⋊C₈	D₈	semidirect
167. C₃×C₈	C₂×SD₁₆	semidirect
169. C₃⋊C₈	C₂×D₈	semidirect
193. C₂×C₈	#215	semidirect
244. C₈	C₂×#215	semidirect
245. C₂×C₃	#1706	semidirect
265. C₃	C₂×#1706	semidirect
267. C₂	#71359	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×C₃×#500	2
C₂×#5085	2
C₂×#5088	2
C₂×#4695	2
#71359	2
C₂×C₈×C₃⋊C₈	2
C₂×#1706	3

Subgroups — C2×#71359 #10782831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#71359 #10782831