Subgroups — C₂×#70494 #10764249

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×#4925	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#70494	C₂	1	8	yes	maximal, direct product factor
5	C₂×#4479	C₂	1	4	yes	maximal, complemented
6	C₂×#1571	—	3	1	no	maximal, Sylow
7	C₂²×C₃×M₁₆	—	2	1	no	—
8	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂³×C₃⋊C₈	—	2	1	no	—
10	C₃×#423	—	2	2	no	—
11	C₃×#423	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊C₈	—	2	1	no	—
13	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×#830	C₄	1	2	yes	—
15	#4925	C₄	1	4	yes	—
16	#4925	C₂²	1	4	yes	—
17	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₂×C₄⋊(C₃⋊C₈)	—	2	2	no	—
19	C₂×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
20	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
21	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
22	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
23	#4479	C₂²	1	16	yes	complemented
24	C₂×#431	—	3	1	no	—
25	C₂×#423	D₃	1	1	yes	characteristic
26	C₂²×C₄⋊C₈	—	3	1	no	—
27	#1571	—	3	8	no	—
28	C₂×#335	—	3	4	no	—
29	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
31	C₂³×C₃×C₄	Q₈	1	1	yes	characteristic
32	C₂×C₃×M₁₆	—	2	4	no	—
33	C₂×C₃×M₁₆	—	4	4	no	—
34	C₂²×C₃×C₈	—	2	2	no	—
35	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄²	D₄	1	2	yes	—
39	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
40	C₂×C₃×C₄²	Q₈	1	2	yes	—
41	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂³	1	2	yes	—
43	C₂²×C₃⋊C₈	—	2	4	no	—
44	C₂²×C₃⋊C₈	—	2	2	no	—
45	C₂²×C₃⋊C₈	—	4	4	no	—
46	C₂²×C₃⋊C₈	C₂×C₄	1	4	yes	—
47	C₃×C₄⋊C₈	—	2	4	no	—
48	C₃×C₄⋊C₈	—	2	2	no	—
49	C₃×C₄⋊C₈	C₂³	1	4	yes	—
50	#830	C₂×C₄	1	8	yes	—
51	C₄⋊(C₃⋊C₈)	—	2	2	no	—
52	C₄⋊(C₃⋊C₈)	—	2	4	no	—
53	C₄⋊(C₃⋊C₈)	C₂×C₄	1	8	yes	—
54	C₄⋊(C₃⋊C₈)	C₂³	1	4	yes	—
55	C₄×C₃⋊C₈	C₂×C₄	1	4	yes	—
56	C₄×C₃⋊C₈	C₂³	1	4	yes	—
57	C₂²×M₁₆	—	2	1	no	—
58	C₂³×C₈	—	6	1	no	—
59	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
60	#431	—	3	4	no	—
61	#431	—	3	4	no	—
62	#423	—	2	2	no	—
63	#423	C₂×D₃	1	4	yes	—
64	C₂×C₄⋊C₈	—	2	1	no	—
65	C₂×C₄⋊C₈	—	3	2	no	—
66	C₂×C₄⋊C₈	—	3	2	no	—
67	C₂×C₄⋊C₈	—	3	4	no	—
68	C₂×C₄⋊C₈	—	6	2	no	—
69	C₂×C₄⋊C₈	—	6	1	no	—
70	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
71	C₂×C₂²⋊C₈	—	3	2	no	—
72	C₂×C₄×C₈	—	3	2	no	—
73	#335	—	3	16	no	—
74	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	—	2	4	no	—
76	C₂²×C₃×C₄	—	4	2	no	—
77	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
80	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
81	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
82	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₃×C₄	D₈	1	4	yes	—
84	C₂²×C₃×C₄	Q₁₆	1	4	yes	—
85	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
91	C₃×M₁₆	—	4	8	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	—	4	4	no	—
94	C₃×C₄²	C₄⋊C₄	1	2	yes	—
95	C₃×C₄²	C₄⋊C₄	1	2	yes	—
96	C₃×C₄²	C₄⋊C₄	1	2	yes	—
97	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
98	C₃×C₄²	C₂²×C₄	1	2	yes	—
99	C₃×C₄²	C₂×D₄	1	2	yes	—
100	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
101	C₃×C₄²	C₂×Q₈	1	2	yes	—
102	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
103	C₃×C₄²	C₂⁴	1	1	yes	characteristic
104	C₂×C₃⋊C₈	—	2	8	no	—
105	C₂×C₃⋊C₈	—	2	4	no	—
106	C₂×C₃⋊C₈	—	4	4	no	—
107	C₂×C₃⋊C₈	—	4	8	no	—
108	C₂×C₃⋊C₈	C₂²×C₄	1	8	yes	—
109	C₂³×C₄	Dic₆	1	1	yes	characteristic
110	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂³×C₄	#67	1	1	yes	characteristic
112	C₂×M₁₆	—	2	4	no	—
113	C₂×M₁₆	—	4	4	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	3	4	no	—
116	C₂²×C₈	—	6	4	no	—
117	C₂²×C₈	—	6	2	no	—
118	C₂²×C₈	—	12	4	no	—
119	C₂×C₄²	Dic₆	1	1	yes	characteristic
120	C₂×C₄²	Dic₆	1	2	yes	—
121	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
122	C₂×C₄²	C₄×D₃	1	2	yes	—
123	C₂×C₄²	#67	1	2	yes	—
124	C₂×C₄²	#67	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₃	1	2	yes	—
126	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	3	8	no	—
130	C₄⋊C₈	—	3	4	no	—
131	C₄⋊C₈	—	6	2	no	—
132	C₄⋊C₈	—	6	4	no	—
133	C₄⋊C₈	C₂²×D₃	1	4	yes	—
134	C₂²⋊C₈	—	3	8	no	—
135	C₄×C₈	—	3	4	no	—
136	C₄×C₈	—	3	4	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	—	2	2	no	—
139	C₂³×C₃	#107	1	4	yes	—
140	C₂³×C₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	#107	1	8	yes	—
150	C₂×C₃×C₄	#107	1	4	yes	—
151	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	#131	1	4	yes	—
158	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
159	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
160	C₂×C₃×C₄	C₂×Q₁₆	1	4	yes	—
161	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
162	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
165	C₃×C₈	—	4	4	no	—
166	C₃⋊C₈	—	2	8	no	—
167	C₃⋊C₈	—	4	4	no	—
168	C₂⁴	Dic₃⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
173	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
175	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	Dic₃⋊C₄	1	2	yes	—
177	C₂²×C₄	#213	1	4	yes	—
178	C₂²×C₄	#216	1	4	yes	—
179	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₆	1	2	yes	—
181	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
182	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×#67	1	2	yes	—
184	C₂²×C₄	C₂×#67	1	1	yes	characteristic
185	M₁₆	—	4	8	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	3	8	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	6	8	no	—
190	C₂×C₈	—	6	4	no	—
191	C₂×C₈	—	12	4	no	—
192	C₂×C₈	—	12	8	no	—
193	C₄²	Dic₃⋊C₄	1	2	yes	—
194	C₄²	Dic₃⋊C₄	1	2	yes	—
195	C₄²	Dic₃⋊C₄	1	2	yes	—
196	C₄²	C₂×Dic₆	1	2	yes	—
197	C₄²	C₂×Dic₆	1	1	yes	characteristic
198	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
199	C₄²	C₂×C₄×D₃	1	2	yes	—
200	C₄²	C₂×#67	1	2	yes	—
201	C₄²	C₂×#67	1	1	yes	characteristic
202	C₄²	C₂³×D₃	1	1	yes	characteristic
203	C₂²×C₃	—	2	8	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	#424	1	2	yes	—
207	C₂²×C₃	C₂×#107	1	4	yes	—
208	C₂²×C₃	C₂×#107	1	2	yes	—
209	C₂²×C₃	#430	1	2	yes	—
210	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	#424	1	2	yes	—
216	C₃×C₄	C₂×#107	1	2	yes	—
217	C₃×C₄	C₂×#107	1	4	yes	—
218	C₃×C₄	C₂×#131	1	2	yes	—
219	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
220	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	#807	1	4	yes	—
224	C₂³	C₂×Dic₃⋊C₄	1	2	yes	—
225	C₂³	C₂×Dic₃⋊C₄	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	#807	1	8	yes	—
234	C₂×C₄	#807	1	4	yes	—
235	C₂×C₄	#907	1	4	yes	—
236	C₂×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
237	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
239	C₂×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
240	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×#213	1	2	yes	—
243	C₂×C₄	C₂×#213	1	4	yes	—
244	C₂×C₄	C₂×#216	1	4	yes	—
245	C₂×C₄	C₂×#216	1	2	yes	—
246	C₂×C₄	C₂²×Dic₆	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×#67	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	6	8	no	—
251	C₈	—	12	4	no	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	#1571	1	4	yes	complemented
254	C₂×C₃	C₂×#424	1	1	yes	characteristic
255	C₂×C₃	C₂²×#107	1	1	yes	characteristic
256	C₂×C₃	C₂×#430	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×#807	1	4	yes	—
261	C₂²	C₂×#807	1	2	yes	—
262	C₂²	#5119	1	2	yes	—
263	C₂²	#5124	1	2	yes	—
264	C₂²	C₂²×Dic₃⋊C₄	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	C₂×#807	1	4	yes	—
270	C₄	C₂×#807	1	2	yes	—
271	C₄	#5119	1	2	yes	—
272	C₄	C₂×#907	1	2	yes	—
273	C₄	C₂²×#213	1	1	yes	characteristic
274	C₄	C₂²×#216	1	1	yes	characteristic
275	C₃	C₂×#1571	1	1	yes	characteristic, complemented, Sylow
276	C₂	—	2	4	no	—
277	C₂	#70494	1	4	yes	direct product factor
278	C₂	C₂²×#807	1	1	yes	characteristic
279	C₂	C₂×#5119	1	1	yes	characteristic
280	C₂	C₂×#5124	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₄²	C₂³×D₃
Fitting	C₂×C₃×#423	C₂
socle	C₂³×C₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂⁴
hypercenter	C₂×#423	D₃
hypocenter	C₃	C₂×#1571
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #70494	C₂	direct (N × H)
5. C₂×#4479	C₂	semidirect
23. #4479	C₂²	semidirect
253. C₂×C₃	#1571	semidirect
275. C₃	C₂×#1571	semidirect
277. C₂	#70494	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×#423	2
C₂×#4925	2
C₂²×C₄⋊(C₃⋊C₈)	2
#70494	2
C₂×#4479	2
C₂×#1571	3

Subgroups — C2×#70494 #10764249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#70494 #10764249