Subgroups — C₂×#3510 #62808

Order 256.

3651 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3510	C₂	1	32	yes	maximal, direct product factor
2	C₂×#559	C₂	1	4	yes	maximal, complemented
3	C₂×#556	C₂	1	4	yes	maximal, complemented
4	C₂×#551	C₂	1	8	yes	maximal, complemented
5	C₂×#542	C₂	1	4	yes	maximal, complemented
6	C₂×#536	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#535	C₂	1	1	yes	maximal, characteristic
8	C₂×#533	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#529	C₂	1	4	yes	maximal, complemented
10	C₂²×#124	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#519	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#518	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41	C₂²	1	1	yes	characteristic
15	C₂³×Q₈	C₂²	1	1	yes	characteristic
16	C₂³×D₄	C₂²	1	1	yes	characteristic
17	#559	C₂²	1	64	yes	complemented
18	#556	C₂²	1	64	yes	complemented
19	#551	C₂²	1	128	yes	complemented
20	#542	C₂²	1	64	yes	complemented
21	#536	C₂²	1	16	yes	complemented
22	#535	C₂²	1	16	yes	—
23	#533	C₂²	1	16	yes	complemented
24	C₂×C₄⋊Q₈	C₂²	1	2	yes	complemented
25	C₂×C₄⋊Q₈	C₂²	1	2	yes	complemented
26	C₂×#127	C₂²	1	2	yes	—
27	#529	C₂²	1	64	yes	complemented
28	C₂×#126	C₂²	1	8	yes	complemented
29	C₂×#126	C₂²	1	8	yes	complemented
30	C₂×#125	C₂²	1	2	yes	complemented
31	C₂×#124	C₂²	1	2	yes	—
32	C₂×#124	C₂²	1	2	yes	—
33	C₂×#124	C₂²	1	4	yes	—
34	C₂×#124	C₂²	1	2	yes	—
35	C₂×#124	C₂²	1	16	yes	complemented
36	C₂×#124	C₂²	1	2	yes	complemented
37	C₂×#124	C₂²	1	8	yes	complemented
38	C₂×#123	C₂²	1	4	yes	—
39	C₂×#123	C₂²	1	8	yes	complemented
40	C₂×#123	C₂²	1	8	yes	complemented
41	C₂×C₂²⋊Q₈	C₂²	1	4	yes	—
42	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
43	C₂×C₂²⋊Q₈	C₂²	1	8	yes	complemented
44	C₂×#121	C₂²	1	4	yes	—
45	C₂×#120	C₂²	1	4	yes	—
46	#519	C₂²	1	16	yes	complemented
47	#518	C₂²	1	16	yes	complemented
48	C₄×#41	C₂²	1	16	yes	complemented
49	C₂×C₄×Q₈	C₂²	1	2	yes	complemented
50	C₂×C₄×Q₈	C₂²	1	4	yes	complemented
51	C₂×C₄×Q₈	C₂²	1	4	yes	complemented
52	C₂×C₄×Q₈	C₂²	1	4	yes	complemented
53	C₂×C₄×D₄	C₂²	1	4	yes	—
54	C₂×C₄×D₄	C₂²	1	4	yes	—
55	C₂×C₄×D₄	C₂²	1	4	yes	—
56	C₂×C₄×D₄	C₂²	1	2	yes	complemented
57	C₂×C₄×D₄	C₂²	1	8	yes	complemented
58	C₂×#117	C₂²	1	1	yes	characteristic
59	C₂×#117	C₂²	1	1	yes	characteristic
60	C₂×#117	C₂²	1	1	yes	characteristic
61	C₂×#117	C₂²	1	4	yes	—
62	C₂×#117	C₂²	1	2	yes	complemented
63	C₂×#117	C₂²	1	8	yes	complemented
64	C₂²×C₂²⋊C₄	C₂²	1	4	yes	—
65	C₂²×C₂²⋊C₄	C₂²	1	4	yes	—
66	C₂²×C₄²	C₂²	1	1	yes	characteristic
67	C₂²×C₄²	C₂²	1	2	yes	complemented
68	C₂⁵	C₂³	1	2	yes	—
69	C₂×#41	—	2	8	no	—
70	C₂×#41	C₂³	1	8	yes	—
71	C₂²×Q₈	—	2	8	no	—
72	C₂²×Q₈	C₂³	1	2	yes	—
73	C₂²×Q₈	C₂³	1	2	yes	—
74	C₂²×Q₈	C₂³	1	1	yes	characteristic
75	C₂²×Q₈	C₂³	1	8	yes	—
76	C₂²×Q₈	C₂³	1	4	yes	—
77	C₂²×D₄	—	2	8	no	—
78	C₂²×D₄	C₂³	1	2	yes	—
79	C₂²×D₄	C₂³	1	2	yes	—
80	C₂²×D₄	C₂³	1	2	yes	—
81	C₂²×D₄	C₂³	1	1	yes	characteristic
82	C₂²×D₄	C₂³	1	8	yes	—
83	C₂²×D₄	C₂³	1	8	yes	—
84	C₂³×C₄	C₂³	1	2	yes	—
85	C₂³×C₄	C₂³	1	2	yes	—
86	C₂³×C₄	C₂³	1	2	yes	—
87	C₂³×C₄	C₂³	1	8	yes	—
88	C₂³×C₄	C₂³	1	4	yes	—
89	C₂³×C₄	C₂³	1	1	yes	characteristic
90	C₄⋊Q₈	C₂³	1	16	yes	complemented
91	C₄⋊Q₈	C₂³	1	16	yes	complemented
92	#127	C₂³	1	16	yes	—
93	#126	C₂³	1	64	yes	complemented
94	#126	C₂³	1	64	yes	complemented
95	#125	C₂³	1	16	yes	complemented
96	#124	C₂³	1	16	yes	—
97	#124	C₂³	1	16	yes	—
98	#124	C₂³	1	32	yes	—
99	#124	C₂³	1	16	yes	—
100	#124	C₂³	1	16	yes	complemented
101	#124	C₂³	1	64	yes	complemented
102	#123	C₂³	1	32	yes	—
103	#123	C₂³	1	64	yes	complemented
104	#123	C₂³	1	64	yes	complemented
105	C₂²⋊Q₈	C₂³	1	32	yes	—
106	C₂²⋊Q₈	C₂³	1	32	yes	complemented
107	C₂²⋊Q₈	C₂³	1	64	yes	complemented
108	#121	C₂³	1	32	yes	—
109	#120	C₂³	1	32	yes	—
110	C₄×Q₈	C₂³	1	16	yes	complemented
111	C₄×Q₈	C₂³	1	32	yes	complemented
112	C₄×Q₈	C₂³	1	32	yes	complemented
113	C₄×Q₈	C₂³	1	32	yes	complemented
114	C₄×D₄	C₂³	1	32	yes	—
115	C₄×D₄	C₂³	1	32	yes	—
116	C₄×D₄	C₂³	1	32	yes	—
117	C₄×D₄	C₂³	1	16	yes	complemented
118	C₄×D₄	C₂³	1	64	yes	complemented
119	#117	C₂³	1	8	yes	—
120	#117	C₂³	1	8	yes	—
121	#117	C₂³	1	8	yes	—
122	#117	C₂³	1	32	yes	—
123	#117	C₂³	1	16	yes	complemented
124	#117	C₂³	1	64	yes	complemented
125	C₂×C₄⋊C₄	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄	C₂³	1	2	yes	—
127	C₂×C₄⋊C₄	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄	C₂³	1	8	yes	—
129	C₂×C₄⋊C₄	C₂³	1	4	yes	—
130	C₂×C₄⋊C₄	C₂³	1	4	yes	—
131	C₂×C₄⋊C₄	C₂³	1	8	yes	—
132	C₂×C₄⋊C₄	C₂³	1	4	yes	—
133	C₂×C₄⋊C₄	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
137	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
138	C₂×C₂²⋊C₄	C₂³	1	32	yes	—
139	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
140	C₂×C₂²⋊C₄	C₂³	1	32	yes	—
141	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
143	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
144	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
145	C₂×C₄²	C₂³	1	2	yes	—
146	C₂×C₄²	C₂³	1	8	yes	—
147	C₂×C₄²	C₂³	1	2	yes	—
148	C₂×C₄²	C₂³	1	2	yes	—
149	C₂×C₄²	C₂³	1	4	yes	—
150	C₂×C₄²	C₂³	1	1	yes	characteristic
151	C₂×C₄²	C₂³	1	2	yes	—
152	C₂×C₄²	C₂³	1	8	yes	—
153	C₂×C₄²	C₂³	1	1	yes	characteristic
154	C₂×C₄²	C₂³	1	16	yes	complemented
155	C₂⁴	—	2	8	no	—
156	C₂⁴	—	4	8	no	—
157	C₂⁴	C₂⁴	1	4	yes	—
158	C₂⁴	C₂⁴	1	2	yes	—
159	C₂⁴	C₂⁴	1	1	yes	characteristic
160	C₂⁴	C₂⁴	1	8	yes	—
161	#41	—	2	32	no	—
162	C₂×Q₈	—	2	8	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	2	32	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	8	no	—
167	C₂×Q₈	C₂⁴	1	8	yes	—
168	C₂×Q₈	C₂⁴	1	8	yes	—
169	C₂×Q₈	C₂⁴	1	4	yes	—
170	C₂×Q₈	C₂⁴	1	16	yes	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	—	2	8	no	—
173	C₂×D₄	—	2	8	no	—
174	C₂×D₄	—	2	32	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	16	no	—
177	C₂×D₄	C₂⁴	1	8	yes	—
178	C₂×D₄	C₂⁴	1	8	yes	—
179	C₂×D₄	C₂⁴	1	8	yes	—
180	C₂×D₄	C₂⁴	1	4	yes	—
181	C₂×D₄	C₂⁴	1	32	yes	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	—	4	16	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	#41	1	8	yes	—
188	C₂²×C₄	C₂⁴	1	8	yes	—
189	C₂²×C₄	C₂⁴	1	8	yes	—
190	C₂²×C₄	C₂⁴	1	4	yes	—
191	C₂²×C₄	C₂⁴	1	2	yes	—
192	C₂²×C₄	C₂⁴	1	4	yes	—
193	C₂²×C₄	C₂⁴	1	4	yes	—
194	C₂²×C₄	C₂⁴	1	2	yes	—
195	C₂²×C₄	C₂⁴	1	2	yes	—
196	C₂²×C₄	C₂⁴	1	4	yes	—
197	C₂²×C₄	C₂⁴	1	8	yes	—
198	C₂²×C₄	C₂⁴	1	16	yes	—
199	C₂²×C₄	C₂⁴	1	32	yes	—
200	C₂²×C₄	C₂⁴	1	4	yes	—
201	C₂²×C₄	C₂⁴	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴	1	1	yes	characteristic
203	C₄⋊C₄	C₂⁴	1	8	yes	—
204	C₄⋊C₄	C₂⁴	1	8	yes	—
205	C₄⋊C₄	C₂⁴	1	8	yes	—
206	C₄⋊C₄	C₂⁴	1	32	yes	—
207	C₄⋊C₄	C₂⁴	1	16	yes	—
208	C₄⋊C₄	C₂⁴	1	16	yes	—
209	C₄⋊C₄	C₂⁴	1	32	yes	—
210	C₄⋊C₄	C₂⁴	1	16	yes	—
211	C₄⋊C₄	C₂⁴	1	16	yes	—
212	C₂²⋊C₄	C₂⁴	1	8	yes	—
213	C₂²⋊C₄	C₂⁴	1	8	yes	—
214	C₂²⋊C₄	C₂⁴	1	32	yes	—
215	C₂²⋊C₄	C₂⁴	1	32	yes	—
216	C₂²⋊C₄	C₂⁴	1	32	yes	—
217	C₂²⋊C₄	C₂⁴	1	8	yes	—
218	C₂²⋊C₄	C₂⁴	1	16	yes	—
219	C₂²⋊C₄	C₂⁴	1	32	yes	—
220	C₂²⋊C₄	C₂⁴	1	32	yes	—
221	C₄²	C₂⁴	1	4	yes	—
222	C₄²	C₂⁴	1	4	yes	—
223	C₄²	C₂⁴	1	8	yes	—
224	C₄²	C₂⁴	1	8	yes	—
225	C₄²	C₂⁴	1	16	yes	—
226	C₄²	C₂⁴	1	32	yes	—
227	C₄²	C₂⁴	1	8	yes	—
228	C₄²	C₂⁴	1	8	yes	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	8	no	—
232	C₂³	—	2	8	no	—
233	C₂³	—	2	16	no	—
234	C₂³	—	2	16	no	—
235	C₂³	—	4	16	no	—
236	C₂³	—	8	16	no	—
237	C₂³	C₂×#41	1	4	yes	—
238	C₂³	C₂⁵	1	8	yes	—
239	C₂³	C₂⁵	1	4	yes	—
240	C₂³	C₂⁵	1	2	yes	—
241	C₂³	C₂⁵	1	1	yes	characteristic
242	Q₈	—	2	16	no	—
243	Q₈	—	2	16	no	—
244	Q₈	—	2	8	no	—
245	Q₈	—	4	16	no	—
246	D₄	—	2	16	no	—
247	D₄	—	2	16	no	—
248	D₄	—	2	16	no	—
249	D₄	—	2	8	no	—
250	D₄	—	4	32	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	4	32	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	C₂×#41	1	4	yes	—
264	C₂×C₄	C₂×#41	1	4	yes	—
265	C₂×C₄	C₂×#41	1	16	yes	—
266	C₂×C₄	C₂⁵	1	4	yes	—
267	C₂×C₄	C₂⁵	1	8	yes	—
268	C₂×C₄	C₂⁵	1	8	yes	—
269	C₂×C₄	C₂⁵	1	8	yes	—
270	C₂×C₄	C₂⁵	1	4	yes	—
271	C₂×C₄	C₂⁵	1	4	yes	—
272	C₂×C₄	C₂⁵	1	8	yes	—
273	C₂×C₄	C₂⁵	1	2	yes	—
274	C₂×C₄	C₂⁵	1	2	yes	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	16	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	16	no	—
283	C₂²	—	8	16	no	—
284	C₂²	C₂²×#41	1	4	yes	—
285	C₂²	C₂²×#41	1	2	yes	—
286	C₂²	#585	1	4	yes	—
287	C₂²	C₂⁶	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	8	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	8	no	—
292	C₄	—	2	8	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	8	no	—
295	C₄	C₂²×#41	1	4	yes	—
296	C₄	C₂²×#41	1	4	yes	—
297	C₂	—	2	4	no	—
298	C₂	—	4	4	no	—
299	C₂	—	4	8	no	—
300	C₂	#3510	1	4	yes	direct product factor
301	C₂	C₂³×#41	1	1	yes	characteristic
302	C₂	C₂×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂×#535	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3510	C₂	direct (N × H)
2. C₂×#559	C₂	semidirect
3. C₂×#556	C₂	semidirect
4. C₂×#551	C₂	semidirect
5. C₂×#542	C₂	semidirect
6. C₂×#536	C₂	semidirect
8. C₂×#533	C₂	semidirect
9. C₂×#529	C₂	semidirect
10. C₂²×#124	C₂	semidirect
11. C₂×#519	C₂	semidirect
12. C₂×#518	C₂	semidirect
13. C₂×C₄×#41	C₂	semidirect
17. #559	C₂²	semidirect
18. #556	C₂²	semidirect
19. #551	C₂²	semidirect
20. #542	C₂²	semidirect
21. #536	C₂²	semidirect
23. #533	C₂²	semidirect
24. C₂×C₄⋊Q₈	C₂²	semidirect
25. C₂×C₄⋊Q₈	C₂²	semidirect
27. #529	C₂²	semidirect
28. C₂×#126	C₂²	semidirect
29. C₂×#126	C₂²	semidirect
30. C₂×#125	C₂²	semidirect
35. C₂×#124	C₂²	semidirect
36. C₂×#124	C₂²	semidirect
37. C₂×#124	C₂²	semidirect
39. C₂×#123	C₂²	semidirect
40. C₂×#123	C₂²	semidirect
42. C₂×C₂²⋊Q₈	C₂²	semidirect
43. C₂×C₂²⋊Q₈	C₂²	semidirect
46. #519	C₂²	semidirect
47. #518	C₂²	semidirect
48. C₄×#41	C₂²	semidirect
49. C₂×C₄×Q₈	C₂²	semidirect
50. C₂×C₄×Q₈	C₂²	semidirect
51. C₂×C₄×Q₈	C₂²	semidirect
52. C₂×C₄×Q₈	C₂²	semidirect
56. C₂×C₄×D₄	C₂²	semidirect
57. C₂×C₄×D₄	C₂²	semidirect
62. C₂×#117	C₂²	semidirect
63. C₂×#117	C₂²	semidirect
67. C₂²×C₄²	C₂²	semidirect
90. C₄⋊Q₈	C₂³	semidirect
91. C₄⋊Q₈	C₂³	semidirect
93. #126	C₂³	semidirect
94. #126	C₂³	semidirect
95. #125	C₂³	semidirect
100. #124	C₂³	semidirect
101. #124	C₂³	semidirect
103. #123	C₂³	semidirect
104. #123	C₂³	semidirect
106. C₂²⋊Q₈	C₂³	semidirect
107. C₂²⋊Q₈	C₂³	semidirect
110. C₄×Q₈	C₂³	semidirect
111. C₄×Q₈	C₂³	semidirect
112. C₄×Q₈	C₂³	semidirect
113. C₄×Q₈	C₂³	semidirect
117. C₄×D₄	C₂³	semidirect
118. C₄×D₄	C₂³	semidirect
123. #117	C₂³	semidirect
124. #117	C₂³	semidirect
154. C₂×C₄²	C₂³	semidirect
300. C₂	#3510	direct (N × H)

Maximal subgroups

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

subgroup	index
#3510	2
C₂×#559	2
C₂×#556	2
C₂×#551	2
C₂×#542	2
C₂×#536	2
C₂×#535	2
C₂×#533	2
C₂×#529	2
C₂²×#124	2
C₂×#519	2
C₂×#518	2
C₂×C₄×#41	2

Subgroups — C2×#3510 #62808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3510 #62808