Subgroups — C₂×#3484 #62763

Order 256.

7219 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#3484	C₂	1	32	yes	maximal, direct product factor
4	C₂×#546	C₂	1	8	yes	maximal, complemented
5	C₂×D₄²	C₂	1	8	yes	maximal, complemented
6	C₂×#535	C₂	1	2	yes	maximal, complemented
7	C₂×#534	C₂	1	4	yes	maximal, complemented
8	C₂×#525	C₂	1	2	yes	maximal, complemented
9	C₂²×#120	C₂	1	4	yes	maximal, complemented
10	C₂×#518	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁶	C₂²	1	2	yes	complemented
12	C₂×H(5,𝔽₂)	—	2	16	no	—
13	C₂×H(5,𝔽₂)	C₂²	1	16	yes	complemented
14	C₂²×#41	C₂²	1	2	yes	complemented
15	C₂²×#41	C₂²	1	4	yes	complemented
16	C₂³×D₄	—	2	16	no	—
17	C₂³×D₄	C₂²	1	1	yes	characteristic
18	C₂³×D₄	C₂²	1	2	yes	complemented
19	C₂³×D₄	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₄	C₂²	1	8	yes	complemented
21	C₂³×D₄	C₂²	1	4	yes	complemented
22	C₂³×D₄	C₂²	1	16	yes	complemented
23	C₂³×D₄	C₂²	1	4	yes	complemented
24	C₂³×D₄	C₂²	1	8	yes	complemented
25	C₂⁴×C₄	C₂²	1	1	yes	characteristic, complemented
26	#546	C₂²	1	128	yes	complemented
27	D₄²	C₂²	1	128	yes	complemented
28	#535	C₂²	1	32	yes	complemented
29	#534	C₂²	1	64	yes	complemented
30	C₂×#127	C₂²	1	4	yes	complemented
31	C₂×#124	C₂²	1	4	yes	complemented
32	#525	C₂²	1	32	yes	complemented
33	C₂×#123	C₂²	1	4	yes	complemented
34	C₂×#123	C₂²	1	8	yes	complemented
35	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
36	C₂×#121	C₂²	1	4	yes	complemented
37	C₂×#121	C₂²	1	16	yes	complemented
38	C₂×#121	C₂²	1	8	yes	complemented
39	C₂×#120	C₂²	1	4	yes	complemented
40	C₂×#120	C₂²	1	8	yes	complemented
41	C₂×#120	C₂²	1	16	yes	complemented
42	C₂×#120	C₂²	1	8	yes	complemented
43	C₂×#120	C₂²	1	64	yes	complemented
44	#518	C₂²	1	16	yes	complemented
45	C₂×C₄×D₄	C₂²	1	8	yes	complemented
46	C₂×C₄×D₄	C₂²	1	8	yes	complemented
47	C₂×#117	C₂²	1	2	yes	complemented
48	C₂²×C₂²⋊C₄	C₂²	1	4	yes	complemented
49	C₂²×C₂²⋊C₄	C₂²	1	8	yes	complemented
50	C₂⁵	—	2	16	no	—
51	C₂⁵	—	4	16	no	—
52	C₂⁵	C₂³	1	1	yes	characteristic
53	C₂⁵	C₂³	1	8	yes	—
54	C₂⁵	C₂³	1	2	yes	—
55	C₂⁵	C₂³	1	2	yes	—
56	C₂⁵	C₂³	1	16	yes	complemented
57	C₂⁵	C₂³	1	4	yes	complemented
58	H(5,𝔽₂)	—	2	128	no	—
59	C₂×#41	—	2	32	no	—
60	C₂×#41	—	2	16	no	—
61	C₂×#41	C₂³	1	16	yes	complemented
62	C₂×#41	C₂³	1	32	yes	complemented
63	C₂²×Q₈	C₂³	1	2	yes	complemented
64	C₂²×D₄	—	2	16	no	—
65	C₂²×D₄	—	2	32	no	—
66	C₂²×D₄	—	2	64	no	—
67	C₂²×D₄	—	2	128	no	—
68	C₂²×D₄	—	2	32	no	—
69	C₂²×D₄	—	2	8	no	—
70	C₂²×D₄	—	4	32	no	—
71	C₂²×D₄	D₄	1	16	yes	complemented
72	C₂²×D₄	C₂³	1	8	yes	—
73	C₂²×D₄	C₂³	1	4	yes	—
74	C₂²×D₄	C₂³	1	8	yes	—
75	C₂²×D₄	C₂³	1	4	yes	—
76	C₂²×D₄	C₂³	1	4	yes	—
77	C₂²×D₄	C₂³	1	8	yes	—
78	C₂²×D₄	C₂³	1	16	yes	complemented
79	C₂²×D₄	C₂³	1	8	yes	complemented
80	C₂²×D₄	C₂³	1	64	yes	complemented
81	C₂²×D₄	C₂³	1	8	yes	complemented
82	C₂²×D₄	C₂³	1	2	yes	complemented
83	C₂²×D₄	C₂³	1	16	yes	complemented
84	C₂²×D₄	C₂³	1	32	yes	complemented
85	C₂²×D₄	C₂³	1	32	yes	complemented
86	C₂²×D₄	C₂³	1	64	yes	complemented
87	C₂³×C₄	—	2	8	no	—
88	C₂³×C₄	C₂³	1	4	yes	—
89	C₂³×C₄	C₂³	1	1	yes	characteristic
90	C₂³×C₄	C₂³	1	1	yes	characteristic
91	C₂³×C₄	C₂³	1	8	yes	complemented
92	C₂³×C₄	C₂³	1	4	yes	complemented
93	C₂³×C₄	C₂³	1	4	yes	complemented
94	C₂³×C₄	C₂³	1	8	yes	complemented
95	#127	C₂³	1	32	yes	complemented
96	#124	C₂³	1	32	yes	complemented
97	#123	C₂³	1	32	yes	complemented
98	#123	C₂³	1	64	yes	complemented
99	C₂²⋊Q₈	C₂³	1	32	yes	complemented
100	#121	C₂³	1	32	yes	complemented
101	#121	C₂³	1	128	yes	complemented
102	#121	C₂³	1	64	yes	complemented
103	#120	C₂³	1	32	yes	complemented
104	#120	C₂³	1	64	yes	complemented
105	#120	C₂³	1	128	yes	complemented
106	#120	C₂³	1	64	yes	complemented
107	C₄×D₄	C₂³	1	64	yes	complemented
108	C₄×D₄	C₂³	1	64	yes	complemented
109	#117	C₂³	1	16	yes	complemented
110	C₂×C₄⋊C₄	C₂³	1	4	yes	complemented
111	C₂×C₄⋊C₄	C₂³	1	8	yes	complemented
112	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
114	C₂×C₂²⋊C₄	C₂³	1	32	yes	complemented
115	C₂×C₂²⋊C₄	C₂³	1	64	yes	complemented
116	C₂×C₂²⋊C₄	C₂³	1	8	yes	complemented
117	C₂×C₂²⋊C₄	C₂³	1	16	yes	complemented
118	C₂×C₂²⋊C₄	C₂³	1	8	yes	complemented
119	C₂×C₄²	C₂³	1	4	yes	complemented
120	C₂⁴	—	2	4	no	—
121	C₂⁴	—	2	8	no	—
122	C₂⁴	—	2	32	no	—
123	C₂⁴	—	2	8	no	—
124	C₂⁴	—	2	64	no	—
125	C₂⁴	—	2	32	no	—
126	C₂⁴	—	2	16	no	—
127	C₂⁴	—	4	8	no	—
128	C₂⁴	—	4	16	no	—
129	C₂⁴	—	4	8	no	—
130	C₂⁴	—	4	64	no	—
131	C₂⁴	—	4	16	no	—
132	C₂⁴	—	4	128	no	—
133	C₂⁴	C₂×D₄	1	16	yes	complemented
134	C₂⁴	C₂⁴	1	4	yes	—
135	C₂⁴	C₂⁴	1	4	yes	—
136	C₂⁴	C₂⁴	1	4	yes	—
137	C₂⁴	C₂⁴	1	32	yes	—
138	C₂⁴	C₂⁴	1	1	yes	characteristic
139	C₂⁴	C₂⁴	1	4	yes	—
140	C₂⁴	C₂⁴	1	2	yes	—
141	C₂⁴	C₂⁴	1	8	yes	—
142	C₂⁴	C₂⁴	1	8	yes	—
143	C₂⁴	C₂⁴	1	16	yes	complemented
144	#41	—	2	128	no	—
145	#41	—	2	64	no	—
146	C₂×Q₈	—	2	8	no	—
147	C₂×Q₈	C₂⁴	1	8	yes	complemented
148	C₂×D₄	—	2	32	no	—
149	C₂×D₄	—	2	16	no	—
150	C₂×D₄	—	2	32	no	—
151	C₂×D₄	—	2	64	no	—
152	C₂×D₄	—	2	128	no	—
153	C₂×D₄	—	2	32	no	—
154	C₂×D₄	—	2	256	no	—
155	C₂×D₄	—	2	128	no	—
156	C₂×D₄	—	2	16	no	—
157	C₂×D₄	—	2	8	no	—
158	C₂×D₄	—	2	32	no	—
159	C₂×D₄	—	4	32	no	—
160	C₂×D₄	—	4	128	no	—
161	C₂×D₄	C₂×D₄	1	32	yes	complemented
162	C₂×D₄	C₂×D₄	1	64	yes	complemented
163	C₂×D₄	C₂⁴	1	32	yes	—
164	C₂×D₄	C₂⁴	1	16	yes	—
165	C₂×D₄	C₂⁴	1	32	yes	—
166	C₂×D₄	C₂⁴	1	16	yes	—
167	C₂×D₄	C₂⁴	1	16	yes	—
168	C₂×D₄	C₂⁴	1	32	yes	complemented
169	C₂×D₄	C₂⁴	1	8	yes	complemented
170	C₂×D₄	C₂⁴	1	64	yes	complemented
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	16	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	16	no	—
175	C₂²×C₄	—	2	32	no	—
176	C₂²×C₄	—	4	16	no	—
177	C₂²×C₄	C₂×D₄	1	8	yes	complemented
178	C₂²×C₄	C₂⁴	1	16	yes	—
179	C₂²×C₄	C₂⁴	1	4	yes	—
180	C₂²×C₄	C₂⁴	1	8	yes	—
181	C₂²×C₄	C₂⁴	1	4	yes	—
182	C₂²×C₄	C₂⁴	1	2	yes	—
183	C₂²×C₄	C₂⁴	1	2	yes	—
184	C₂²×C₄	C₂⁴	1	4	yes	—
185	C₂²×C₄	C₂⁴	1	16	yes	complemented
186	C₂²×C₄	C₂⁴	1	16	yes	complemented
187	C₂²×C₄	C₂⁴	1	32	yes	complemented
188	C₄⋊C₄	C₂⁴	1	16	yes	complemented
189	C₄⋊C₄	C₂⁴	1	32	yes	complemented
190	C₂²⋊C₄	C₂⁴	1	16	yes	—
191	C₂²⋊C₄	C₂⁴	1	32	yes	—
192	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
193	C₂²⋊C₄	C₂⁴	1	64	yes	complemented
194	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
195	C₄²	C₂⁴	1	16	yes	complemented
196	C₂³	—	2	32	no	—
197	C₂³	—	2	8	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	16	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	2	16	no	—
203	C₂³	—	2	8	no	—
204	C₂³	—	2	64	no	—
205	C₂³	—	2	16	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	16	no	—
208	C₂³	—	2	16	no	—
209	C₂³	—	2	64	no	—
210	C₂³	—	2	32	no	—
211	C₂³	—	4	32	no	—
212	C₂³	—	4	16	no	—
213	C₂³	—	4	32	no	—
214	C₂³	—	4	16	no	—
215	C₂³	—	4	32	no	—
216	C₂³	—	4	128	no	—
217	C₂³	—	4	256	no	—
218	C₂³	—	4	32	no	—
219	C₂³	—	8	32	no	—
220	C₂³	C₂²×D₄	1	4	yes	—
221	C₂³	C₂²×D₄	1	16	yes	—
222	C₂³	C₂²×D₄	1	32	yes	complemented
223	C₂³	H(5,𝔽₂)	1	8	yes	complemented
224	C₂³	C₂⁵	1	8	yes	—
225	C₂³	C₂⁵	1	8	yes	—
226	C₂³	C₂⁵	1	2	yes	—
227	C₂³	C₂⁵	1	8	yes	—
228	C₂³	C₂⁵	1	4	yes	—
229	C₂³	C₂⁵	1	1	yes	characteristic
230	Q₈	—	2	16	no	—
231	D₄	—	2	64	no	—
232	D₄	—	2	32	no	—
233	D₄	—	2	64	no	—
234	D₄	—	2	64	no	—
235	D₄	—	2	32	no	—
236	D₄	—	2	16	no	—
237	D₄	—	4	64	no	—
238	D₄	C₂²×D₄	1	64	yes	complemented
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	32	no	—
243	C₂×C₄	—	2	16	no	—
244	C₂×C₄	—	2	32	no	—
245	C₂×C₄	—	2	32	no	—
246	C₂×C₄	—	4	32	no	—
247	C₂×C₄	C₂²×D₄	1	8	yes	—
248	C₂×C₄	C₂²×D₄	1	16	yes	complemented
249	C₂×C₄	C₂⁵	1	16	yes	—
250	C₂×C₄	C₂⁵	1	8	yes	—
251	C₂×C₄	C₂⁵	1	4	yes	—
252	C₂×C₄	C₂⁵	1	4	yes	—
253	C₂²	—	2	32	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	16	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	16	no	—
260	C₂²	—	2	32	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	2	16	no	—
264	C₂²	—	2	16	no	—
265	C₂²	—	4	32	no	—
266	C₂²	—	4	128	no	—
267	C₂²	—	4	32	no	—
268	C₂²	—	4	16	no	—
269	C₂²	—	4	16	no	—
270	C₂²	—	8	32	no	—
271	C₂²	C₂³×D₄	1	4	yes	—
272	C₂²	C₂³×D₄	1	16	yes	—
273	C₂²	C₂³×D₄	1	2	yes	—
274	C₂²	C₂×H(5,𝔽₂)	1	4	yes	—
275	C₂²	C₂×H(5,𝔽₂)	1	8	yes	complemented
276	C₂²	C₂⁶	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	2	8	no	—
279	C₄	—	2	16	no	—
280	C₄	C₂³×D₄	1	8	yes	—
281	C₂	—	2	4	no	—
282	C₂	—	2	8	no	—
283	C₂	—	2	16	no	—
284	C₂	—	4	8	no	—
285	C₂	—	4	8	no	—
286	C₂	#3484	1	4	yes	direct product factor
287	C₂	C₂⁴×D₄	1	1	yes	characteristic
288	C₂	C₂²×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
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. C₂²×H(5,𝔽₂)	C₂	semidirect
2. C₂⁴×D₄	C₂	semidirect
3. #3484	C₂	direct (N × H)
4. C₂×#546	C₂	semidirect
5. C₂×D₄²	C₂	semidirect
6. C₂×#535	C₂	semidirect
7. C₂×#534	C₂	semidirect
8. C₂×#525	C₂	semidirect
9. C₂²×#120	C₂	semidirect
10. C₂×#518	C₂	semidirect
11. C₂⁶	C₂²	semidirect
13. C₂×H(5,𝔽₂)	C₂²	semidirect
14. C₂²×#41	C₂²	semidirect
15. C₂²×#41	C₂²	semidirect
18. C₂³×D₄	C₂²	semidirect
19. C₂³×D₄	C₂²	semidirect
20. C₂³×D₄	C₂²	semidirect
21. C₂³×D₄	C₂²	semidirect
22. C₂³×D₄	C₂²	semidirect
23. C₂³×D₄	C₂²	semidirect
24. C₂³×D₄	C₂²	semidirect
25. C₂⁴×C₄	C₂²	semidirect
26. #546	C₂²	semidirect
27. D₄²	C₂²	semidirect
28. #535	C₂²	semidirect
29. #534	C₂²	semidirect
30. C₂×#127	C₂²	semidirect
31. C₂×#124	C₂²	semidirect
32. #525	C₂²	semidirect
33. C₂×#123	C₂²	semidirect
34. C₂×#123	C₂²	semidirect
35. C₂×C₂²⋊Q₈	C₂²	semidirect
36. C₂×#121	C₂²	semidirect
37. C₂×#121	C₂²	semidirect
38. C₂×#121	C₂²	semidirect
39. C₂×#120	C₂²	semidirect
40. C₂×#120	C₂²	semidirect
41. C₂×#120	C₂²	semidirect
42. C₂×#120	C₂²	semidirect
43. C₂×#120	C₂²	semidirect
44. #518	C₂²	semidirect
45. C₂×C₄×D₄	C₂²	semidirect
46. C₂×C₄×D₄	C₂²	semidirect
47. C₂×#117	C₂²	semidirect
48. C₂²×C₂²⋊C₄	C₂²	semidirect
49. C₂²×C₂²⋊C₄	C₂²	semidirect
56. C₂⁵	C₂³	semidirect
57. C₂⁵	C₂³	semidirect
61. C₂×#41	C₂³	semidirect
62. C₂×#41	C₂³	semidirect
63. C₂²×Q₈	C₂³	semidirect
71. C₂²×D₄	D₄	semidirect
78. C₂²×D₄	C₂³	semidirect
79. C₂²×D₄	C₂³	semidirect
80. C₂²×D₄	C₂³	semidirect
81. C₂²×D₄	C₂³	semidirect
82. C₂²×D₄	C₂³	semidirect
83. C₂²×D₄	C₂³	semidirect
84. C₂²×D₄	C₂³	semidirect
85. C₂²×D₄	C₂³	semidirect
86. C₂²×D₄	C₂³	semidirect
91. C₂³×C₄	C₂³	semidirect
92. C₂³×C₄	C₂³	semidirect
93. C₂³×C₄	C₂³	semidirect
94. C₂³×C₄	C₂³	semidirect
95. #127	C₂³	semidirect
96. #124	C₂³	semidirect
97. #123	C₂³	semidirect
98. #123	C₂³	semidirect
99. C₂²⋊Q₈	C₂³	semidirect
100. #121	C₂³	semidirect
101. #121	C₂³	semidirect
102. #121	C₂³	semidirect
103. #120	C₂³	semidirect
104. #120	C₂³	semidirect
105. #120	C₂³	semidirect
106. #120	C₂³	semidirect
107. C₄×D₄	C₂³	semidirect
108. C₄×D₄	C₂³	semidirect
109. #117	C₂³	semidirect
110. C₂×C₄⋊C₄	C₂³	semidirect
111. C₂×C₄⋊C₄	C₂³	semidirect
114. C₂×C₂²⋊C₄	C₂³	semidirect
115. C₂×C₂²⋊C₄	C₂³	semidirect
116. C₂×C₂²⋊C₄	C₂³	semidirect
117. C₂×C₂²⋊C₄	C₂³	semidirect
118. C₂×C₂²⋊C₄	C₂³	semidirect
119. C₂×C₄²	C₂³	semidirect
133. C₂⁴	C₂×D₄	semidirect
143. C₂⁴	C₂⁴	semidirect
147. C₂×Q₈	C₂⁴	semidirect
161. C₂×D₄	C₂×D₄	semidirect
162. C₂×D₄	C₂×D₄	semidirect
168. C₂×D₄	C₂⁴	semidirect
169. C₂×D₄	C₂⁴	semidirect
170. C₂×D₄	C₂⁴	semidirect
177. C₂²×C₄	C₂×D₄	semidirect
185. C₂²×C₄	C₂⁴	semidirect
186. C₂²×C₄	C₂⁴	semidirect
187. C₂²×C₄	C₂⁴	semidirect
188. C₄⋊C₄	C₂⁴	semidirect
189. C₄⋊C₄	C₂⁴	semidirect
192. C₂²⋊C₄	C₂⁴	semidirect
193. C₂²⋊C₄	C₂⁴	semidirect
194. C₂²⋊C₄	C₂⁴	semidirect
195. C₄²	C₂⁴	semidirect
222. C₂³	C₂²×D₄	semidirect
223. C₂³	H(5,𝔽₂)	semidirect
238. D₄	C₂²×D₄	semidirect
248. C₂×C₄	C₂²×D₄	semidirect
275. C₂²	C₂×H(5,𝔽₂)	semidirect
286. C₂	#3484	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)	2
C₂⁴×D₄	2
#3484	2
C₂×#546	2
C₂×D₄²	2
C₂×#535	2
C₂×#534	2
C₂×#525	2
C₂²×#120	2
C₂×#518	2

Subgroups — C2×#3484 #62763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3484 #62763