Subgroups — C₂²×#529 #62663

Order 256.

4335 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#126	C₂	1	2	yes	maximal, complemented
3	C₂²×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#123	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#121	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#117	C₂	1	2	yes	maximal, complemented
12	C₂³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈	C₂²	1	1	yes	characteristic
14	C₂³×D₄	C₂²	1	2	yes	—
15	C₂³×D₄	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄	C₂²	1	2	yes	—
20	#529	C₂²	1	256	yes	direct product factor
21	C₂×#126	C₂²	1	48	yes	complemented
22	C₂×#125	C₂²	1	24	yes	complemented
23	C₂×#124	C₂²	1	24	yes	complemented
24	C₂×#123	C₂²	1	48	yes	complemented
25	C₂×C₂²⋊Q₈	C₂²	1	24	yes	complemented
26	C₂×#121	C₂²	1	24	yes	—
27	C₂×C₄×Q₈	C₂²	1	24	yes	complemented
28	C₂×C₄×D₄	C₂²	1	48	yes	complemented
29	C₂×C₄×D₄	C₂²	1	24	yes	complemented
30	C₂×#117	C₂²	1	48	yes	complemented
31	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
38	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
39	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
40	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂²×C₄²	C₂²	1	1	yes	characteristic
43	C₂²×C₄²	C₂²	1	1	yes	characteristic
44	C₂²×C₄²	C₂²	1	1	yes	characteristic
45	C₂²×C₄²	C₂²	1	1	yes	characteristic
46	C₂²×C₄²	C₂²	1	2	yes	—
47	C₂²×C₄²	C₂²	1	24	yes	complemented
48	C₂⁵	C₂³	1	2	yes	—
49	C₂⁵	C₂³	1	1	yes	characteristic
50	C₂²×Q₈	—	2	8	no	—
51	C₂²×Q₈	C₂³	1	12	yes	—
52	C₂²×D₄	—	2	16	no	—
53	C₂²×D₄	—	2	8	no	—
54	C₂²×D₄	C₂³	1	24	yes	—
55	C₂²×D₄	C₂³	1	12	yes	—
56	C₂³×C₄	—	2	8	no	—
57	C₂³×C₄	—	2	8	no	—
58	C₂³×C₄	—	2	16	no	—
59	C₂³×C₄	—	2	8	no	—
60	C₂³×C₄	C₂³	1	2	yes	—
61	C₂³×C₄	C₂³	1	2	yes	—
62	C₂³×C₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄	C₂³	1	1	yes	characteristic
64	C₂³×C₄	C₂³	1	1	yes	characteristic
65	C₂³×C₄	C₂³	1	1	yes	characteristic
66	C₂³×C₄	C₂³	1	2	yes	—
67	C₂³×C₄	C₂³	1	24	yes	—
68	C₂³×C₄	C₂³	1	12	yes	—
69	C₂³×C₄	C₂³	1	12	yes	—
70	C₂³×C₄	C₂³	1	12	yes	—
71	C₂³×C₄	C₂³	1	1	yes	characteristic
72	C₂³×C₄	C₂³	1	1	yes	characteristic
73	#126	C₂³	1	128	yes	complemented
74	#125	C₂³	1	64	yes	complemented
75	#124	C₂³	1	64	yes	complemented
76	#123	C₂³	1	128	yes	complemented
77	C₂²⋊Q₈	C₂³	1	64	yes	complemented
78	#121	C₂³	1	64	yes	—
79	C₄×Q₈	C₂³	1	64	yes	complemented
80	C₄×D₄	C₂³	1	64	yes	complemented
81	C₄×D₄	C₂³	1	128	yes	complemented
82	#117	C₂³	1	128	yes	complemented
83	C₂×C₄⋊C₄	C₂³	1	24	yes	—
84	C₂×C₄⋊C₄	C₂³	1	24	yes	—
85	C₂×C₄⋊C₄	C₂³	1	12	yes	—
86	C₂×C₄⋊C₄	C₂³	1	24	yes	—
87	C₂×C₄⋊C₄	C₂³	1	12	yes	—
88	C₂×C₄⋊C₄	C₂³	1	24	yes	—
89	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
90	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
91	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
92	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
93	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
94	C₂×C₄²	C₂³	1	12	yes	—
95	C₂×C₄²	C₂³	1	24	yes	—
96	C₂×C₄²	C₂³	1	12	yes	—
97	C₂×C₄²	C₂³	1	12	yes	—
98	C₂×C₄²	C₂³	1	12	yes	—
99	C₂×C₄²	C₂³	1	64	yes	complemented
100	C₂⁴	—	2	4	no	—
101	C₂⁴	—	2	4	no	—
102	C₂⁴	—	2	8	no	—
103	C₂⁴	—	2	8	no	—
104	C₂⁴	—	2	8	no	—
105	C₂⁴	#41	1	8	yes	—
106	C₂⁴	C₂⁴	1	12	yes	—
107	C₂⁴	C₂⁴	1	6	yes	—
108	C₂⁴	C₂⁴	1	1	yes	characteristic
109	C₂×Q₈	—	2	48	no	—
110	C₂×Q₈	C₂⁴	1	16	yes	—
111	C₂×D₄	—	2	96	no	—
112	C₂×D₄	—	2	48	no	—
113	C₂×D₄	C₂⁴	1	32	yes	—
114	C₂×D₄	C₂⁴	1	16	yes	—
115	C₂²×C₄	—	2	48	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	—	2	48	no	—
124	C₂²×C₄	—	2	96	no	—
125	C₂²×C₄	—	2	48	no	—
126	C₂²×C₄	#41	1	8	yes	—
127	C₂²×C₄	#41	1	8	yes	—
128	C₂²×C₄	C₂⁴	1	6	yes	—
129	C₂²×C₄	C₂⁴	1	6	yes	—
130	C₂²×C₄	C₂⁴	1	6	yes	—
131	C₂²×C₄	C₂⁴	1	6	yes	—
132	C₂²×C₄	C₂⁴	1	12	yes	—
133	C₂²×C₄	C₂⁴	1	12	yes	—
134	C₂²×C₄	C₂⁴	1	16	yes	—
135	C₂²×C₄	C₂⁴	1	16	yes	—
136	C₂²×C₄	C₂⁴	1	16	yes	—
137	C₂²×C₄	C₂⁴	1	12	yes	—
138	C₂²×C₄	C₂⁴	1	32	yes	—
139	C₂²×C₄	C₂⁴	1	6	yes	—
140	C₂²×C₄	C₂⁴	1	6	yes	—
141	C₄⋊C₄	C₂⁴	1	32	yes	—
142	C₄⋊C₄	C₂⁴	1	32	yes	—
143	C₄⋊C₄	C₂⁴	1	16	yes	—
144	C₄⋊C₄	C₂⁴	1	32	yes	—
145	C₄⋊C₄	C₂⁴	1	16	yes	—
146	C₄⋊C₄	C₂⁴	1	32	yes	—
147	C₂²⋊C₄	C₂⁴	1	32	yes	—
148	C₂²⋊C₄	C₂⁴	1	32	yes	—
149	C₂²⋊C₄	C₂⁴	1	32	yes	—
150	C₂²⋊C₄	C₂⁴	1	32	yes	—
151	C₂²⋊C₄	C₂⁴	1	32	yes	—
152	C₄²	C₂⁴	1	16	yes	—
153	C₄²	C₂⁴	1	16	yes	—
154	C₄²	C₂⁴	1	16	yes	—
155	C₄²	C₂⁴	1	16	yes	—
156	C₄²	C₂⁴	1	32	yes	—
157	C₂³	—	2	32	no	—
158	C₂³	—	2	12	no	—
159	C₂³	—	2	12	no	—
160	C₂³	—	2	24	no	—
161	C₂³	—	2	24	no	—
162	C₂³	—	2	24	no	—
163	C₂³	—	4	32	no	—
164	C₂³	C₂×#41	1	24	yes	—
165	C₂³	C₂×#41	1	4	yes	—
166	C₂³	C₂×#41	1	4	yes	—
167	C₂³	C₂×#41	1	4	yes	—
168	C₂³	C₂⁵	1	8	yes	—
169	C₂³	C₂⁵	1	4	yes	—
170	C₂³	C₂⁵	1	3	yes	—
171	Q₈	—	2	32	no	—
172	D₄	—	2	64	no	—
173	D₄	—	2	32	no	—
174	C₂×C₄	—	2	32	no	—
175	C₂×C₄	—	2	12	no	—
176	C₂×C₄	—	2	12	no	—
177	C₂×C₄	—	2	12	no	—
178	C₂×C₄	—	2	12	no	—
179	C₂×C₄	—	2	24	no	—
180	C₂×C₄	—	2	24	no	—
181	C₂×C₄	—	2	24	no	—
182	C₂×C₄	—	2	32	no	—
183	C₂×C₄	—	2	64	no	—
184	C₂×C₄	—	2	32	no	—
185	C₂×C₄	C₂×#41	1	24	yes	—
186	C₂×C₄	C₂×#41	1	24	yes	—
187	C₂×C₄	C₂⁵	1	4	yes	—
188	C₂×C₄	C₂⁵	1	4	yes	—
189	C₂×C₄	C₂⁵	1	4	yes	—
190	C₂×C₄	C₂⁵	1	4	yes	—
191	C₂×C₄	C₂⁵	1	8	yes	—
192	C₂×C₄	C₂⁵	1	8	yes	—
193	C₂×C₄	C₂⁵	1	8	yes	—
194	C₂×C₄	C₂⁵	1	4	yes	—
195	C₂×C₄	C₂⁵	1	4	yes	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	48	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	4	48	no	—
203	C₂²	#529	1	16	yes	direct product factor
204	C₂²	C₂²×#41	1	8	yes	—
205	C₂²	C₂²×#41	1	6	yes	—
206	C₂²	C₂²×#41	1	6	yes	—
207	C₂²	C₂²×#41	1	6	yes	—
208	C₂²	C₂⁶	1	1	yes	characteristic
209	C₄	—	2	4	no	—
210	C₄	—	2	4	no	—
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	8	no	—
214	C₄	—	2	8	no	—
215	C₄	—	2	8	no	—
216	C₄	C₂²×#41	1	8	yes	—
217	C₄	C₂²×#41	1	8	yes	—
218	C₂	—	2	8	no	—
219	C₂	—	4	8	no	—
220	C₂	C₂×#529	1	12	yes	direct product factor
221	C₂	C₂³×#41	1	1	yes	characteristic
222	C₂	C₂³×#41	1	1	yes	characteristic
223	C₂	C₂³×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂⁴	C₂⁴
prime radical	C₂²×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#529	C₂	direct (N × H)
2. C₂²×#126	C₂	semidirect
3. C₂²×#125	C₂	semidirect
4. C₂²×#124	C₂	semidirect
5. C₂²×#123	C₂	semidirect
6. C₂²×C₂²⋊Q₈	C₂	semidirect
8. C₂²×C₄×Q₈	C₂	semidirect
9. C₂²×C₄×D₄	C₂	semidirect
10. C₂²×C₄×D₄	C₂	semidirect
11. C₂²×#117	C₂	semidirect
12. C₂³×C₄²	C₂	semidirect
20. #529	C₂²	direct (N × H)
21. C₂×#126	C₂²	semidirect
22. C₂×#125	C₂²	semidirect
23. C₂×#124	C₂²	semidirect
24. C₂×#123	C₂²	semidirect
25. C₂×C₂²⋊Q₈	C₂²	semidirect
27. C₂×C₄×Q₈	C₂²	semidirect
28. C₂×C₄×D₄	C₂²	semidirect
29. C₂×C₄×D₄	C₂²	semidirect
30. C₂×#117	C₂²	semidirect
47. C₂²×C₄²	C₂²	semidirect
73. #126	C₂³	semidirect
74. #125	C₂³	semidirect
75. #124	C₂³	semidirect
76. #123	C₂³	semidirect
77. C₂²⋊Q₈	C₂³	semidirect
79. C₄×Q₈	C₂³	semidirect
80. C₄×D₄	C₂³	semidirect
81. C₄×D₄	C₂³	semidirect
82. #117	C₂³	semidirect
99. C₂×C₄²	C₂³	semidirect
203. C₂²	#529	direct (N × H)
220. C₂	C₂×#529	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×#529	2
C₂²×#126	2
C₂²×#125	2
C₂²×#124	2
C₂²×#123	2
C₂²×C₂²⋊Q₈	2
C₂²×#121	2
C₂²×C₄×Q₈	2
C₂²×C₄×D₄	2
C₂²×#117	2
C₂³×C₄²	2

Subgroups — C22×#529 #62663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#529 #62663