Subgroups — C₂²×#525 #62659

Order 256.

7167 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁵×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#525	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#123	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂²×#121	C₂	1	2	yes	maximal, complemented
7	C₂²×#120	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×D₄	C₂	1	4	yes	maximal, complemented
9	C₂²×#117	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁶	C₂²	1	1	yes	characteristic
11	C₂²×#41	—	2	16	no	—
12	C₂²×#41	C₂²	1	24	yes	complemented
13	C₂³×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂³×D₄	C₂²	1	2	yes	—
15	C₂³×D₄	C₂²	1	4	yes	—
16	C₂³×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂⁴×C₄	—	2	16	no	—
18	C₂⁴×C₄	C₂²	1	2	yes	—
19	C₂⁴×C₄	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄	C₂²	1	2	yes	—
21	C₂⁴×C₄	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄	C₂²	1	24	yes	complemented
23	C₂⁴×C₄	C₂²	1	2	yes	complemented
24	#525	C₂²	1	256	yes	direct product factor
25	C₂×#123	C₂²	1	48	yes	complemented
26	C₂×C₂²⋊Q₈	C₂²	1	48	yes	complemented
27	C₂×#121	C₂²	1	48	yes	complemented
28	C₂×#120	C₂²	1	48	yes	complemented
29	C₂×C₄×D₄	C₂²	1	96	yes	complemented
30	C₂×#117	C₂²	1	24	yes	complemented
31	C₂²×C₄⋊C₄	C₂²	1	2	yes	complemented
32	C₂²×C₄⋊C₄	C₂²	1	4	yes	complemented
33	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄	C₂²	1	4	yes	—
35	C₂²×C₂²⋊C₄	C₂²	1	4	yes	complemented
36	C₂²×C₄²	C₂²	1	2	yes	complemented
37	C₂⁵	—	2	8	no	—
38	C₂⁵	—	2	16	no	—
39	C₂⁵	C₂³	1	2	yes	—
40	C₂⁵	C₂³	1	12	yes	—
41	C₂⁵	C₂³	1	1	yes	characteristic
42	C₂⁵	C₂³	1	2	yes	—
43	C₂×#41	—	2	192	no	—
44	C₂×#41	C₂³	1	64	yes	complemented
45	C₂²×Q₈	—	2	8	no	—
46	C₂²×Q₈	C₂³	1	12	yes	complemented
47	C₂²×D₄	—	2	16	no	—
48	C₂²×D₄	—	2	32	no	—
49	C₂²×D₄	—	2	8	no	—
50	C₂²×D₄	C₂³	1	48	yes	—
51	C₂²×D₄	C₂³	1	24	yes	—
52	C₂²×D₄	C₂³	1	12	yes	complemented
53	C₂³×C₄	—	2	8	no	—
54	C₂³×C₄	—	2	16	no	—
55	C₂³×C₄	—	2	16	no	—
56	C₂³×C₄	—	2	16	no	—
57	C₂³×C₄	—	2	192	no	—
58	C₂³×C₄	D₄	1	16	yes	complemented
59	C₂³×C₄	C₂³	1	4	yes	—
60	C₂³×C₄	C₂³	1	2	yes	—
61	C₂³×C₄	C₂³	1	2	yes	—
62	C₂³×C₄	C₂³	1	24	yes	—
63	C₂³×C₄	C₂³	1	24	yes	—
64	C₂³×C₄	C₂³	1	12	yes	—
65	C₂³×C₄	C₂³	1	1	yes	characteristic
66	C₂³×C₄	C₂³	1	12	yes	—
67	C₂³×C₄	C₂³	1	1	yes	characteristic
68	C₂³×C₄	C₂³	1	64	yes	complemented
69	C₂³×C₄	C₂³	1	24	yes	complemented
70	#123	C₂³	1	128	yes	complemented
71	C₂²⋊Q₈	C₂³	1	128	yes	complemented
72	#121	C₂³	1	128	yes	complemented
73	#120	C₂³	1	128	yes	complemented
74	C₄×D₄	C₂³	1	256	yes	complemented
75	#117	C₂³	1	64	yes	complemented
76	C₂×C₄⋊C₄	C₂³	1	24	yes	complemented
77	C₂×C₄⋊C₄	C₂³	1	48	yes	complemented
78	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
79	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
80	C₂×C₂²⋊C₄	C₂³	1	48	yes	complemented
81	C₂×C₄²	C₂³	1	24	yes	complemented
82	C₂⁴	—	2	128	no	—
83	C₂⁴	—	2	48	no	—
84	C₂⁴	—	2	8	no	—
85	C₂⁴	—	2	8	no	—
86	C₂⁴	—	2	8	no	—
87	C₂⁴	—	2	96	no	—
88	C₂⁴	—	2	8	no	—
89	C₂⁴	—	2	16	no	—
90	C₂⁴	C₂×D₄	1	8	yes	—
91	C₂⁴	#41	1	16	yes	complemented
92	C₂⁴	C₂⁴	1	12	yes	—
93	C₂⁴	C₂⁴	1	16	yes	—
94	C₂⁴	C₂⁴	1	6	yes	—
95	C₂⁴	C₂⁴	1	12	yes	—
96	C₂⁴	C₂⁴	1	1	yes	characteristic
97	#41	—	2	256	no	—
98	C₂×Q₈	—	2	48	no	—
99	C₂×Q₈	C₂⁴	1	16	yes	complemented
100	C₂×D₄	—	2	96	no	—
101	C₂×D₄	—	2	192	no	—
102	C₂×D₄	—	2	48	no	—
103	C₂×D₄	C₂⁴	1	64	yes	—
104	C₂×D₄	C₂⁴	1	32	yes	—
105	C₂×D₄	C₂⁴	1	16	yes	complemented
106	C₂²×C₄	—	2	48	no	—
107	C₂²×C₄	—	2	8	no	—
108	C₂²×C₄	—	2	8	no	—
109	C₂²×C₄	—	2	16	no	—
110	C₂²×C₄	—	2	96	no	—
111	C₂²×C₄	—	2	96	no	—
112	C₂²×C₄	—	2	96	no	—
113	C₂²×C₄	—	2	256	no	—
114	C₂²×C₄	C₂×D₄	1	8	yes	—
115	C₂²×C₄	C₂×D₄	1	8	yes	—
116	C₂²×C₄	C₂×D₄	1	96	yes	complemented
117	C₂²×C₄	C₂⁴	1	24	yes	—
118	C₂²×C₄	C₂⁴	1	12	yes	—
119	C₂²×C₄	C₂⁴	1	12	yes	—
120	C₂²×C₄	C₂⁴	1	32	yes	—
121	C₂²×C₄	C₂⁴	1	32	yes	—
122	C₂²×C₄	C₂⁴	1	16	yes	—
123	C₂²×C₄	C₂⁴	1	6	yes	—
124	C₂²×C₄	C₂⁴	1	6	yes	—
125	C₂²×C₄	C₂⁴	1	16	yes	—
126	C₂²×C₄	C₂⁴	1	32	yes	complemented
127	C₄⋊C₄	C₂⁴	1	32	yes	complemented
128	C₄⋊C₄	C₂⁴	1	64	yes	complemented
129	C₂²⋊C₄	C₂⁴	1	32	yes	—
130	C₂²⋊C₄	C₂⁴	1	64	yes	—
131	C₂²⋊C₄	C₂⁴	1	64	yes	complemented
132	C₄²	C₂⁴	1	32	yes	complemented
133	C₂³	—	2	384	no	—
134	C₂³	—	2	32	no	—
135	C₂³	—	2	64	no	—
136	C₂³	—	2	32	no	—
137	C₂³	—	2	24	no	—
138	C₂³	—	2	24	no	—
139	C₂³	—	2	24	no	—
140	C₂³	—	2	64	no	—
141	C₂³	—	2	24	no	—
142	C₂³	—	2	48	no	—
143	C₂³	—	4	32	no	—
144	C₂³	C₂²×D₄	1	24	yes	—
145	C₂³	C₂²×D₄	1	4	yes	—
146	C₂³	C₂×#41	1	8	yes	—
147	C₂³	C₂×#41	1	48	yes	complemented
148	C₂³	C₂⁵	1	8	yes	—
149	C₂³	C₂⁵	1	4	yes	—
150	C₂³	C₂⁵	1	8	yes	—
151	C₂³	C₂⁵	1	3	yes	—
152	Q₈	—	2	32	no	—
153	D₄	—	2	64	no	—
154	D₄	—	2	128	no	—
155	D₄	—	2	32	no	—
156	C₂×C₄	—	2	32	no	—
157	C₂×C₄	—	2	24	no	—
158	C₂×C₄	—	2	24	no	—
159	C₂×C₄	—	2	48	no	—
160	C₂×C₄	—	2	64	no	—
161	C₂×C₄	—	2	64	no	—
162	C₂×C₄	—	2	64	no	—
163	C₂×C₄	C₂²×D₄	1	24	yes	—
164	C₂×C₄	C₂²×D₄	1	24	yes	—
165	C₂×C₄	C₂²×D₄	1	64	yes	complemented
166	C₂×C₄	C₂⁵	1	16	yes	—
167	C₂×C₄	C₂⁵	1	8	yes	—
168	C₂×C₄	C₂⁵	1	8	yes	—
169	C₂×C₄	C₂⁵	1	4	yes	—
170	C₂×C₄	C₂⁵	1	4	yes	—
171	C₂²	—	2	128	no	—
172	C₂²	—	2	8	no	—
173	C₂²	—	2	8	no	—
174	C₂²	—	2	8	no	—
175	C₂²	—	2	48	no	—
176	C₂²	—	2	96	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	2	16	no	—
179	C₂²	—	4	48	no	—
180	C₂²	#525	1	16	yes	direct product factor
181	C₂²	C₂³×D₄	1	8	yes	—
182	C₂²	C₂³×D₄	1	6	yes	—
183	C₂²	C₂²×#41	1	12	yes	—
184	C₂²	C₂²×#41	1	16	yes	complemented
185	C₂²	C₂⁶	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	—	2	8	no	—
188	C₄	—	2	16	no	—
189	C₄	C₂³×D₄	1	8	yes	—
190	C₄	C₂³×D₄	1	8	yes	—
191	C₂	—	2	8	no	—
192	C₂	—	2	16	no	—
193	C₂	—	4	8	no	—
194	C₂	C₂×#525	1	12	yes	direct product factor
195	C₂	C₂⁴×D₄	1	1	yes	characteristic
196	C₂	C₂³×#41	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂⁴	C₂⁴
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. C₂³×#41	C₂	semidirect
2. C₂⁵×C₄	C₂	semidirect
3. C₂×#525	C₂	direct (N × H)
4. C₂²×#123	C₂	semidirect
5. C₂²×C₂²⋊Q₈	C₂	semidirect
6. C₂²×#121	C₂	semidirect
7. C₂²×#120	C₂	semidirect
8. C₂²×C₄×D₄	C₂	semidirect
9. C₂²×#117	C₂	semidirect
12. C₂²×#41	C₂²	semidirect
13. C₂³×Q₈	C₂²	semidirect
16. C₂³×D₄	C₂²	semidirect
22. C₂⁴×C₄	C₂²	semidirect
23. C₂⁴×C₄	C₂²	semidirect
24. #525	C₂²	direct (N × H)
25. C₂×#123	C₂²	semidirect
26. C₂×C₂²⋊Q₈	C₂²	semidirect
27. C₂×#121	C₂²	semidirect
28. C₂×#120	C₂²	semidirect
29. C₂×C₄×D₄	C₂²	semidirect
30. C₂×#117	C₂²	semidirect
31. C₂²×C₄⋊C₄	C₂²	semidirect
32. C₂²×C₄⋊C₄	C₂²	semidirect
35. C₂²×C₂²⋊C₄	C₂²	semidirect
36. C₂²×C₄²	C₂²	semidirect
44. C₂×#41	C₂³	semidirect
46. C₂²×Q₈	C₂³	semidirect
52. C₂²×D₄	C₂³	semidirect
58. C₂³×C₄	D₄	semidirect
68. C₂³×C₄	C₂³	semidirect
69. C₂³×C₄	C₂³	semidirect
70. #123	C₂³	semidirect
71. C₂²⋊Q₈	C₂³	semidirect
72. #121	C₂³	semidirect
73. #120	C₂³	semidirect
74. C₄×D₄	C₂³	semidirect
75. #117	C₂³	semidirect
76. C₂×C₄⋊C₄	C₂³	semidirect
77. C₂×C₄⋊C₄	C₂³	semidirect
80. C₂×C₂²⋊C₄	C₂³	semidirect
81. C₂×C₄²	C₂³	semidirect
91. C₂⁴	#41	semidirect
99. C₂×Q₈	C₂⁴	semidirect
105. C₂×D₄	C₂⁴	semidirect
116. C₂²×C₄	C₂×D₄	semidirect
126. C₂²×C₄	C₂⁴	semidirect
127. C₄⋊C₄	C₂⁴	semidirect
128. C₄⋊C₄	C₂⁴	semidirect
131. C₂²⋊C₄	C₂⁴	semidirect
132. C₄²	C₂⁴	semidirect
147. C₂³	C₂×#41	semidirect
165. C₂×C₄	C₂²×D₄	semidirect
180. C₂²	#525	direct (N × H)
184. C₂²	C₂²×#41	semidirect
194. C₂	C₂×#525	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41	2
C₂⁵×C₄	2
C₂×#525	2
C₂²×#123	2
C₂²×C₂²⋊Q₈	2
C₂²×#121	2
C₂²×#120	2
C₂²×C₄×D₄	2
C₂²×#117	2

Subgroups — C22×#525 #62659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#525 #62659