Subgroups — C₂²×#538 #62671

Order 256.

4783 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×#538	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#126	C₂	1	2	yes	maximal, complemented
3	C₂²×#124	C₂	1	2	yes	maximal, complemented
4	C₂²×#123	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#121	C₂	1	2	yes	maximal, complemented
7	C₂²×#121	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#120	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₂³×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁶	C₂²	1	1	yes	characteristic
12	C₂³×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₄	C₂²	1	2	yes	—
14	C₂³×D₄	C₂²	1	2	yes	—
15	C₂³×D₄	C₂²	1	2	yes	—
16	C₂³×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂⁴×C₄	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄	C₂²	1	2	yes	complemented
20	#538	C₂²	1	256	yes	direct product factor
21	C₂×#126	C₂²	1	48	yes	complemented
22	C₂×#124	C₂²	1	48	yes	complemented
23	C₂×#123	C₂²	1	48	yes	complemented
24	C₂×C₂²⋊Q₈	C₂²	1	24	yes	complemented
25	C₂×#121	C₂²	1	48	yes	complemented
26	C₂×#121	C₂²	1	24	yes	complemented
27	C₂×#120	C₂²	1	48	yes	complemented
28	C₂×C₄×D₄	C₂²	1	48	yes	complemented
29	C₂²×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₂²×C₄⋊C₄	C₂²	1	2	yes	complemented
31	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
32	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
33	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
36	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
37	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
38	C₂²×C₂²⋊C₄	C₂²	1	24	yes	complemented
39	C₂²×C₂²⋊C₄	C₂²	1	2	yes	complemented
40	C₂²×C₂²⋊C₄	C₂²	1	2	yes	complemented
41	C₂²×C₄²	C₂²	1	2	yes	complemented
42	C₂⁵	—	2	8	no	—
43	C₂⁵	—	4	4	no	—
44	C₂⁵	—	4	4	no	—
45	C₂⁵	C₂³	1	2	yes	—
46	C₂⁵	C₂³	1	2	yes	—
47	C₂⁵	C₂³	1	1	yes	characteristic
48	C₂⁵	C₂³	1	12	yes	—
49	C₂²×Q₈	—	4	4	no	—
50	C₂²×Q₈	C₂³	1	12	yes	complemented
51	C₂²×D₄	—	2	16	no	—
52	C₂²×D₄	—	4	8	no	—
53	C₂²×D₄	—	4	8	no	—
54	C₂²×D₄	—	4	4	no	—
55	C₂²×D₄	C₂³	1	24	yes	—
56	C₂²×D₄	C₂³	1	24	yes	—
57	C₂²×D₄	C₂³	1	24	yes	—
58	C₂²×D₄	C₂³	1	12	yes	complemented
59	C₂³×C₄	—	4	4	no	—
60	C₂³×C₄	—	4	4	no	—
61	C₂³×C₄	—	4	8	no	—
62	C₂³×C₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄	C₂³	1	2	yes	—
64	C₂³×C₄	C₂³	1	2	yes	—
65	C₂³×C₄	C₂³	1	2	yes	—
66	C₂³×C₄	C₂³	1	1	yes	characteristic
67	C₂³×C₄	C₂³	1	2	yes	—
68	C₂³×C₄	C₂³	1	12	yes	—
69	C₂³×C₄	C₂³	1	12	yes	—
70	C₂³×C₄	C₂³	1	24	yes	complemented
71	#126	C₂³	1	128	yes	complemented
72	#124	C₂³	1	128	yes	complemented
73	#123	C₂³	1	128	yes	complemented
74	C₂²⋊Q₈	C₂³	1	64	yes	complemented
75	#121	C₂³	1	128	yes	complemented
76	#121	C₂³	1	64	yes	complemented
77	#120	C₂³	1	128	yes	complemented
78	C₄×D₄	C₂³	1	128	yes	complemented
79	C₂×C₄⋊C₄	C₂³	1	24	yes	complemented
80	C₂×C₄⋊C₄	C₂³	1	24	yes	complemented
81	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
82	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
83	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
84	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
85	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
86	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
87	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
88	C₂×C₂²⋊C₄	C₂³	1	64	yes	complemented
89	C₂×C₂²⋊C₄	C₂³	1	24	yes	complemented
90	C₂×C₂²⋊C₄	C₂³	1	24	yes	complemented
91	C₂×C₄²	C₂³	1	24	yes	complemented
92	C₂⁴	—	2	4	no	—
93	C₂⁴	—	2	4	no	—
94	C₂⁴	—	2	8	no	—
95	C₂⁴	—	2	48	no	—
96	C₂⁴	—	2	8	no	—
97	C₂⁴	—	2	8	no	—
98	C₂⁴	—	2	8	no	—
99	C₂⁴	—	2	8	no	—
100	C₂⁴	—	2	8	no	—
101	C₂⁴	—	4	24	no	—
102	C₂⁴	—	4	24	no	—
103	C₂⁴	—	4	64	no	—
104	C₂⁴	#41	1	8	yes	—
105	C₂⁴	C₂⁴	1	12	yes	—
106	C₂⁴	C₂⁴	1	12	yes	—
107	C₂⁴	C₂⁴	1	6	yes	—
108	C₂⁴	C₂⁴	1	16	yes	—
109	C₂⁴	C₂⁴	1	1	yes	characteristic
110	C₂×Q₈	—	4	24	no	—
111	C₂×Q₈	C₂⁴	1	16	yes	complemented
112	C₂×D₄	—	2	96	no	—
113	C₂×D₄	—	4	48	no	—
114	C₂×D₄	—	4	48	no	—
115	C₂×D₄	—	4	24	no	—
116	C₂×D₄	C₂⁴	1	32	yes	—
117	C₂×D₄	C₂⁴	1	32	yes	—
118	C₂×D₄	C₂⁴	1	32	yes	—
119	C₂×D₄	C₂⁴	1	16	yes	complemented
120	C₂²×C₄	—	2	8	no	—
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	8	no	—
126	C₂²×C₄	—	4	24	no	—
127	C₂²×C₄	—	4	24	no	—
128	C₂²×C₄	—	4	48	no	—
129	C₂²×C₄	C₂⁴	1	12	yes	—
130	C₂²×C₄	C₂⁴	1	12	yes	—
131	C₂²×C₄	C₂⁴	1	12	yes	—
132	C₂²×C₄	C₂⁴	1	6	yes	—
133	C₂²×C₄	C₂⁴	1	6	yes	—
134	C₂²×C₄	C₂⁴	1	12	yes	—
135	C₂²×C₄	C₂⁴	1	16	yes	—
136	C₂²×C₄	C₂⁴	1	16	yes	—
137	C₂²×C₄	C₂⁴	1	32	yes	complemented
138	C₄⋊C₄	C₂⁴	1	32	yes	complemented
139	C₄⋊C₄	C₂⁴	1	32	yes	complemented
140	C₄⋊C₄	C₂⁴	1	16	yes	complemented
141	C₄⋊C₄	C₂⁴	1	16	yes	complemented
142	C₂²⋊C₄	C₂⁴	1	32	yes	—
143	C₂²⋊C₄	C₂⁴	1	32	yes	—
144	C₂²⋊C₄	C₂⁴	1	32	yes	—
145	C₂²⋊C₄	C₂⁴	1	32	yes	—
146	C₂²⋊C₄	C₂⁴	1	32	yes	—
147	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
148	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
149	C₄²	C₂⁴	1	32	yes	complemented
150	C₂³	—	2	32	no	—
151	C₂³	—	2	12	no	—
152	C₂³	—	2	12	no	—
153	C₂³	—	2	24	no	—
154	C₂³	—	2	32	no	—
155	C₂³	—	2	24	no	—
156	C₂³	—	2	24	no	—
157	C₂³	—	2	24	no	—
158	C₂³	—	2	24	no	—
159	C₂³	—	2	24	no	—
160	C₂³	—	4	16	no	—
161	C₂³	—	4	16	no	—
162	C₂³	—	4	192	no	—
163	C₂³	—	4	32	no	—
164	C₂³	—	4	32	no	—
165	C₂³	C₂×#41	1	24	yes	—
166	C₂³	C₂×#41	1	4	yes	—
167	C₂³	H(5,𝔽₂)	1	4	yes	—
168	C₂³	H(5,𝔽₂)	1	4	yes	—
169	C₂³	C₂⁵	1	8	yes	—
170	C₂³	C₂⁵	1	8	yes	—
171	C₂³	C₂⁵	1	4	yes	—
172	C₂³	C₂⁵	1	3	yes	—
173	Q₈	—	4	16	no	—
174	D₄	—	2	64	no	—
175	D₄	—	4	32	no	—
176	D₄	—	4	32	no	—
177	D₄	—	4	16	no	—
178	C₂×C₄	—	2	24	no	—
179	C₂×C₄	—	2	24	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	24	no	—
182	C₂×C₄	—	2	12	no	—
183	C₂×C₄	—	2	24	no	—
184	C₂×C₄	—	4	16	no	—
185	C₂×C₄	—	4	16	no	—
186	C₂×C₄	—	4	32	no	—
187	C₂×C₄	C₂⁵	1	8	yes	—
188	C₂×C₄	C₂⁵	1	8	yes	—
189	C₂×C₄	C₂⁵	1	8	yes	—
190	C₂×C₄	C₂⁵	1	4	yes	—
191	C₂×C₄	C₂⁵	1	4	yes	—
192	C₂×C₄	C₂⁵	1	8	yes	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	48	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	8	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	4	64	no	—
203	C₂²	—	4	48	no	—
204	C₂²	—	4	48	no	—
205	C₂²	#538	1	16	yes	direct product factor
206	C₂²	C₂²×#41	1	8	yes	—
207	C₂²	C₂²×#41	1	6	yes	—
208	C₂²	C₂×H(5,𝔽₂)	1	6	yes	—
209	C₂²	C₂×H(5,𝔽₂)	1	6	yes	—
210	C₂²	C₂⁶	1	1	yes	characteristic
211	C₄	—	2	8	no	—
212	C₄	—	2	8	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	8	no	—
215	C₄	—	2	4	no	—
216	C₄	—	2	8	no	—
217	C₂	—	2	8	no	—
218	C₂	—	4	8	no	—
219	C₂	—	4	8	no	—
220	C₂	C₂×#538	1	12	yes	direct product factor
221	C₂	C₂³×#41	1	1	yes	characteristic
222	C₂	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
223	C₂	C₂²×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#538	C₂	direct (N × H)
2. C₂²×#126	C₂	semidirect
3. C₂²×#124	C₂	semidirect
4. C₂²×#123	C₂	semidirect
5. C₂²×C₂²⋊Q₈	C₂	semidirect
6. C₂²×#121	C₂	semidirect
7. C₂²×#121	C₂	semidirect
8. C₂²×#120	C₂	semidirect
9. C₂²×C₄×D₄	C₂	semidirect
10. C₂³×C₂²⋊C₄	C₂	semidirect
12. C₂³×Q₈	C₂²	semidirect
16. C₂³×D₄	C₂²	semidirect
19. C₂⁴×C₄	C₂²	semidirect
20. #538	C₂²	direct (N × H)
21. C₂×#126	C₂²	semidirect
22. C₂×#124	C₂²	semidirect
23. C₂×#123	C₂²	semidirect
24. C₂×C₂²⋊Q₈	C₂²	semidirect
25. C₂×#121	C₂²	semidirect
26. C₂×#121	C₂²	semidirect
27. C₂×#120	C₂²	semidirect
28. C₂×C₄×D₄	C₂²	semidirect
29. C₂²×C₄⋊C₄	C₂²	semidirect
30. C₂²×C₄⋊C₄	C₂²	semidirect
31. C₂²×C₄⋊C₄	C₂²	semidirect
32. C₂²×C₄⋊C₄	C₂²	semidirect
38. C₂²×C₂²⋊C₄	C₂²	semidirect
39. C₂²×C₂²⋊C₄	C₂²	semidirect
40. C₂²×C₂²⋊C₄	C₂²	semidirect
41. C₂²×C₄²	C₂²	semidirect
50. C₂²×Q₈	C₂³	semidirect
58. C₂²×D₄	C₂³	semidirect
70. C₂³×C₄	C₂³	semidirect
71. #126	C₂³	semidirect
72. #124	C₂³	semidirect
73. #123	C₂³	semidirect
74. C₂²⋊Q₈	C₂³	semidirect
75. #121	C₂³	semidirect
76. #121	C₂³	semidirect
77. #120	C₂³	semidirect
78. C₄×D₄	C₂³	semidirect
79. C₂×C₄⋊C₄	C₂³	semidirect
80. C₂×C₄⋊C₄	C₂³	semidirect
81. C₂×C₄⋊C₄	C₂³	semidirect
82. C₂×C₄⋊C₄	C₂³	semidirect
88. C₂×C₂²⋊C₄	C₂³	semidirect
89. C₂×C₂²⋊C₄	C₂³	semidirect
90. C₂×C₂²⋊C₄	C₂³	semidirect
91. C₂×C₄²	C₂³	semidirect
111. C₂×Q₈	C₂⁴	semidirect
119. C₂×D₄	C₂⁴	semidirect
137. C₂²×C₄	C₂⁴	semidirect
138. C₄⋊C₄	C₂⁴	semidirect
139. C₄⋊C₄	C₂⁴	semidirect
140. C₄⋊C₄	C₂⁴	semidirect
141. C₄⋊C₄	C₂⁴	semidirect
147. C₂²⋊C₄	C₂⁴	semidirect
148. C₂²⋊C₄	C₂⁴	semidirect
149. C₄²	C₂⁴	semidirect
205. C₂²	#538	direct (N × H)
220. C₂	C₂×#538	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×#538	2
C₂²×#126	2
C₂²×#124	2
C₂²×#123	2
C₂²×C₂²⋊Q₈	2
C₂²×#121	2
C₂²×#120	2
C₂²×C₄×D₄	2
C₂³×C₂²⋊C₄	2

Subgroups — C22×#538 #62671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#538 #62671