Subgroups — C₂²×#546 #62697

Order 256.

7231 conjugacy classes of subgroups.

The subgroup lattice has 273 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	C₂⁴×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#546	C₂	1	48	yes	maximal, direct product factor
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, complemented
8	C₂²×#121	C₂	1	2	yes	maximal, complemented
9	C₂²×#120	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
13	C₂⁶	C₂²	1	2	yes	—
14	C₂²×#41	—	2	16	no	—
15	C₂²×#41	C₂²	1	24	yes	complemented
16	C₂³×Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₄	—	2	16	no	—
18	C₂³×D₄	C₂²	1	2	yes	—
19	C₂³×D₄	C₂²	1	1	yes	characteristic
20	C₂³×D₄	C₂²	1	1	yes	characteristic
21	C₂³×D₄	C₂²	1	1	yes	characteristic
22	C₂³×D₄	C₂²	1	24	yes	complemented
23	C₂³×D₄	C₂²	1	2	yes	complemented
24	C₂³×D₄	C₂²	1	2	yes	complemented
25	C₂⁴×C₄	C₂²	1	1	yes	characteristic
26	C₂⁴×C₄	C₂²	1	2	yes	complemented
27	C₂⁴×C₄	C₂²	1	1	yes	characteristic, complemented
28	C₂⁴×C₄	C₂²	1	2	yes	complemented
29	#546	C₂²	1	256	yes	direct product factor
30	C₂×#124	C₂²	1	24	yes	complemented
31	C₂×#123	C₂²	1	48	yes	complemented
32	C₂×C₂²⋊Q₈	C₂²	1	24	yes	complemented
33	C₂×#121	C₂²	1	24	yes	complemented
34	C₂×#121	C₂²	1	48	yes	complemented
35	C₂×#120	C₂²	1	48	yes	complemented
36	C₂×C₄×D₄	C₂²	1	24	yes	complemented
37	C₂×C₄×D₄	C₂²	1	24	yes	complemented
38	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
39	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
40	C₂²×C₄⋊C₄	C₂²	1	2	yes	complemented
41	C₂²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
42	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
43	C₂²×C₂²⋊C₄	C₂²	1	48	yes	complemented
44	C₂²×C₂²⋊C₄	C₂²	1	2	yes	complemented
45	C₂²×C₂²⋊C₄	C₂²	1	2	yes	complemented
46	C₂²×C₂²⋊C₄	C₂²	1	2	yes	complemented
47	C₂²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
48	C₂²×C₂²⋊C₄	C₂²	1	2	yes	complemented
49	C₂²×C₄²	C₂²	1	1	yes	characteristic, complemented
50	C₂⁵	—	2	16	no	—
51	C₂⁵	—	2	16	no	—
52	C₂⁵	—	4	8	no	—
53	C₂⁵	C₂³	1	2	yes	—
54	C₂⁵	C₂³	1	1	yes	characteristic
55	C₂⁵	C₂³	1	24	yes	—
56	C₂⁵	C₂³	1	2	yes	—
57	C₂⁵	C₂³	1	1	yes	characteristic
58	C₂×#41	—	2	192	no	—
59	C₂×#41	C₂³	1	64	yes	complemented
60	C₂²×Q₈	—	2	8	no	—
61	C₂²×Q₈	C₂³	1	12	yes	complemented
62	C₂²×D₄	—	2	16	no	—
63	C₂²×D₄	—	2	16	no	—
64	C₂²×D₄	—	2	8	no	—
65	C₂²×D₄	—	2	192	no	—
66	C₂²×D₄	—	2	8	no	—
67	C₂²×D₄	—	4	8	no	—
68	C₂²×D₄	D₄	1	16	yes	complemented
69	C₂²×D₄	C₂³	1	12	yes	—
70	C₂²×D₄	C₂³	1	24	yes	—
71	C₂²×D₄	C₂³	1	12	yes	—
72	C₂²×D₄	C₂³	1	12	yes	—
73	C₂²×D₄	C₂³	1	64	yes	complemented
74	C₂²×D₄	C₂³	1	24	yes	complemented
75	C₂²×D₄	C₂³	1	24	yes	complemented
76	C₂³×C₄	—	2	16	no	—
77	C₂³×C₄	—	2	8	no	—
78	C₂³×C₄	—	2	8	no	—
79	C₂³×C₄	—	4	8	no	—
80	C₂³×C₄	C₂³	1	12	yes	—
81	C₂³×C₄	C₂³	1	1	yes	characteristic
82	C₂³×C₄	C₂³	1	2	yes	—
83	C₂³×C₄	C₂³	1	1	yes	characteristic
84	C₂³×C₄	C₂³	1	2	yes	—
85	C₂³×C₄	C₂³	1	1	yes	characteristic
86	C₂³×C₄	C₂³	1	1	yes	characteristic
87	C₂³×C₄	C₂³	1	1	yes	characteristic
88	C₂³×C₄	C₂³	1	24	yes	complemented
89	C₂³×C₄	C₂³	1	12	yes	complemented
90	C₂³×C₄	C₂³	1	24	yes	complemented
91	#124	C₂³	1	64	yes	complemented
92	#123	C₂³	1	128	yes	complemented
93	C₂²⋊Q₈	C₂³	1	64	yes	complemented
94	#121	C₂³	1	64	yes	complemented
95	#121	C₂³	1	128	yes	complemented
96	#120	C₂³	1	128	yes	complemented
97	C₄×D₄	C₂³	1	64	yes	complemented
98	C₄×D₄	C₂³	1	64	yes	complemented
99	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
100	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
101	C₂×C₄⋊C₄	C₂³	1	24	yes	complemented
102	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
103	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
104	C₂×C₂²⋊C₄	C₂³	1	128	yes	complemented
105	C₂×C₂²⋊C₄	C₂³	1	24	yes	complemented
106	C₂×C₂²⋊C₄	C₂³	1	24	yes	complemented
107	C₂×C₂²⋊C₄	C₂³	1	24	yes	complemented
108	C₂×C₂²⋊C₄	C₂³	1	12	yes	complemented
109	C₂×C₂²⋊C₄	C₂³	1	24	yes	complemented
110	C₂×C₄²	C₂³	1	12	yes	complemented
111	C₂⁴	—	2	4	no	—
112	C₂⁴	—	2	8	no	—
113	C₂⁴	—	2	8	no	—
114	C₂⁴	—	2	4	no	—
115	C₂⁴	—	2	4	no	—
116	C₂⁴	—	2	96	no	—
117	C₂⁴	—	2	8	no	—
118	C₂⁴	—	2	4	no	—
119	C₂⁴	—	2	8	no	—
120	C₂⁴	—	2	96	no	—
121	C₂⁴	—	2	4	no	—
122	C₂⁴	—	2	8	no	—
123	C₂⁴	—	4	48	no	—
124	C₂⁴	—	4	128	no	—
125	C₂⁴	C₂×D₄	1	16	yes	—
126	C₂⁴	#41	1	8	yes	complemented
127	C₂⁴	C₂⁴	1	12	yes	—
128	C₂⁴	C₂⁴	1	6	yes	—
129	C₂⁴	C₂⁴	1	32	yes	—
130	C₂⁴	C₂⁴	1	12	yes	—
131	C₂⁴	C₂⁴	1	6	yes	—
132	C₂⁴	C₂⁴	1	1	yes	characteristic
133	#41	—	2	256	no	—
134	C₂×Q₈	—	2	48	no	—
135	C₂×Q₈	C₂⁴	1	16	yes	complemented
136	C₂×D₄	—	2	96	no	—
137	C₂×D₄	—	2	96	no	—
138	C₂×D₄	—	2	48	no	—
139	C₂×D₄	—	2	256	no	—
140	C₂×D₄	—	2	48	no	—
141	C₂×D₄	—	4	48	no	—
142	C₂×D₄	C₂×D₄	1	96	yes	complemented
143	C₂×D₄	C₂⁴	1	16	yes	—
144	C₂×D₄	C₂⁴	1	32	yes	—
145	C₂×D₄	C₂⁴	1	16	yes	—
146	C₂×D₄	C₂⁴	1	16	yes	—
147	C₂×D₄	C₂⁴	1	32	yes	complemented
148	C₂×D₄	C₂⁴	1	32	yes	complemented
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	96	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	48	no	—
157	C₂²×C₄	—	2	48	no	—
158	C₂²×C₄	—	4	48	no	—
159	C₂²×C₄	C₂×D₄	1	8	yes	—
160	C₂²×C₄	C₂⁴	1	6	yes	—
161	C₂²×C₄	C₂⁴	1	6	yes	—
162	C₂²×C₄	C₂⁴	1	12	yes	—
163	C₂²×C₄	C₂⁴	1	6	yes	—
164	C₂²×C₄	C₂⁴	1	12	yes	—
165	C₂²×C₄	C₂⁴	1	6	yes	—
166	C₂²×C₄	C₂⁴	1	16	yes	—
167	C₂²×C₄	C₂⁴	1	6	yes	—
168	C₂²×C₄	C₂⁴	1	32	yes	complemented
169	C₂²×C₄	C₂⁴	1	16	yes	complemented
170	C₂²×C₄	C₂⁴	1	32	yes	complemented
171	C₄⋊C₄	C₂⁴	1	16	yes	complemented
172	C₄⋊C₄	C₂⁴	1	16	yes	complemented
173	C₄⋊C₄	C₂⁴	1	32	yes	complemented
174	C₂²⋊C₄	C₂⁴	1	16	yes	—
175	C₂²⋊C₄	C₂⁴	1	32	yes	—
176	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
177	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
178	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
179	C₂²⋊C₄	C₂⁴	1	16	yes	complemented
180	C₂²⋊C₄	C₂⁴	1	32	yes	complemented
181	C₄²	C₂⁴	1	16	yes	complemented
182	C₂³	—	2	32	no	—
183	C₂³	—	2	64	no	—
184	C₂³	—	2	12	no	—
185	C₂³	—	2	24	no	—
186	C₂³	—	2	24	no	—
187	C₂³	—	2	12	no	—
188	C₂³	—	2	12	no	—
189	C₂³	—	2	64	no	—
190	C₂³	—	2	24	no	—
191	C₂³	—	2	12	no	—
192	C₂³	—	2	24	no	—
193	C₂³	—	2	64	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	2	24	no	—
196	C₂³	—	4	32	no	—
197	C₂³	—	4	384	no	—
198	C₂³	—	4	16	no	—
199	C₂³	—	4	32	no	—
200	C₂³	C₂²×D₄	1	48	yes	—
201	C₂³	C₂²×D₄	1	4	yes	—
202	C₂³	C₂×#41	1	4	yes	—
203	C₂³	C₂×#41	1	24	yes	complemented
204	C₂³	H(5,𝔽₂)	1	4	yes	—
205	C₂³	C₂⁵	1	8	yes	—
206	C₂³	C₂⁵	1	4	yes	—
207	C₂³	C₂⁵	1	8	yes	—
208	C₂³	C₂⁵	1	4	yes	—
209	C₂³	C₂⁵	1	3	yes	—
210	Q₈	—	2	32	no	—
211	D₄	—	2	64	no	—
212	D₄	—	2	64	no	—
213	D₄	—	2	32	no	—
214	D₄	—	2	32	no	—
215	D₄	—	4	32	no	—
216	D₄	C₂²×D₄	1	64	yes	complemented
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	24	no	—
220	C₂×C₄	—	2	64	no	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	12	no	—
223	C₂×C₄	—	2	24	no	—
224	C₂×C₄	—	2	32	no	—
225	C₂×C₄	—	2	32	no	—
226	C₂×C₄	—	4	32	no	—
227	C₂×C₄	C₂²×D₄	1	24	yes	—
228	C₂×C₄	C₂⁵	1	4	yes	—
229	C₂×C₄	C₂⁵	1	4	yes	—
230	C₂×C₄	C₂⁵	1	8	yes	—
231	C₂×C₄	C₂⁵	1	4	yes	—
232	C₂×C₄	C₂⁵	1	8	yes	—
233	C₂×C₄	C₂⁵	1	4	yes	—
234	C₂×C₄	C₂⁵	1	4	yes	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	8	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	48	no	—
240	C₂²	—	2	96	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	8	no	—
247	C₂²	—	4	128	no	—
248	C₂²	—	4	24	no	—
249	C₂²	—	4	48	no	—
250	C₂²	#546	1	16	yes	direct product factor
251	C₂²	C₂³×D₄	1	16	yes	—
252	C₂²	C₂³×D₄	1	6	yes	—
253	C₂²	C₂²×#41	1	6	yes	—
254	C₂²	C₂²×#41	1	8	yes	complemented
255	C₂²	C₂×H(5,𝔽₂)	1	6	yes	—
256	C₂²	C₂⁶	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	8	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	C₂³×D₄	1	8	yes	—
264	C₂	—	2	8	no	—
265	C₂	—	2	16	no	—
266	C₂	—	4	4	no	—
267	C₂	—	4	8	no	—
268	C₂	C₂×#546	1	12	yes	direct product factor
269	C₂	C₂⁴×D₄	1	1	yes	characteristic
270	C₂	C₂³×#41	1	1	yes	characteristic
271	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. C₂³×#41	C₂	semidirect
2. C₂⁴×D₄	C₂	semidirect
3. C₂×#546	C₂	direct (N × H)
4. C₂²×#124	C₂	semidirect
5. C₂²×#123	C₂	semidirect
6. C₂²×C₂²⋊Q₈	C₂	semidirect
7. C₂²×#121	C₂	semidirect
8. C₂²×#121	C₂	semidirect
9. C₂²×#120	C₂	semidirect
10. C₂²×C₄×D₄	C₂	semidirect
11. C₂²×C₄×D₄	C₂	semidirect
12. C₂³×C₂²⋊C₄	C₂	semidirect
15. C₂²×#41	C₂²	semidirect
16. C₂³×Q₈	C₂²	semidirect
22. C₂³×D₄	C₂²	semidirect
23. C₂³×D₄	C₂²	semidirect
24. C₂³×D₄	C₂²	semidirect
26. C₂⁴×C₄	C₂²	semidirect
27. C₂⁴×C₄	C₂²	semidirect
28. C₂⁴×C₄	C₂²	semidirect
29. #546	C₂²	direct (N × H)
30. C₂×#124	C₂²	semidirect
31. C₂×#123	C₂²	semidirect
32. C₂×C₂²⋊Q₈	C₂²	semidirect
33. C₂×#121	C₂²	semidirect
34. C₂×#121	C₂²	semidirect
35. C₂×#120	C₂²	semidirect
36. C₂×C₄×D₄	C₂²	semidirect
37. C₂×C₄×D₄	C₂²	semidirect
38. C₂²×C₄⋊C₄	C₂²	semidirect
39. C₂²×C₄⋊C₄	C₂²	semidirect
40. C₂²×C₄⋊C₄	C₂²	semidirect
43. C₂²×C₂²⋊C₄	C₂²	semidirect
44. C₂²×C₂²⋊C₄	C₂²	semidirect
45. C₂²×C₂²⋊C₄	C₂²	semidirect
46. C₂²×C₂²⋊C₄	C₂²	semidirect
47. C₂²×C₂²⋊C₄	C₂²	semidirect
48. C₂²×C₂²⋊C₄	C₂²	semidirect
49. C₂²×C₄²	C₂²	semidirect
59. C₂×#41	C₂³	semidirect
61. C₂²×Q₈	C₂³	semidirect
68. C₂²×D₄	D₄	semidirect
73. C₂²×D₄	C₂³	semidirect
74. C₂²×D₄	C₂³	semidirect
75. C₂²×D₄	C₂³	semidirect
88. C₂³×C₄	C₂³	semidirect
89. C₂³×C₄	C₂³	semidirect
90. C₂³×C₄	C₂³	semidirect
91. #124	C₂³	semidirect
92. #123	C₂³	semidirect
93. C₂²⋊Q₈	C₂³	semidirect
94. #121	C₂³	semidirect
95. #121	C₂³	semidirect
96. #120	C₂³	semidirect
97. C₄×D₄	C₂³	semidirect
98. C₄×D₄	C₂³	semidirect
99. C₂×C₄⋊C₄	C₂³	semidirect
100. C₂×C₄⋊C₄	C₂³	semidirect
101. C₂×C₄⋊C₄	C₂³	semidirect
104. C₂×C₂²⋊C₄	C₂³	semidirect
105. C₂×C₂²⋊C₄	C₂³	semidirect
106. C₂×C₂²⋊C₄	C₂³	semidirect
107. C₂×C₂²⋊C₄	C₂³	semidirect
108. C₂×C₂²⋊C₄	C₂³	semidirect
109. C₂×C₂²⋊C₄	C₂³	semidirect
110. C₂×C₄²	C₂³	semidirect
126. C₂⁴	#41	semidirect
135. C₂×Q₈	C₂⁴	semidirect
142. C₂×D₄	C₂×D₄	semidirect
147. C₂×D₄	C₂⁴	semidirect
148. C₂×D₄	C₂⁴	semidirect
168. C₂²×C₄	C₂⁴	semidirect
169. C₂²×C₄	C₂⁴	semidirect
170. C₂²×C₄	C₂⁴	semidirect
171. C₄⋊C₄	C₂⁴	semidirect
172. C₄⋊C₄	C₂⁴	semidirect
173. C₄⋊C₄	C₂⁴	semidirect
176. C₂²⋊C₄	C₂⁴	semidirect
177. C₂²⋊C₄	C₂⁴	semidirect
178. C₂²⋊C₄	C₂⁴	semidirect
179. C₂²⋊C₄	C₂⁴	semidirect
180. C₂²⋊C₄	C₂⁴	semidirect
181. C₄²	C₂⁴	semidirect
203. C₂³	C₂×#41	semidirect
216. D₄	C₂²×D₄	semidirect
250. C₂²	#546	direct (N × H)
254. C₂²	C₂²×#41	semidirect
268. C₂	C₂×#546	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
C₂⁴×D₄	2
C₂×#546	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×#546 #62697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#546 #62697