Subgroups — #62987

Order 256.

5798 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	#3500	C₂	1	6	yes	maximal, complemented
4	#3496	C₂	1	9	yes	maximal, complemented
5	#3489	C₂	1	6	yes	maximal, complemented
6	#3488	C₂	1	6	yes	maximal, complemented
7	#3485	C₂	1	9	yes	maximal, complemented
8	#3484	C₂	1	9	yes	maximal, complemented
9	D₄×#41	C₂	1	6	yes	maximal, complemented
10	C₂×#546	C₂	1	9	yes	maximal, complemented
11	C₄×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
12	#585	—	2	16	no	—
13	C₂×2₋¹⁺⁴	C₂²	1	6	yes	complemented
14	C₂×H(5,𝔽₂)	—	2	16	no	—
15	C₂×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
16	C₂×H(5,𝔽₂)	C₂²	1	9	yes	complemented
17	C₂×H(5,𝔽₂)	C₂²	1	6	yes	complemented
18	C₂×H(5,𝔽₂)	C₂²	1	9	yes	complemented
19	C₂²×#41	C₂²	1	6	yes	complemented
20	C₂²×#41	C₂²	1	9	yes	complemented
21	C₂²×#41	C₂²	1	6	yes	complemented
22	C₂³×D₄	C₂²	1	9	yes	complemented
23	#550	C₂²	1	6	yes	complemented
24	D₄×Q₈	C₂²	1	6	yes	complemented
25	#548	C₂²	1	2	yes	complemented
26	#548	C₂²	1	6	yes	complemented
27	#548	C₂²	1	12	yes	complemented
28	#547	C₂²	1	18	yes	complemented
29	#547	C₂²	1	36	yes	complemented
30	#546	C₂²	1	18	yes	complemented
31	#546	C₂²	1	36	yes	complemented
32	#546	C₂²	1	36	yes	complemented
33	#546	C₂²	1	36	yes	complemented
34	#546	C₂²	1	36	yes	complemented
35	#546	C₂²	1	18	yes	complemented
36	D₄²	C₂²	1	18	yes	complemented
37	D₄²	C₂²	1	36	yes	complemented
38	#537	C₂²	1	18	yes	complemented
39	#536	C₂²	1	18	yes	complemented
40	#535	C₂²	1	18	yes	complemented
41	#534	C₂²	1	36	yes	complemented
42	#532	C₂²	1	18	yes	complemented
43	C₂×#124	C₂²	1	9	yes	complemented
44	#525	C₂²	1	18	yes	complemented
45	#525	C₂²	1	18	yes	complemented
46	C₂×#123	C₂²	1	18	yes	complemented
47	C₂×C₂²⋊Q₈	C₂²	1	9	yes	complemented
48	C₂×#121	C₂²	1	9	yes	complemented
49	C₂×#121	C₂²	1	18	yes	complemented
50	C₂×#120	C₂²	1	18	yes	complemented
51	#520	C₂²	1	6	yes	complemented
52	#518	C₂²	1	9	yes	complemented
53	C₄×#41	C₂²	1	6	yes	complemented
54	C₂×C₄×D₄	C₂²	1	9	yes	complemented
55	C₂×C₄×D₄	C₂²	1	9	yes	complemented
56	C₂²×C₂²⋊C₄	C₂²	1	6	yes	complemented
57	C₂⁵	C₂³	1	6	yes	—
58	2₋¹⁺⁴	—	2	48	no	—
59	H(5,𝔽₂)	—	2	48	no	—
60	H(5,𝔽₂)	—	2	72	no	—
61	H(5,𝔽₂)	—	2	72	no	—
62	H(5,𝔽₂)	D₄	1	16	yes	complemented
63	C₂×#41	—	2	48	no	—
64	C₂×#41	—	2	48	no	—
65	C₂×#41	—	2	72	no	—
66	C₂×#41	C₂³	1	18	yes	—
67	C₂×#41	C₂³	1	18	yes	—
68	C₂×#41	C₂³	1	6	yes	—
69	C₂×#41	C₂³	1	6	yes	—
70	C₂×#41	C₂³	1	18	yes	complemented
71	C₂×#41	C₂³	1	18	yes	complemented
72	C₂×#41	C₂³	1	36	yes	complemented
73	C₂×#41	C₂³	1	2	yes	complemented
74	C₂²×Q₈	C₂³	1	2	yes	—
75	C₂²×Q₈	C₂³	1	9	yes	complemented
76	C₂²×Q₈	C₂³	1	6	yes	complemented
77	C₂²×D₄	—	2	72	no	—
78	C₂²×D₄	C₂³	1	18	yes	—
79	C₂²×D₄	C₂³	1	9	yes	—
80	C₂²×D₄	C₂³	1	18	yes	—
81	C₂²×D₄	C₂³	1	36	yes	—
82	C₂²×D₄	C₂³	1	9	yes	—
83	C₂²×D₄	C₂³	1	9	yes	—
84	C₂²×D₄	C₂³	1	18	yes	complemented
85	C₂²×D₄	C₂³	1	18	yes	complemented
86	C₂²×D₄	C₂³	1	18	yes	complemented
87	C₂³×C₄	C₂³	1	9	yes	—
88	C₂³×C₄	C₂³	1	6	yes	complemented
89	C₂³×C₄	C₂³	1	9	yes	complemented
90	C₂³×C₄	C₂³	1	6	yes	complemented
91	C₄⋊Q₈	C₂³	1	18	yes	complemented
92	#127	C₂³	1	18	yes	complemented
93	#124	C₂³	1	18	yes	complemented
94	#124	C₂³	1	36	yes	complemented
95	#124	C₂³	1	6	yes	complemented
96	#123	C₂³	1	36	yes	complemented
97	#123	C₂³	1	72	yes	complemented
98	#123	C₂³	1	36	yes	complemented
99	C₂²⋊Q₈	C₂³	1	18	yes	—
100	C₂²⋊Q₈	C₂³	1	36	yes	complemented
101	C₂²⋊Q₈	C₂³	1	6	yes	complemented
102	C₂²⋊Q₈	C₂³	1	18	yes	complemented
103	C₂²⋊Q₈	C₂³	1	18	yes	complemented
104	#121	C₂³	1	6	yes	—
105	#121	C₂³	1	18	yes	complemented
106	#121	C₂³	1	36	yes	complemented
107	#121	C₂³	1	72	yes	complemented
108	#121	C₂³	1	36	yes	complemented
109	#121	C₂³	1	18	yes	complemented
110	#121	C₂³	1	18	yes	complemented
111	#121	C₂³	1	36	yes	complemented
112	#120	C₂³	1	36	yes	—
113	#120	C₂³	1	72	yes	complemented
114	#120	C₂³	1	36	yes	complemented
115	C₄×Q₈	C₂³	1	2	yes	complemented
116	C₄×Q₈	C₂³	1	6	yes	complemented
117	C₄×D₄	C₂³	1	6	yes	—
118	C₄×D₄	C₂³	1	18	yes	—
119	C₄×D₄	C₂³	1	18	yes	complemented
120	C₄×D₄	C₂³	1	36	yes	complemented
121	C₄×D₄	C₂³	1	18	yes	complemented
122	C₄×D₄	C₂³	1	18	yes	complemented
123	C₄×D₄	C₂³	1	18	yes	complemented
124	C₄×D₄	C₂³	1	36	yes	complemented
125	#117	C₂³	1	18	yes	complemented
126	C₂×C₄⋊C₄	C₂³	1	9	yes	complemented
127	C₂×C₄⋊C₄	C₂³	1	9	yes	complemented
128	C₂×C₄⋊C₄	C₂³	1	18	yes	complemented
129	C₂×C₂²⋊C₄	C₂³	1	9	yes	—
130	C₂×C₂²⋊C₄	C₂³	1	18	yes	—
131	C₂×C₂²⋊C₄	C₂³	1	18	yes	complemented
132	C₂×C₂²⋊C₄	C₂³	1	18	yes	complemented
133	C₂×C₂²⋊C₄	C₂³	1	18	yes	complemented
134	C₂×C₂²⋊C₄	C₂³	1	9	yes	complemented
135	C₂×C₂²⋊C₄	C₂³	1	18	yes	complemented
136	C₂×C₄²	C₂³	1	9	yes	complemented
137	C₂⁴	—	2	24	no	—
138	C₂⁴	—	4	12	no	—
139	C₂⁴	—	8	6	no	—
140	C₂⁴	C₂⁴	1	9	yes	—
141	C₂⁴	C₂⁴	1	18	yes	—
142	C₂⁴	C₂⁴	1	9	yes	—
143	C₂⁴	C₂⁴	1	6	yes	—
144	#41	—	2	8	no	—
145	#41	—	2	24	no	—
146	#41	—	2	72	no	—
147	#41	—	2	72	no	—
148	#41	—	2	144	no	—
149	#41	—	2	72	no	—
150	#41	—	2	72	no	—
151	#41	C₂×D₄	1	48	yes	complemented
152	C₂×Q₈	—	2	8	no	—
153	C₂×Q₈	—	2	24	no	—
154	C₂×Q₈	—	2	36	no	—
155	C₂×Q₈	C₂⁴	1	18	yes	—
156	C₂×Q₈	C₂⁴	1	6	yes	—
157	C₂×Q₈	C₂⁴	1	2	yes	—
158	C₂×D₄	—	2	72	no	—
159	C₂×D₄	—	2	72	no	—
160	C₂×D₄	—	2	72	no	—
161	C₂×D₄	—	2	144	no	—
162	C₂×D₄	—	2	72	no	—
163	C₂×D₄	—	2	36	no	—
164	C₂×D₄	—	4	18	no	—
165	C₂×D₄	—	8	18	no	—
166	C₂×D₄	C₂×D₄	1	72	yes	complemented
167	C₂×D₄	C₂⁴	1	6	yes	—
168	C₂×D₄	C₂⁴	1	36	yes	—
169	C₂×D₄	C₂⁴	1	36	yes	—
170	C₂×D₄	C₂⁴	1	6	yes	—
171	C₂×D₄	C₂⁴	1	18	yes	—
172	C₂×D₄	C₂⁴	1	18	yes	—
173	C₂×D₄	C₂⁴	1	36	yes	—
174	C₂×D₄	C₂⁴	1	18	yes	—
175	C₂²×C₄	—	2	24	no	—
176	C₂²×C₄	—	2	36	no	—
177	C₂²×C₄	—	2	36	no	—
178	C₂²×C₄	—	8	6	no	—
179	C₂²×C₄	C₂⁴	1	9	yes	—
180	C₂²×C₄	C₂⁴	1	18	yes	—
181	C₂²×C₄	C₂⁴	1	9	yes	—
182	C₂²×C₄	C₂⁴	1	6	yes	—
183	C₂²×C₄	C₂⁴	1	6	yes	—
184	C₂²×C₄	C₂⁴	1	18	yes	—
185	C₂²×C₄	C₂⁴	1	18	yes	—
186	C₂²×C₄	C₂⁴	1	9	yes	—
187	C₂²×C₄	C₂⁴	1	9	yes	—
188	C₂²×C₄	C₂⁴	1	18	yes	—
189	C₂²×C₄	C₂⁴	1	9	yes	—
190	C₄⋊C₄	C₂⁴	1	6	yes	—
191	C₄⋊C₄	C₂⁴	1	18	yes	—
192	C₄⋊C₄	C₂⁴	1	36	yes	—
193	C₄⋊C₄	C₂⁴	1	18	yes	—
194	C₄⋊C₄	C₂⁴	1	6	yes	—
195	C₂²⋊C₄	C₂⁴	1	1	yes	characteristic
196	C₂²⋊C₄	C₂⁴	1	9	yes	—
197	C₂²⋊C₄	C₂⁴	1	9	yes	—
198	C₂²⋊C₄	C₂⁴	1	9	yes	—
199	C₂²⋊C₄	C₂⁴	1	36	yes	—
200	C₂²⋊C₄	C₂⁴	1	6	yes	—
201	C₂²⋊C₄	C₂⁴	1	36	yes	—
202	C₂²⋊C₄	C₂⁴	1	36	yes	—
203	C₂²⋊C₄	C₂⁴	1	18	yes	—
204	C₂²⋊C₄	C₂⁴	1	12	yes	—
205	C₄²	C₂⁴	1	6	yes	—
206	C₄²	C₂⁴	1	18	yes	—
207	C₂³	—	2	18	no	—
208	C₂³	—	2	18	no	—
209	C₂³	—	2	18	no	—
210	C₂³	—	2	36	no	—
211	C₂³	—	4	6	no	—
212	C₂³	—	4	6	no	—
213	C₂³	—	4	9	no	—
214	C₂³	—	4	9	no	—
215	C₂³	—	4	9	no	—
216	C₂³	—	4	18	no	—
217	C₂³	—	4	18	no	—
218	C₂³	—	8	48	no	—
219	C₂³	C₂²×D₄	1	24	yes	—
220	C₂³	C₂⁵	1	9	yes	—
221	C₂³	C₂⁵	1	6	yes	—
222	C₂³	C₂⁵	1	9	yes	—
223	C₂³	C₂⁵	1	1	yes	characteristic
224	Q₈	—	2	36	no	—
225	Q₈	—	2	12	no	—
226	Q₈	C₂²×D₄	1	8	yes	—
227	D₄	—	2	72	no	—
228	D₄	—	2	72	no	—
229	D₄	—	2	12	no	—
230	D₄	—	2	36	no	—
231	D₄	—	2	12	no	—
232	D₄	—	4	18	no	—
233	D₄	—	4	36	no	—
234	D₄	C₂²×D₄	1	72	yes	—
235	C₂×C₄	—	2	18	no	—
236	C₂×C₄	—	2	18	no	—
237	C₂×C₄	—	2	36	no	—
238	C₂×C₄	—	2	12	no	—
239	C₂×C₄	—	2	12	no	—
240	C₂×C₄	—	2	36	no	—
241	C₂×C₄	—	2	36	no	—
242	C₂×C₄	—	4	9	no	—
243	C₂×C₄	—	4	18	no	—
244	C₂×C₄	—	4	9	no	—
245	C₂×C₄	C₂²×D₄	1	36	yes	—
246	C₂×C₄	C₂⁵	1	1	yes	characteristic
247	C₂×C₄	C₂⁵	1	6	yes	—
248	C₂×C₄	C₂⁵	1	9	yes	—
249	C₂×C₄	C₂⁵	1	1	yes	characteristic
250	C₂×C₄	C₂⁵	1	9	yes	—
251	C₂×C₄	C₂⁵	1	6	yes	—
252	C₂×C₄	C₂⁵	1	6	yes	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	9	no	—
255	C₂²	—	2	9	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	6	no	—
258	C₂²	—	2	9	no	—
259	C₂²	—	2	6	no	—
260	C₂²	—	2	9	no	—
261	C₂²	—	2	6	no	—
262	C₂²	—	2	9	no	—
263	C₂²	—	4	24	no	—
264	C₂²	—	4	36	no	—
265	C₂²	—	8	18	no	—
266	C₂²	—	8	36	no	—
267	C₂²	C₂³×D₄	1	18	yes	—
268	C₂²	#585	1	2	yes	complemented
269	C₂²	C₂⁶	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	6	no	—
272	C₄	—	2	9	no	—
273	C₄	—	2	6	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	9	no	—
276	C₄	C₂³×D₄	1	12	yes	—
277	C₂	—	2	2	no	—
278	C₂	—	2	18	no	—
279	C₂	—	4	9	no	—
280	C₂	—	4	6	no	—
281	C₂	C₂⁴×D₄	1	1	yes	characteristic
282	C₂	C₂×#585	1	1	yes	characteristic
283	C₂	H(7,𝔽₂)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×#585	C₂	semidirect
2. C₂²×H(5,𝔽₂)	C₂	semidirect
3. #3500	C₂	semidirect
4. #3496	C₂	semidirect
5. #3489	C₂	semidirect
6. #3488	C₂	semidirect
7. #3485	C₂	semidirect
8. #3484	C₂	semidirect
9. D₄×#41	C₂	semidirect
10. C₂×#546	C₂	semidirect
11. C₄×H(5,𝔽₂)	C₂	semidirect
13. C₂×2₋¹⁺⁴	C₂²	semidirect
16. C₂×H(5,𝔽₂)	C₂²	semidirect
17. C₂×H(5,𝔽₂)	C₂²	semidirect
18. C₂×H(5,𝔽₂)	C₂²	semidirect
19. C₂²×#41	C₂²	semidirect
20. C₂²×#41	C₂²	semidirect
21. C₂²×#41	C₂²	semidirect
22. C₂³×D₄	C₂²	semidirect
23. #550	C₂²	semidirect
24. D₄×Q₈	C₂²	semidirect
25. #548	C₂²	semidirect
26. #548	C₂²	semidirect
27. #548	C₂²	semidirect
28. #547	C₂²	semidirect
29. #547	C₂²	semidirect
30. #546	C₂²	semidirect
31. #546	C₂²	semidirect
32. #546	C₂²	semidirect
33. #546	C₂²	semidirect
34. #546	C₂²	semidirect
35. #546	C₂²	semidirect
36. D₄²	C₂²	semidirect
37. D₄²	C₂²	semidirect
38. #537	C₂²	semidirect
39. #536	C₂²	semidirect
40. #535	C₂²	semidirect
41. #534	C₂²	semidirect
42. #532	C₂²	semidirect
43. C₂×#124	C₂²	semidirect
44. #525	C₂²	semidirect
45. #525	C₂²	semidirect
46. C₂×#123	C₂²	semidirect
47. C₂×C₂²⋊Q₈	C₂²	semidirect
48. C₂×#121	C₂²	semidirect
49. C₂×#121	C₂²	semidirect
50. C₂×#120	C₂²	semidirect
51. #520	C₂²	semidirect
52. #518	C₂²	semidirect
53. C₄×#41	C₂²	semidirect
54. C₂×C₄×D₄	C₂²	semidirect
55. C₂×C₄×D₄	C₂²	semidirect
56. C₂²×C₂²⋊C₄	C₂²	semidirect
62. H(5,𝔽₂)	D₄	semidirect
70. C₂×#41	C₂³	semidirect
71. C₂×#41	C₂³	semidirect
72. C₂×#41	C₂³	semidirect
73. C₂×#41	C₂³	semidirect
75. C₂²×Q₈	C₂³	semidirect
76. C₂²×Q₈	C₂³	semidirect
84. C₂²×D₄	C₂³	semidirect
85. C₂²×D₄	C₂³	semidirect
86. C₂²×D₄	C₂³	semidirect
88. C₂³×C₄	C₂³	semidirect
89. C₂³×C₄	C₂³	semidirect
90. C₂³×C₄	C₂³	semidirect
91. C₄⋊Q₈	C₂³	semidirect
92. #127	C₂³	semidirect
93. #124	C₂³	semidirect
94. #124	C₂³	semidirect
95. #124	C₂³	semidirect
96. #123	C₂³	semidirect
97. #123	C₂³	semidirect
98. #123	C₂³	semidirect
100. C₂²⋊Q₈	C₂³	semidirect
101. C₂²⋊Q₈	C₂³	semidirect
102. C₂²⋊Q₈	C₂³	semidirect
103. C₂²⋊Q₈	C₂³	semidirect
105. #121	C₂³	semidirect
106. #121	C₂³	semidirect
107. #121	C₂³	semidirect
108. #121	C₂³	semidirect
109. #121	C₂³	semidirect
110. #121	C₂³	semidirect
111. #121	C₂³	semidirect
113. #120	C₂³	semidirect
114. #120	C₂³	semidirect
115. C₄×Q₈	C₂³	semidirect
116. C₄×Q₈	C₂³	semidirect
119. C₄×D₄	C₂³	semidirect
120. C₄×D₄	C₂³	semidirect
121. C₄×D₄	C₂³	semidirect
122. C₄×D₄	C₂³	semidirect
123. C₄×D₄	C₂³	semidirect
124. C₄×D₄	C₂³	semidirect
125. #117	C₂³	semidirect
126. C₂×C₄⋊C₄	C₂³	semidirect
127. C₂×C₄⋊C₄	C₂³	semidirect
128. C₂×C₄⋊C₄	C₂³	semidirect
131. C₂×C₂²⋊C₄	C₂³	semidirect
132. C₂×C₂²⋊C₄	C₂³	semidirect
133. C₂×C₂²⋊C₄	C₂³	semidirect
134. C₂×C₂²⋊C₄	C₂³	semidirect
135. C₂×C₂²⋊C₄	C₂³	semidirect
136. C₂×C₄²	C₂³	semidirect
151. #41	C₂×D₄	semidirect
166. C₂×D₄	C₂×D₄	semidirect
268. C₂²	#585	semidirect

Maximal subgroups

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

subgroup	index
C₂×#585	2
C₂²×H(5,𝔽₂)	2
#3500	2
#3496	2
#3489	2
#3488	2
#3485	2
#3484	2
D₄×#41	2
C₂×#546	2
C₄×H(5,𝔽₂)	2