Subgroups — #20564

Order 256.

485 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×#435	C₂	1	1	yes	maximal, characteristic
3	#2290	C₂	1	1	yes	maximal, characteristic
4	#1771	C₂	1	1	yes	maximal, characteristic, complemented
5	#1770	C₂	1	1	yes	maximal, characteristic, complemented
6	#1768	C₂	1	1	yes	maximal, characteristic, complemented
7	#1753	C₂	1	1	yes	maximal, characteristic, complemented
8	#1753	C₂	1	1	yes	maximal, characteristic, complemented
9	#1753	C₂	1	1	yes	maximal, characteristic, complemented
10	#1752	C₂	1	1	yes	maximal, characteristic, complemented
11	#1750	C₂	1	1	yes	maximal, characteristic, complemented
12	#1749	C₂	1	1	yes	maximal, characteristic, complemented
13	#1746	C₂	1	1	yes	maximal, characteristic, complemented
14	#1729	C₂	1	1	yes	maximal, characteristic, complemented
15	#1728	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆	C₂²	1	1	yes	characteristic
19	C₂³×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
21	C₂×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄²	C₂²	1	1	yes	characteristic
23	#436	C₄	1	8	yes	complemented
24	#435	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×#97	C₂²	1	1	yes	characteristic
32	C₂×#97	C₂²	1	1	yes	characteristic
33	C₂×#97	C₂²	1	1	yes	characteristic
34	C₂×#97	C₂²	1	1	yes	characteristic
35	C₂×#97	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#388	C₄	1	2	yes	—
40	#387	C₄	1	2	yes	—
41	#386	C₄	1	2	yes	—
42	#385	C₄	1	2	yes	—
43	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
46	#376	C₂²	1	1	yes	characteristic
47	#336	C₂²	1	1	yes	characteristic
48	#336	C₂²	1	1	yes	characteristic
49	#336	C₂²	1	1	yes	characteristic
50	#336	C₂²	1	1	yes	characteristic
51	#336	C₂²	1	1	yes	characteristic
52	#336	C₂²	1	1	yes	characteristic
53	#336	C₂²	1	1	yes	characteristic
54	#336	C₂²	1	1	yes	characteristic
55	#336	C₂²	1	1	yes	characteristic
56	#336	C₂²	1	1	yes	characteristic
57	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄	C₂×C₄	1	2	yes	—
59	C₂³×C₄	C₂³	1	1	yes	characteristic
60	C₂³×C₄	C₂³	1	1	yes	characteristic
61	C₂×M₁₆	—	4	1	no	—
62	C₂×M₁₆	—	4	1	no	—
63	C₂×M₁₆	—	4	1	no	—
64	C₂×M₁₆	—	4	1	no	—
65	C₂×M₁₆	—	4	1	no	—
66	C₂×M₁₆	—	4	1	no	—
67	C₂×M₁₆	C₂×C₄	1	4	yes	—
68	C₂×M₁₆	C₂×C₄	1	4	yes	—
69	C₂×M₁₆	C₂×C₄	1	4	yes	complemented
70	C₂²×C₈	—	4	1	no	—
71	C₂²×C₈	—	4	1	no	—
72	C₂²×C₈	C₂×C₄	1	4	yes	complemented
73	C₂²×C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₈	C₂³	1	1	yes	characteristic
78	C₂²×C₈	C₂³	1	1	yes	characteristic
79	C₂²×C₈	C₂³	1	1	yes	characteristic
80	C₂²×C₈	C₂³	1	1	yes	characteristic
81	C₄×D₄	C₂×C₄	1	4	yes	—
82	C₄×D₄	C₂×C₄	1	4	yes	—
83	#117	—	4	1	no	—
84	#117	—	4	1	no	—
85	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
87	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
89	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
90	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²	—	4	1	no	—
94	C₂×C₄²	—	4	1	no	—
95	C₂×C₄²	C₂×C₄	1	2	yes	—
96	C₂×C₄²	C₂³	1	1	yes	characteristic
97	C₂×C₄²	C₂³	1	1	yes	characteristic
98	C₂×C₄²	C₂³	1	1	yes	characteristic
99	C₂×C₄²	C₂³	1	1	yes	characteristic
100	C₂×C₄²	C₂³	1	1	yes	characteristic
101	C₄⋊C₈	C₂×C₄	1	4	yes	complemented
102	C₄⋊C₈	C₂×C₄	1	4	yes	complemented
103	C₂²⋊C₈	C₂×C₄	1	4	yes	—
104	C₂²⋊C₈	C₂×C₄	1	4	yes	—
105	C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
106	C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
107	#97	—	4	1	no	—
108	#97	—	4	1	no	—
109	#97	—	4	1	no	—
110	#97	—	4	1	no	—
111	#97	C₂×C₄	1	4	yes	complemented
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	C₂×C₄	1	4	yes	complemented
115	#95	C₂×C₄	1	2	yes	—
116	#95	C₂×C₄	1	2	yes	—
117	#95	C₂×C₄	1	2	yes	—
118	#95	C₂×C₄	1	2	yes	—
119	#95	C₂×C₄	1	2	yes	—
120	C₂⁴	C₂²×C₄	1	1	yes	characteristic
121	C₂⁴	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄	—	4	2	no	—
123	C₂×D₄	C₄²	1	4	yes	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	C₂²×C₄	1	2	yes	—
136	C₂²×C₄	C₂²×C₄	1	2	yes	—
137	C₂²×C₄	C₂²×C₄	1	2	yes	—
138	C₂²×C₄	C₂²×C₄	1	2	yes	—
139	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄	1	2	yes	—
144	C₂²×C₄	C₂²×C₄	1	2	yes	—
145	C₂²×C₄	C₂²×C₄	1	2	yes	—
146	C₂²×C₄	C₂²×C₄	1	2	yes	—
147	C₂²×C₄	C₂⁴	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	4	no	—
150	M₁₆	—	4	4	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	C₂²×C₄	1	2	yes	—
169	C₂×C₈	C₂²×C₄	1	2	yes	—
170	C₂×C₈	C₂²×C₄	1	2	yes	—
171	C₂×C₈	C₂²×C₄	1	2	yes	—
172	C₂×C₈	C₂²×C₄	1	2	yes	—
173	C₂×C₈	C₂²×C₄	1	2	yes	—
174	C₂×C₈	C₂²×C₄	1	2	yes	—
175	C₂×C₈	C₂²×C₄	1	2	yes	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	C₄²	1	4	yes	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	C₄²	1	4	yes	—
183	C₂²⋊C₄	C₄²	1	4	yes	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	4	2	no	—
193	C₄²	C₂²×C₄	1	2	yes	—
194	C₄²	C₂²×C₄	1	2	yes	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	C₂×C₄²	1	2	yes	—
204	C₂³	C₂×C₄²	1	2	yes	—
205	C₂³	C₂³×C₄	1	1	yes	characteristic
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	C₂×C₄²	1	2	yes	—
235	C₂×C₄	C₂×C₄²	1	2	yes	—
236	C₂×C₄	C₂×C₄²	1	2	yes	—
237	C₂×C₄	C₂×C₄²	1	2	yes	—
238	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	C₂²×C₄²	1	1	yes	characteristic
263	C₂²	#518	1	1	yes	characteristic
264	C₂²	#520	1	1	yes	characteristic
265	C₂²	#568	1	2	yes	—
266	C₂²	#568	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	2	no	—
275	C₄	#518	1	1	yes	characteristic
276	C₄	#518	1	1	yes	characteristic
277	C₄	#520	1	1	yes	characteristic
278	C₄	#520	1	1	yes	characteristic
279	C₂	—	4	2	no	—
280	C₂	—	4	2	no	—
281	C₂	#2275	1	1	yes	characteristic
282	C₂	#2873	1	1	yes	characteristic
283	C₂	#2875	1	1	yes	characteristic
284	C₂	#2978	1	2	yes	—
285	C₂	#2995	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂²	C₂²×C₄²
socle	C₂³	C₂³×C₄
prime radical	C₂²×D₄	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1771	C₂	semidirect
5. #1770	C₂	semidirect
6. #1768	C₂	semidirect
7. #1753	C₂	semidirect
8. #1753	C₂	semidirect
9. #1753	C₂	semidirect
10. #1752	C₂	semidirect
11. #1750	C₂	semidirect
12. #1749	C₂	semidirect
13. #1746	C₂	semidirect
14. #1729	C₂	semidirect
15. #1728	C₂	semidirect
23. #436	C₄	semidirect
24. #435	C₄	semidirect
69. C₂×M₁₆	C₂×C₄	semidirect
72. C₂²×C₈	C₂×C₄	semidirect
101. C₄⋊C₈	C₂×C₄	semidirect
102. C₄⋊C₈	C₂×C₄	semidirect
105. C₂²⋊C₈	C₂×C₄	semidirect
106. C₂²⋊C₈	C₂×C₄	semidirect
111. #97	C₂×C₄	semidirect
114. C₄×C₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436	2
C₂×#435	2
#2290	2
#1771	2
#1770	2
#1768	2
#1753	2
#1752	2
#1750	2
#1749	2
#1746	2
#1729	2
#1728	2