Subgroups — #20490

Order 256.

569 conjugacy classes of subgroups.

The subgroup lattice has 293 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	C₂×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#2290	C₂	1	1	yes	maximal, characteristic
6	#1771	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#1768	C₂	1	1	yes	maximal, characteristic, complemented
9	#1753	C₂	1	1	yes	maximal, characteristic, complemented
10	#1753	C₂	1	1	yes	maximal, characteristic, complemented
11	#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	#1750	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	#1726	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	#405	—	2	4	no	—
32	C₄×M₁₆	—	2	4	no	—
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×#97	C₂²	1	1	yes	characteristic
37	C₂×#97	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
45	#388	C₄	1	2	yes	—
46	#387	C₄	1	2	yes	—
47	#386	C₄	1	2	yes	—
48	#385	C₄	1	2	yes	—
49	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
51	C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
52	#376	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	#336	C₂²	1	1	yes	characteristic
58	#336	C₂²	1	1	yes	characteristic
59	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄	C₂×C₄	1	2	yes	—
61	C₂³×C₄	C₂³	1	1	yes	characteristic
62	C₂³×C₄	C₂³	1	1	yes	characteristic
63	C₂×M₁₆	—	2	2	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	—	2	2	no	—
69	C₂×M₁₆	C₂×C₄	1	4	yes	—
70	C₂×M₁₆	C₂×C₄	1	4	yes	—
71	C₂×M₁₆	C₂×C₄	1	4	yes	complemented
72	C₂²×C₈	—	2	2	no	—
73	C₂²×C₈	—	2	2	no	—
74	C₂²×C₈	C₂×C₄	1	4	yes	complemented
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	C₂²×C₈	C₂³	1	1	yes	characteristic
82	C₂²×C₈	C₂³	1	1	yes	characteristic
83	C₄×D₄	C₂×C₄	1	4	yes	—
84	C₄×D₄	C₂×C₄	1	4	yes	—
85	#117	—	2	2	no	—
86	#117	—	2	2	no	—
87	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
88	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
89	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
92	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
93	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²	—	2	2	no	—
96	C₂×C₄²	—	2	2	no	—
97	C₂×C₄²	C₂×C₄	1	2	yes	—
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	C₂×C₄²	C₂³	1	1	yes	characteristic
102	C₂×C₄²	C₂³	1	1	yes	characteristic
103	C₄⋊C₈	C₂×C₄	1	4	yes	complemented
104	C₄⋊C₈	C₂×C₄	1	4	yes	complemented
105	C₂²⋊C₈	C₂×C₄	1	4	yes	—
106	C₂²⋊C₈	C₂×C₄	1	4	yes	—
107	C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
108	C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
109	#97	—	2	2	no	—
110	#97	—	2	2	no	—
111	#97	—	2	2	no	—
112	#97	C₂×C₄	1	4	yes	complemented
113	#97	D₄	1	4	yes	complemented
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₂×C₄	1	4	yes	complemented
121	#95	C₂×C₄	1	2	yes	—
122	#95	C₂×C₄	1	2	yes	—
123	#95	C₂×C₄	1	2	yes	—
124	#95	C₂×C₄	1	2	yes	—
125	#95	C₂×C₄	1	2	yes	—
126	C₂⁴	C₂²×C₄	1	1	yes	characteristic
127	C₂⁴	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	C₄²	1	4	yes	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂²×C₄	1	2	yes	—
142	C₂²×C₄	C₂²×C₄	1	2	yes	—
143	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄	1	2	yes	—
148	C₂²×C₄	C₂²×C₄	1	2	yes	—
149	C₂²×C₄	C₂²×C₄	1	2	yes	—
150	C₂²×C₄	C₂²×C₄	1	2	yes	—
151	C₂²×C₄	C₂²×C₄	1	2	yes	—
152	C₂²×C₄	C₂²×C₄	1	2	yes	—
153	C₂²×C₄	C₂⁴	1	1	yes	characteristic
154	M₁₆	—	2	8	no	—
155	M₁₆	—	2	8	no	—
156	M₁₆	—	2	8	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	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	8	no	—
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	C₂×C₈	C₂²×C₄	1	2	yes	—
177	C₂×C₈	C₂²×C₄	1	2	yes	—
178	C₂×C₈	C₂×D₄	1	2	yes	—
179	C₂×C₈	C₂×D₄	1	2	yes	—
180	C₂×C₈	#41	1	2	yes	—
181	C₂×C₈	#41	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	C₄²	1	4	yes	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	C₄²	1	4	yes	—
189	C₂²⋊C₄	C₄²	1	4	yes	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	C₂²×C₄	1	2	yes	—
198	C₄²	C₂²×C₄	1	2	yes	—
199	C₄²	C₂×D₄	1	2	yes	—
200	C₄²	#41	1	2	yes	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	C₂×C₄²	1	2	yes	—
210	C₂³	C₂×C₄²	1	2	yes	—
211	C₂³	C₂³×C₄	1	1	yes	characteristic
212	D₄	—	4	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	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	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	C₂×C₄²	1	2	yes	—
239	C₂×C₄	C₂×C₄²	1	2	yes	—
240	C₂×C₄	C₂×C₄²	1	2	yes	—
241	C₂×C₄	C₂×C₄²	1	2	yes	—
242	C₂×C₄	C₄×D₄	1	4	yes	—
243	C₂×C₄	#131	1	4	yes	—
244	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄	C₂×#41	1	1	yes	characteristic
248	C₂×C₄	C₂×#41	1	1	yes	characteristic
249	C₂×C₄	C₂×#41	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₈	—	2	2	no	—
256	C₈	C₄×D₄	1	4	yes	—
257	C₈	C₄×D₄	1	4	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	C₂²×C₄²	1	1	yes	characteristic
269	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
270	C₂²	C₄×#41	1	1	yes	characteristic
271	C₂²	C₂×#131	1	2	yes	—
272	C₂²	#568	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	2	no	—
281	C₄	C₂×C₄×D₄	1	1	yes	characteristic
282	C₄	C₂×C₄×D₄	1	1	yes	characteristic
283	C₄	C₄×#41	1	1	yes	characteristic
284	C₄	C₄×#41	1	1	yes	characteristic
285	C₂	—	4	2	no	—
286	C₂	—	4	2	no	—
287	C₂	C₄²×D₄	1	1	yes	characteristic
288	C₂	#2873	1	1	yes	characteristic
289	C₂	C₄×#131	1	1	yes	characteristic
290	C₂	#2933	1	2	yes	—
291	C₂	#2969	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
3. C₂×#405	C₂	semidirect
4. C₂×C₄×M₁₆	C₂	semidirect
6. #1771	C₂	semidirect
7. C₈×C₄⋊C₄	C₂	semidirect
8. #1768	C₂	semidirect
9. #1753	C₂	semidirect
10. #1753	C₂	semidirect
11. #1753	C₂	semidirect
12. C₈×C₂²⋊C₄	C₂	semidirect
13. #1750	C₂	semidirect
14. C₄×C₂²⋊C₈	C₂	semidirect
15. #1726	C₂	semidirect
23. #436	C₄	semidirect
24. #435	C₄	semidirect
71. C₂×M₁₆	C₂×C₄	semidirect
74. C₂²×C₈	C₂×C₄	semidirect
103. C₄⋊C₈	C₂×C₄	semidirect
104. C₄⋊C₈	C₂×C₄	semidirect
107. C₂²⋊C₈	C₂×C₄	semidirect
108. C₂²⋊C₈	C₂×C₄	semidirect
112. #97	C₂×C₄	semidirect
113. #97	D₄	semidirect
120. 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
C₂×#405	2
C₂×C₄×M₁₆	2
#2290	2
#1771	2
C₈×C₄⋊C₄	2
#1768	2
#1753	2
C₈×C₂²⋊C₄	2
#1750	2
C₄×C₂²⋊C₈	2
#1726	2