Subgroups — C₄×#2263 #512762

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2263	C₂	1	1	yes	maximal, characteristic
2	C₄×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#13654	C₂	1	2	yes	maximal, complemented
5	#13652	C₂	1	2	yes	maximal, complemented
6	C₄×SD₆₄	C₂	1	2	yes	maximal, complemented
7	C₄×D₃₂	C₂	1	2	yes	maximal, complemented
8	#13634	C₂	1	1	yes	maximal, characteristic, complemented
9	#13617	C₂	1	1	yes	maximal, characteristic, complemented
10	#13616	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#508	C₂²	1	1	yes	characteristic
13	C₂²×D₁₆	C₂²	1	1	yes	characteristic
14	C₄×#135	—	2	1	no	—
15	C₂×C₄×D₈	—	2	1	no	—
16	#2263	C₄	1	8	yes	direct product factor
17	C₂×SD₆₄	C₂²	1	2	yes	—
18	C₂×D₃₂	C₂²	1	2	yes	—
19	C₂×M₆₄	C₂²	1	1	yes	characteristic
20	C₂×M₆₄	C₂²	1	1	yes	characteristic
21	C₄×Q₃₂	C₂²	1	1	yes	characteristic, complemented
22	C₄×Q₃₂	C₂²	1	1	yes	characteristic, complemented
23	C₄×SD₃₂	—	2	1	no	—
24	C₄×SD₃₂	—	2	1	no	—
25	C₄×D₁₆	—	2	1	no	—
26	C₄×D₁₆	—	2	1	no	—
27	C₄×D₁₆	C₂²	1	2	yes	—
28	C₄×D₁₆	C₂²	1	2	yes	—
29	C₄×D₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₁₆	C₂²	1	1	yes	characteristic, complemented
31	#2158	C₂²	1	1	yes	characteristic
32	C₂×#366	C₂²	1	1	yes	characteristic
33	#2138	—	2	1	no	—
34	C₂×#357	—	2	1	no	—
35	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
36	#1424	C₂²	1	2	yes	complemented
37	#1423	C₂²	1	2	yes	complemented
38	#1416	C₂²	1	2	yes	complemented
39	#1415	C₂²	1	2	yes	—
40	#1415	C₂²	1	2	yes	—
41	#1415	C₂²	1	2	yes	complemented
42	#1397	C₂²	1	2	yes	complemented
43	C₄×C₃₂	C₂²	1	2	yes	complemented
44	C₂×#135	—	2	1	no	—
45	C₂²×D₈	—	2	1	no	—
46	C₄×#41	—	4	1	no	—
47	C₂×C₄×D₄	—	4	1	no	—
48	#508	—	2	2	no	—
49	#508	C₂×C₄	1	4	yes	complemented
50	C₂×Q₃₂	C₂³	1	1	yes	characteristic
51	C₂×SD₃₂	—	2	1	no	—
52	C₂×D₁₆	—	2	1	no	—
53	C₂×D₁₆	—	2	2	no	—
54	C₂×D₁₆	C₂×C₄	1	4	yes	complemented
55	C₂×D₁₆	C₂³	1	2	yes	—
56	C₂×D₁₆	C₂³	1	1	yes	characteristic
57	C₂²×C₁₆	D₄	1	1	yes	characteristic
58	C₂²×C₁₆	C₂³	1	1	yes	characteristic
59	C₄×Q₁₆	—	2	1	no	—
60	C₄×Q₁₆	—	2	1	no	—
61	C₄×SD₁₆	—	4	1	no	—
62	C₄×SD₁₆	—	4	1	no	—
63	C₄×D₈	—	2	2	no	—
64	C₄×D₈	—	2	2	no	—
65	C₄×D₈	—	2	1	no	—
66	C₄×D₈	—	2	1	no	—
67	C₄×D₈	—	4	1	no	—
68	C₄×D₈	—	4	1	no	—
69	#427	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	#416	—	4	1	no	—
72	C₂×#102	—	4	1	no	—
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	SD₆₄	C₂×C₄	1	8	yes	complemented
75	D₃₂	C₂×C₄	1	8	yes	complemented
76	M₆₄	—	2	2	no	—
77	M₆₄	C₂×C₄	1	4	yes	complemented
78	C₂×C₃₂	C₂³	1	2	yes	—
79	C₂×C₃₂	C₂³	1	2	yes	—
80	#367	—	2	1	no	—
81	#366	—	2	1	no	—
82	#366	C₂³	1	2	yes	—
83	#366	C₂³	1	1	yes	characteristic
84	#366	C₂³	1	1	yes	characteristic
85	#358	—	2	1	no	—
86	#358	—	2	1	no	—
87	#357	—	2	2	no	—
88	#357	—	2	2	no	—
89	#357	—	2	1	no	—
90	#357	—	2	1	no	—
91	C₄×C₁₆	D₄	1	1	yes	characteristic
92	C₄×C₁₆	D₄	1	1	yes	characteristic
93	C₄×C₁₆	C₂³	1	1	yes	characteristic
94	C₄×C₁₆	C₂³	1	1	yes	characteristic
95	C₂×#41	—	4	1	no	—
96	C₂²×D₄	—	4	1	no	—
97	C₂³×C₄	—	8	1	no	—
98	#135	—	2	4	no	—
99	#135	—	4	2	no	—
100	C₂×Q₁₆	—	2	1	no	—
101	C₂×SD₁₆	—	4	1	no	—
102	C₂×D₈	—	2	4	no	—
103	C₂×D₈	—	2	2	no	—
104	C₂×D₈	—	2	1	no	—
105	C₂×D₈	—	4	1	no	—
106	C₂×D₈	—	4	2	no	—
107	C₂²×C₈	D₈	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₄×Q₈	—	4	1	no	—
110	C₄×Q₈	—	4	1	no	—
111	C₄×D₄	—	4	2	no	—
112	C₄×D₄	—	4	2	no	—
113	C₄×D₄	—	4	1	no	—
114	C₄×D₄	—	4	1	no	—
115	C₄×D₄	—	8	1	no	—
116	C₄×D₄	—	8	1	no	—
117	C₄×D₄	—	8	1	no	—
118	C₄×D₄	—	8	1	no	—
119	#117	—	4	1	no	—
120	#117	—	8	1	no	—
121	C₂×C₄⋊C₄	—	4	1	no	—
122	C₂×C₂²⋊C₄	—	8	1	no	—
123	C₂×C₄²	—	8	1	no	—
124	C₂×C₄²	D₈	1	1	yes	characteristic
125	Q₃₂	—	2	1	no	—
126	Q₃₂	C₂²×C₄	1	2	yes	complemented
127	SD₃₂	—	2	4	no	—
128	D₁₆	—	2	4	no	—
129	D₁₆	—	2	2	no	—
130	D₁₆	—	2	1	no	—
131	D₁₆	C₂²×C₄	1	4	yes	—
132	D₁₆	C₂²×C₄	1	2	yes	complemented
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	C₂²×C₄	1	2	yes	—
135	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
136	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
137	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
138	C₂×C₁₆	#41	1	2	yes	—
139	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
140	#107	—	2	2	no	—
141	#107	—	2	1	no	—
142	#107	—	2	1	no	—
143	#107	—	4	1	no	—
144	#106	—	4	1	no	—
145	#103	—	4	1	no	—
146	#103	—	4	1	no	—
147	#102	—	4	2	no	—
148	#102	—	4	2	no	—
149	#102	—	4	1	no	—
150	#102	—	4	1	no	—
151	C₄×C₈	D₈	1	1	yes	characteristic
152	C₄×C₈	D₈	1	1	yes	characteristic
153	C₄×C₈	C₂×D₄	1	1	yes	characteristic
154	C₄×C₈	C₂×D₄	1	1	yes	characteristic
155	C₃₂	—	2	2	no	—
156	C₃₂	C₂²×C₄	1	4	yes	complemented
157	C₂⁴	—	8	1	no	—
158	#41	—	4	4	no	—
159	#41	—	8	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	1	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	—	8	1	no	—
165	C₂×D₄	—	8	1	no	—
166	C₂×D₄	—	8	2	no	—
167	C₂²×C₄	—	8	1	no	—
168	C₂²×C₄	—	8	1	no	—
169	C₂²×C₄	—	8	1	no	—
170	C₂²×C₄	—	8	2	no	—
171	C₂²×C₄	—	8	1	no	—
172	C₂²×C₄	—	8	2	no	—
173	C₂²×C₄	—	16	4	no	—
174	C₂²×C₄	D₁₆	1	1	yes	characteristic
175	C₂²×C₄	D₁₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
177	Q₁₆	—	2	2	no	—
178	Q₁₆	—	4	1	no	—
179	SD₁₆	—	4	4	no	—
180	D₈	—	2	4	no	—
181	D₈	—	2	2	no	—
182	D₈	—	4	2	no	—
183	D₈	—	4	1	no	—
184	D₈	—	4	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₄×D₄	1	2	yes	—
187	C₂×C₈	C₂×D₈	1	1	yes	characteristic
188	C₂×C₈	C₂×D₈	1	1	yes	characteristic
189	C₂×C₈	C₂×D₈	1	1	yes	characteristic
190	C₂×C₈	#135	1	2	yes	—
191	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	8	1	no	—
196	C₄⋊C₄	—	8	1	no	—
197	C₄⋊C₄	—	8	1	no	—
198	C₂²⋊C₄	—	8	1	no	—
199	C₂²⋊C₄	—	8	2	no	—
200	C₂²⋊C₄	—	8	2	no	—
201	C₂²⋊C₄	—	8	1	no	—
202	C₂²⋊C₄	—	8	1	no	—
203	C₄²	—	8	1	no	—
204	C₄²	—	8	1	no	—
205	C₄²	—	8	1	no	—
206	C₄²	D₁₆	1	1	yes	characteristic
207	C₄²	D₁₆	1	1	yes	characteristic
208	C₄²	C₂×D₈	1	1	yes	characteristic
209	C₄²	C₂×D₈	1	1	yes	characteristic
210	C₁₆	—	2	1	no	—
211	C₁₆	—	2	1	no	—
212	C₁₆	C₄×D₄	1	2	yes	—
213	C₁₆	C₂³×C₄	1	1	yes	characteristic
214	C₁₆	C₂×#41	1	1	yes	characteristic
215	C₂³	—	8	1	no	—
216	C₂³	—	8	2	no	—
217	C₂³	—	8	4	no	—
218	C₂³	—	16	2	no	—
219	C₂³	—	16	2	no	—
220	C₂³	C₂×D₁₆	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	8	1	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	2	no	—
225	D₄	—	8	4	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	1	no	—
228	D₄	—	8	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	8	1	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	16	2	no	—
237	C₂×C₄	—	16	4	no	—
238	C₂×C₄	—	16	2	no	—
239	C₂×C₄	—	16	2	no	—
240	C₂×C₄	—	16	4	no	—
241	C₂×C₄	C₄×D₈	1	2	yes	—
242	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
243	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
244	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
245	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
246	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
247	C₂×C₄	#508	1	2	yes	—
248	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	C₄×D₈	1	2	yes	—
252	C₈	C₂×C₄×D₄	1	1	yes	characteristic
253	C₈	C₂×#135	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	16	8	no	—
259	C₂²	—	16	2	no	—
260	C₂²	—	16	1	no	—
261	C₂²	—	16	1	no	—
262	C₂²	—	16	2	no	—
263	C₂²	C₄×D₁₆	1	2	yes	—
264	C₂²	C₂²×D₁₆	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	2	no	—
269	C₄	—	16	1	no	—
270	C₄	—	16	2	no	—
271	C₄	—	16	1	no	—
272	C₄	C₄×D₁₆	1	2	yes	—
273	C₄	#2263	1	2	yes	direct product factor
274	C₄	C₂×C₄×D₈	1	1	yes	characteristic
275	C₄	C₂×#508	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	16	2	no	—
278	C₂	—	16	4	no	—
279	C₂	C₂×C₄×D₁₆	1	1	yes	characteristic
280	C₂	C₂×#2263	1	1	yes	characteristic
281	C₂	#33981	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂²×D₁₆
prime radical	C₂×#2263	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×#508	C₂	semidirect
3. C₂×C₄×D₁₆	C₂	semidirect
4. #13654	C₂	semidirect
5. #13652	C₂	semidirect
6. C₄×SD₆₄	C₂	semidirect
7. C₄×D₃₂	C₂	semidirect
8. #13634	C₂	semidirect
9. #13617	C₂	semidirect
10. #13616	C₂	semidirect
11. C₄×M₆₄	C₂	semidirect
16. #2263	C₄	direct (N × H)
21. C₄×Q₃₂	C₂²	semidirect
22. C₄×Q₃₂	C₂²	semidirect
29. C₄×D₁₆	C₂²	semidirect
30. C₄×D₁₆	C₂²	semidirect
36. #1424	C₂²	semidirect
37. #1423	C₂²	semidirect
38. #1416	C₂²	semidirect
41. #1415	C₂²	semidirect
42. #1397	C₂²	semidirect
43. C₄×C₃₂	C₂²	semidirect
49. #508	C₂×C₄	semidirect
54. C₂×D₁₆	C₂×C₄	semidirect
74. SD₆₄	C₂×C₄	semidirect
75. D₃₂	C₂×C₄	semidirect
77. M₆₄	C₂×C₄	semidirect
126. Q₃₂	C₂²×C₄	semidirect
132. D₁₆	C₂²×C₄	semidirect
156. C₃₂	C₂²×C₄	semidirect
273. C₄	#2263	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×#2263	2
C₄×#508	2
C₂×C₄×D₁₆	2
#13654	2
#13652	2
C₄×SD₆₄	2
C₄×D₃₂	2
#13634	2
#13617	2
#13616	2
C₄×M₆₄	2

Subgroups — C4×#2263 #512762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#2263 #512762