Subgroups — D₄×M₆₄ #512753

Order 512.

462 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₆₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#13646	C₂	1	2	yes	maximal, complemented
4	#13645	C₂	1	4	yes	maximal, complemented
5	D₄×C₃₂	C₂	1	2	yes	maximal, complemented
6	#13627	C₂	1	1	yes	maximal, characteristic, complemented
7	#13553	C₂	1	2	yes	maximal, complemented
8	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₁₆	C₂²	1	2	yes	—
10	C₂×C₈×D₄	C₄	1	1	yes	characteristic
11	C₂×M₆₄	—	2	8	no	—
12	C₂×M₆₄	C₂²	1	2	yes	—
13	C₂×M₆₄	C₂²	1	1	yes	characteristic
14	C₂×M₆₄	C₂²	1	1	yes	characteristic
15	C₂×M₆₄	C₂²	1	4	yes	complemented
16	C₂²×C₃₂	C₂²	1	4	yes	complemented
17	D₄×C₁₆	C₄	1	2	yes	—
18	D₄×C₁₆	C₄	1	1	yes	characteristic
19	D₄×C₁₆	C₄	1	1	yes	characteristic
20	D₄×C₁₆	C₂²	1	1	yes	characteristic
21	D₄×C₁₆	C₂²	1	2	yes	—
22	D₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
24	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
25	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
26	C₄⋊C₃₂	C₂²	1	2	yes	complemented
27	C₄⋊C₃₂	C₂²	1	2	yes	complemented
28	C₂²⋊C₃₂	C₂²	1	4	yes	complemented
29	C₂²⋊C₃₂	C₂²	1	4	yes	complemented
30	#1397	C₂²	1	2	yes	complemented
31	C₄×C₃₂	C₂²	1	2	yes	complemented
32	C₂³×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₄	C₈	1	1	yes	characteristic
34	C₂²×C₁₆	—	2	4	no	—
35	C₂²×C₁₆	—	2	4	no	—
36	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₆	C₂×C₄	1	2	yes	—
38	C₂²×C₁₆	C₂×C₄	1	2	yes	—
39	C₂²×C₁₆	C₂×C₄	1	2	yes	—
40	C₂²×C₁₆	C₂³	1	2	yes	—
41	C₂²×C₁₆	C₂³	1	2	yes	—
42	C₂²×C₁₆	C₂³	1	1	yes	characteristic
43	C₈×D₄	C₈	1	2	yes	—
44	C₈×D₄	C₈	1	1	yes	characteristic
45	C₈×D₄	C₈	1	1	yes	characteristic
46	C₈×D₄	C₂×C₄	1	1	yes	characteristic
47	C₈×D₄	C₂×C₄	1	2	yes	—
48	C₈×D₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
50	C₂×C₂²⋊C₈	C₈	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	M₆₄	—	2	4	no	—
53	M₆₄	—	2	2	no	—
54	M₆₄	—	2	8	no	—
55	M₆₄	D₄	1	4	yes	direct product factor
56	C₂×C₃₂	—	2	8	no	—
57	C₂×C₃₂	C₂³	1	2	yes	—
58	C₂×C₃₂	C₂³	1	4	yes	—
59	C₂×C₃₂	C₂³	1	2	yes	—
60	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
62	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
64	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
65	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
66	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
67	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
68	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
69	C₄×C₁₆	C₂³	1	1	yes	characteristic
70	C₄×C₁₆	C₂³	1	1	yes	characteristic
71	C₂²×D₄	C₁₆	1	1	yes	characteristic
72	C₂³×C₄	C₂×C₈	1	2	yes	—
73	C₂²×C₈	—	2	4	no	—
74	C₂²×C₈	—	2	4	no	—
75	C₂²×C₈	C₂×C₈	1	2	yes	—
76	C₂²×C₈	C₂×C₈	1	2	yes	—
77	C₂²×C₈	C₂×C₈	1	2	yes	—
78	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	2	yes	—
80	C₂²×C₈	C₂²×C₄	1	2	yes	—
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₄	C₁₆	1	2	yes	—
83	C₄×D₄	C₁₆	1	1	yes	characteristic
84	C₄×D₄	C₁₆	1	1	yes	characteristic
85	C₄×D₄	C₂×C₈	1	2	yes	—
86	C₄×D₄	C₂×C₈	1	1	yes	characteristic
87	C₄×D₄	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₁₆	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	C₁₆	1	2	yes	—
90	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
91	C₂×C₁₆	—	2	4	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	2	4	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₁₆	C₂²×C₄	1	2	yes	—
102	C₂×C₁₆	C₂²×C₄	1	2	yes	—
103	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₁₆	C₂×D₄	1	2	yes	—
105	C₂×C₁₆	#41	1	2	yes	—
106	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₈	C₂×C₈	1	2	yes	—
108	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
109	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
110	C₂²⋊C₈	C₂×C₈	1	2	yes	—
111	C₂²⋊C₈	C₂×C₈	1	2	yes	—
112	C₂²⋊C₈	C₂×C₈	1	2	yes	—
113	C₂²⋊C₈	C₂×C₈	1	2	yes	—
114	C₄×C₈	C₂×C₈	1	1	yes	characteristic
115	C₄×C₈	C₂×C₈	1	1	yes	characteristic
116	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₃₂	—	2	4	no	—
119	C₃₂	—	2	2	no	—
120	C₃₂	C₂×D₄	1	4	yes	complemented
121	C₂⁴	C₂×C₁₆	1	2	yes	—
122	C₂×D₄	—	2	4	no	—
123	C₂×D₄	C₂×C₁₆	1	2	yes	—
124	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
125	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₁₆	1	2	yes	—
130	C₂²×C₄	C₂×C₁₆	1	2	yes	—
131	C₂²×C₄	C₂×C₁₆	1	2	yes	—
132	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
133	C₂²×C₄	C₂²×C₈	1	2	yes	—
134	C₂²×C₄	C₂²×C₈	1	2	yes	—
135	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₄×D₄	1	2	yes	—
145	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₈	C₂²×C₈	1	2	yes	—
148	C₂×C₈	C₂²×C₈	1	2	yes	—
149	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₈	#131	1	2	yes	—
151	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
152	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
153	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
155	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
156	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
157	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
158	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
159	C₄²	C₂×C₁₆	1	1	yes	characteristic
160	C₄²	C₂×C₁₆	1	1	yes	characteristic
161	C₄²	C₂²×C₈	1	1	yes	characteristic
162	C₄²	C₂²×C₈	1	1	yes	characteristic
163	C₁₆	—	2	2	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	—	2	1	no	—
167	C₁₆	C₄×D₄	1	2	yes	—
168	C₁₆	C₂²×D₄	1	1	yes	characteristic
169	C₁₆	C₂×#41	1	1	yes	characteristic
170	C₂³	—	2	4	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	4	2	no	—
173	C₂³	C₂²×C₁₆	1	2	yes	—
174	C₂³	C₂²×C₁₆	1	1	yes	characteristic
175	C₂³	C₂²×C₁₆	1	2	yes	—
176	D₄	—	2	4	no	—
177	D₄	—	2	2	no	—
178	D₄	M₆₄	1	4	yes	direct product factor
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	C₈×D₄	1	2	yes	—
189	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
192	C₂×C₄	C₂²×C₁₆	1	2	yes	—
193	C₂×C₄	C₂²×C₁₆	1	2	yes	—
194	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
196	C₂×C₄	#504	1	2	yes	—
197	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	C₈×D₄	1	2	yes	—
203	C₈	C₂×C₄×D₄	1	1	yes	characteristic
204	C₈	C₂×#131	1	1	yes	characteristic
205	C₂²	—	2	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	4	8	no	—
213	C₂²	D₄×C₁₆	1	2	yes	—
214	C₂²	C₂×M₆₄	1	4	yes	complemented
215	C₂²	C₂³×C₁₆	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	D₄×C₁₆	1	2	yes	—
222	C₄	C₂×M₆₄	1	2	yes	complemented
223	C₄	C₂×C₈×D₄	1	1	yes	characteristic
224	C₄	C₂×#504	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	2	4	no	—
227	C₂	—	4	2	no	—
228	C₂	C₂×D₄×C₁₆	1	1	yes	characteristic
229	C₂	C₂²×M₆₄	1	1	yes	characteristic
230	C₂	#33974	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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
1. C₂²×M₆₄	C₂	semidirect
3. #13646	C₂	semidirect
4. #13645	C₂	semidirect
5. D₄×C₃₂	C₂	semidirect
6. #13627	C₂	semidirect
7. #13553	C₂	semidirect
8. C₄×M₆₄	C₂	semidirect
15. C₂×M₆₄	C₂²	semidirect
16. C₂²×C₃₂	C₂²	semidirect
26. C₄⋊C₃₂	C₂²	semidirect
27. C₄⋊C₃₂	C₂²	semidirect
28. C₂²⋊C₃₂	C₂²	semidirect
29. C₂²⋊C₃₂	C₂²	semidirect
30. #1397	C₂²	semidirect
31. C₄×C₃₂	C₂²	semidirect
55. M₆₄	D₄	direct (N × H)
120. C₃₂	C₂×D₄	semidirect
178. D₄	M₆₄	direct (N × H)
214. C₂²	C₂×M₆₄	semidirect
222. C₄	C₂×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₆₄	2
C₂×D₄×C₁₆	2
#13646	2
#13645	2
D₄×C₃₂	2
#13627	2
#13553	2
C₄×M₆₄	2

Subgroups — D4×M64 #512753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×M₆₄ #512753