Subgroups — #512798

Order 512.

394 conjugacy classes of subgroups.

The subgroup lattice has 223 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#13646	C₂	1	2	yes	maximal, complemented
3	#13645	C₂	1	4	yes	maximal, complemented
4	D₄×C₃₂	C₂	1	2	yes	maximal, complemented
5	#13640	C₂	1	1	yes	maximal, characteristic, complemented
6	#13639	C₂	1	1	yes	maximal, characteristic, complemented
7	#13553	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₃₂	C₂	1	2	yes	maximal, complemented
9	C₂³×C₁₆	C₂²	1	2	yes	—
10	C₂×C₈×D₄	C₄	1	1	yes	characteristic
11	C₂×M₆₄	C₂²	1	4	yes	complemented
12	C₂²×C₃₂	C₂²	1	4	yes	complemented
13	D₄×C₁₆	C₄	1	2	yes	—
14	D₄×C₁₆	C₄	1	1	yes	characteristic
15	D₄×C₁₆	C₄	1	1	yes	characteristic
16	D₄×C₁₆	C₂²	1	1	yes	characteristic
17	D₄×C₁₆	C₂²	1	2	yes	—
18	D₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₄⋊C₃₂	C₂²	1	2	yes	complemented
23	C₄⋊C₃₂	C₂²	1	2	yes	complemented
24	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
25	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
26	C₂²⋊C₃₂	C₂²	1	2	yes	—
27	C₂²⋊C₃₂	C₂²	1	4	yes	complemented
28	C₂²⋊C₃₂	C₂²	1	4	yes	complemented
29	#1397	C₂²	1	2	yes	complemented
30	C₄×C₃₂	C₂²	1	2	yes	complemented
31	C₂³×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₄	C₈	1	1	yes	characteristic
33	C₂²×C₁₆	—	4	4	no	—
34	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₁₆	C₂×C₄	1	2	yes	—
37	C₂²×C₁₆	C₂×C₄	1	2	yes	—
38	C₂²×C₁₆	C₂³	1	1	yes	characteristic
39	C₂²×C₁₆	C₂³	1	2	yes	—
40	C₂²×C₁₆	C₂³	1	2	yes	—
41	C₈×D₄	C₈	1	2	yes	—
42	C₈×D₄	C₈	1	1	yes	characteristic
43	C₈×D₄	C₈	1	1	yes	characteristic
44	C₈×D₄	C₂×C₄	1	1	yes	characteristic
45	C₈×D₄	C₂×C₄	1	2	yes	—
46	C₈×D₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₈	C₈	1	2	yes	—
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	M₆₄	—	4	4	no	—
51	C₂×C₃₂	—	4	4	no	—
52	C₂×C₃₂	C₂³	1	2	yes	—
53	C₂×C₃₂	C₂³	1	2	yes	—
54	C₂×C₃₂	C₂³	1	4	yes	—
55	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
57	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
59	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
60	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
61	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
62	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
64	C₄×C₁₆	C₂³	1	1	yes	characteristic
65	C₄×C₁₆	C₂³	1	1	yes	characteristic
66	C₂²×D₄	C₁₆	1	1	yes	characteristic
67	C₂³×C₄	C₂×C₈	1	2	yes	—
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	—	4	2	no	—
70	C₂²×C₈	C₂×C₈	1	2	yes	—
71	C₂²×C₈	C₂×C₈	1	2	yes	—
72	C₂²×C₈	C₂×C₈	1	2	yes	—
73	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₄	1	2	yes	—
76	C₂²×C₈	C₂²×C₄	1	2	yes	—
77	C₄×D₄	C₁₆	1	2	yes	—
78	C₄×D₄	C₁₆	1	1	yes	characteristic
79	C₄×D₄	C₁₆	1	1	yes	characteristic
80	C₄×D₄	C₂×C₈	1	2	yes	—
81	C₄×D₄	C₂×C₈	1	1	yes	characteristic
82	C₄×D₄	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₁₆	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	C₁₆	1	2	yes	—
85	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
86	C₂×C₁₆	—	2	4	no	—
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	4	2	no	—
90	C₂×C₁₆	—	4	2	no	—
91	C₂×C₁₆	—	4	2	no	—
92	C₂×C₁₆	—	4	1	no	—
93	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₁₆	C₂²×C₄	1	2	yes	—
97	C₂×C₁₆	C₂²×C₄	1	2	yes	—
98	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₈	C₂×C₈	1	2	yes	—
100	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
102	C₂²⋊C₈	C₂×C₈	1	2	yes	—
103	C₂²⋊C₈	C₂×C₈	1	2	yes	—
104	C₂²⋊C₈	C₂×C₈	1	2	yes	—
105	C₂²⋊C₈	C₂×C₈	1	2	yes	—
106	C₄×C₈	C₂×C₈	1	1	yes	characteristic
107	C₄×C₈	C₂×C₈	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₃₂	—	2	2	no	—
111	C₃₂	—	2	4	no	—
112	C₃₂	—	2	2	no	—
113	C₂⁴	C₂×C₁₆	1	2	yes	—
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	C₂×C₁₆	1	2	yes	—
116	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
117	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₁₆	1	2	yes	—
123	C₂²×C₄	C₂×C₁₆	1	2	yes	—
124	C₂²×C₄	C₂×C₁₆	1	2	yes	—
125	C₂²×C₄	C₂²×C₈	1	2	yes	—
126	C₂²×C₄	C₂²×C₈	1	2	yes	—
127	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₈	C₂²×C₈	1	2	yes	—
142	C₂×C₈	C₂²×C₈	1	2	yes	—
143	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
144	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
145	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
147	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
148	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
149	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
150	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
151	C₄²	C₂×C₁₆	1	1	yes	characteristic
152	C₄²	C₂×C₁₆	1	1	yes	characteristic
153	C₄²	C₂²×C₈	1	1	yes	characteristic
154	C₄²	C₂²×C₈	1	1	yes	characteristic
155	C₁₆	—	2	1	no	—
156	C₁₆	—	2	2	no	—
157	C₁₆	—	2	2	no	—
158	C₁₆	—	2	1	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	H(5,𝔽₂)	1	2	yes	—
161	C₂³	—	2	4	no	—
162	C₂³	—	2	4	no	—
163	C₂³	—	4	2	no	—
164	C₂³	C₂²×C₁₆	1	2	yes	—
165	C₂³	C₂²×C₁₆	1	2	yes	—
166	C₂³	C₂²×C₁₆	1	1	yes	characteristic
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	D₄	M₆₄	1	4	yes	complemented
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₁₆	1	2	yes	—
186	C₂×C₄	C₂²×C₁₆	1	2	yes	—
187	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
188	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	#518	1	1	yes	characteristic
195	C₈	#568	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	C₂×M₆₄	1	4	yes	—
205	C₂²	#2258	1	2	yes	—
206	C₂²	C₂³×C₁₆	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	C₂×M₆₄	1	2	yes	—
214	C₄	#2972	1	1	yes	characteristic
215	C₄	#3407	1	1	yes	characteristic
216	C₂	—	2	2	no	—
217	C₂	—	2	4	no	—
218	C₂	—	4	2	no	—
219	C₂	#33486	1	1	yes	characteristic
220	C₂	C₂²×M₆₄	1	1	yes	characteristic
221	C₂	C₂×#2258	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
2. #13646	C₂	semidirect
3. #13645	C₂	semidirect
4. D₄×C₃₂	C₂	semidirect
5. #13640	C₂	semidirect
6. #13639	C₂	semidirect
7. #13553	C₂	semidirect
8. C₂×C₂²⋊C₃₂	C₂	semidirect
11. C₂×M₆₄	C₂²	semidirect
12. C₂²×C₃₂	C₂²	semidirect
22. C₄⋊C₃₂	C₂²	semidirect
23. C₄⋊C₃₂	C₂²	semidirect
27. C₂²⋊C₃₂	C₂²	semidirect
28. C₂²⋊C₃₂	C₂²	semidirect
29. #1397	C₂²	semidirect
30. C₄×C₃₂	C₂²	semidirect
169. D₄	M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₁₆	2
#13646	2
#13645	2
D₄×C₃₂	2
#13640	2
#13639	2
#13553	2
C₂×C₂²⋊C₃₂	2