Subgroups — #512751

Order 512.

419 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33974	C₂	1	2	yes	maximal, complemented
2	C₂×#2170	C₂	1	1	yes	maximal, characteristic
3	#13648	C₂	1	4	yes	maximal, complemented
4	#13647	C₂	1	4	yes	maximal, complemented
5	C₂×#1422	C₂	1	1	yes	maximal, characteristic, complemented
6	#13558	C₂	1	2	yes	maximal, complemented
7	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#504	C₂²	1	2	yes	—
9	C₂×#443	C₄	1	1	yes	characteristic
10	#2258	—	2	4	no	—
11	#2258	C₂²	1	8	yes	complemented
12	C₂×M₆₄	—	2	4	no	—
13	C₂×M₆₄	C₂²	1	2	yes	—
14	C₂×M₆₄	C₂²	1	1	yes	characteristic
15	C₂×M₆₄	C₂²	1	1	yes	characteristic
16	#2170	C₄	1	2	yes	—
17	#2170	C₄	1	2	yes	—
18	#2170	C₂²	1	2	yes	—
19	#2170	C₂²	1	2	yes	—
20	C₂×#364	C₄	1	1	yes	characteristic
21	C₂×#350	C₄	1	2	yes	—
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1422	C₂²	1	2	yes	complemented
24	#1422	C₂²	1	2	yes	complemented
25	#1401	C₂²	1	8	yes	complemented
26	#1397	C₂²	1	2	yes	complemented
27	C₄×C₃₂	C₂²	1	2	yes	complemented
28	C₂×#135	C₈	1	1	yes	characteristic
29	C₂×#131	C₂×C₄	1	2	yes	—
30	#504	—	2	2	no	—
31	#504	—	2	2	no	—
32	#504	C₂×C₄	1	4	yes	—
33	#504	C₂³	1	4	yes	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	C₂×C₄	1	2	yes	—
36	C₂²×C₁₆	—	2	2	no	—
37	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₁₆	C₂³	1	1	yes	characteristic
39	#443	C₈	1	2	yes	—
40	#443	C₈	1	2	yes	—
41	#443	C₂×C₄	1	2	yes	—
42	#443	C₂×C₄	1	2	yes	—
43	C₂×#108	C₈	1	1	yes	characteristic
44	C₂×#104	C₈	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	M₆₄	—	2	2	no	—
47	M₆₄	—	2	1	no	—
48	M₆₄	—	2	1	no	—
49	M₆₄	—	2	8	no	—
50	M₆₄	D₄	1	2	yes	—
51	M₆₄	D₄	1	2	yes	—
52	M₆₄	C₂³	1	4	yes	—
53	C₂×C₃₂	—	2	2	no	—
54	C₂×C₃₂	—	2	8	no	—
55	C₂×C₃₂	C₂³	1	2	yes	—
56	C₂×C₃₂	C₂³	1	2	yes	—
57	#364	C₂×C₄	1	2	yes	—
58	#364	C₂×C₄	1	2	yes	—
59	#350	C₂×C₄	1	4	yes	—
60	#350	C₂×C₄	1	4	yes	—
61	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₆	C₂³	1	1	yes	characteristic
64	C₄×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×#41	C₂×C₈	1	2	yes	—
66	#135	C₁₆	1	2	yes	—
67	#135	C₁₆	1	2	yes	—
68	#135	C₂×C₈	1	2	yes	—
69	#135	C₂×C₈	1	2	yes	—
70	C₂×Q₁₆	C₁₆	1	1	yes	characteristic
71	C₂×SD₁₆	C₁₆	1	2	yes	—
72	C₂×D₈	C₁₆	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	#131	C₂×C₈	1	4	yes	—
76	#131	C₂²×C₄	1	4	yes	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	C₂×C₈	1	2	yes	—
79	C₂²×C₈	—	2	2	no	—
80	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
83	M₃₂	—	2	2	no	—
84	M₃₂	—	2	4	no	—
85	M₃₂	—	2	4	no	—
86	M₃₂	C₂²×C₄	1	4	yes	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	—	2	4	no	—
92	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
96	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
97	C₂×C₁₆	#41	1	1	yes	characteristic
98	C₂×C₁₆	#41	1	1	yes	characteristic
99	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
100	#108	C₂×C₈	1	2	yes	—
101	#108	C₂×C₈	1	2	yes	—
102	#104	C₂×C₈	1	4	yes	—
103	#104	C₂×C₈	1	4	yes	—
104	C₄×C₈	C₂×C₈	1	1	yes	characteristic
105	C₄×C₈	C₂×C₈	1	1	yes	characteristic
106	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₃₂	—	2	4	no	—
109	C₃₂	—	2	2	no	—
110	C₃₂	C₂×D₄	1	4	yes	—
111	#41	—	2	2	no	—
112	#41	—	2	2	no	—
113	#41	C₂×C₁₆	1	4	yes	—
114	#41	C₂²×C₈	1	4	yes	—
115	C₂×Q₈	C₂×C₁₆	1	2	yes	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	C₂×C₁₆	1	2	yes	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
121	Q₁₆	C₂×C₁₆	1	2	yes	—
122	Q₁₆	C₂×C₁₆	1	2	yes	—
123	SD₁₆	C₂×C₁₆	1	4	yes	—
124	SD₁₆	C₂×C₁₆	1	4	yes	—
125	D₈	C₂×C₁₆	1	2	yes	—
126	D₈	C₂×C₁₆	1	2	yes	—
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	4	no	—
129	M₁₆	—	2	4	no	—
130	M₁₆	C₂²×C₈	1	4	yes	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
135	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
136	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
137	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
138	C₂×C₈	C₄×D₄	1	1	yes	characteristic
139	C₂×C₈	C₄×D₄	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₈	#131	1	1	yes	characteristic
144	C₂×C₈	#131	1	1	yes	characteristic
145	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
146	C₄²	—	2	1	no	—
147	C₄²	C₂²×C₈	1	1	yes	characteristic
148	C₄²	C₂²×C₈	1	1	yes	characteristic
149	C₁₆	—	2	4	no	—
150	C₁₆	—	2	1	no	—
151	C₁₆	—	2	1	no	—
152	C₁₆	C₄×D₄	1	1	yes	characteristic
153	C₁₆	C₄×D₄	1	1	yes	characteristic
154	C₁₆	C₂²×D₄	1	1	yes	characteristic
155	C₁₆	C₂×#41	1	1	yes	characteristic
156	C₂³	—	2	2	no	—
157	C₂³	#504	1	1	yes	characteristic
158	Q₈	—	2	2	no	—
159	Q₈	C₂²×C₁₆	1	4	yes	—
160	D₄	—	2	4	no	—
161	D₄	—	2	4	no	—
162	D₄	—	2	2	no	—
163	D₄	C₂²×C₁₆	1	4	yes	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	C₈×D₄	1	1	yes	characteristic
171	C₂×C₄	C₈×D₄	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
174	C₂×C₄	#504	1	1	yes	characteristic
175	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
176	C₈	—	2	4	no	—
177	C₈	C₈×D₄	1	1	yes	characteristic
178	C₈	C₈×D₄	1	1	yes	characteristic
179	C₈	C₂²×C₁₆	1	1	yes	characteristic
180	C₈	C₂²×C₁₆	1	1	yes	characteristic
181	C₈	C₂²×C₁₆	1	1	yes	characteristic
182	C₈	C₂²×C₁₆	1	1	yes	characteristic
183	C₈	C₂×C₄×D₄	1	1	yes	characteristic
184	C₈	C₂×#131	1	1	yes	characteristic
185	C₂²	—	2	4	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	8	2	no	—
188	C₂²	D₄×C₁₆	1	1	yes	characteristic
189	C₂²	D₄×C₁₆	1	1	yes	characteristic
190	C₂²	C₂×#504	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	—	4	1	no	—
194	C₄	D₄×C₁₆	1	1	yes	characteristic
195	C₄	D₄×C₁₆	1	1	yes	characteristic
196	C₄	C₂×C₈×D₄	1	1	yes	characteristic
197	C₄	C₂³×C₁₆	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	4	no	—
202	C₂	C₂×D₄×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²×D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×D₄×C₁₆
prime radical	C₂×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄ ◁ 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. #33974	C₂	semidirect
3. #13648	C₂	semidirect
4. #13647	C₂	semidirect
5. C₂×#1422	C₂	semidirect
6. #13558	C₂	semidirect
7. C₄×M₆₄	C₂	semidirect
11. #2258	C₂²	semidirect
23. #1422	C₂²	semidirect
24. #1422	C₂²	semidirect
25. #1401	C₂²	semidirect
26. #1397	C₂²	semidirect
27. C₄×C₃₂	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#33974	2
C₂×#2170	2
#13648	2
#13647	2
C₂×#1422	2
#13558	2
C₄×M₆₄	2