Subgroups — #512792

Order 512.

327 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33480	C₂	1	1	yes	maximal, characteristic
2	#13660	C₂	1	2	yes	maximal, complemented
3	#13646	C₂	1	2	yes	maximal, complemented
4	#13645	C₂	1	2	yes	maximal, complemented
5	#13645	C₂	1	2	yes	maximal, complemented
6	#13641	C₂	1	2	yes	maximal, complemented
7	#13640	C₂	1	1	yes	maximal, characteristic, complemented
8	#13549	C₂	1	2	yes	maximal, complemented
9	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#2966	C₄	1	1	yes	characteristic
11	C₂×M₆₄	C₂²	1	2	yes	—
12	C₂×M₆₄	C₂²	1	2	yes	—
13	C₂²×C₃₂	C₂²	1	2	yes	—
14	#2183	C₄	1	1	yes	characteristic
15	Q₈×C₁₆	C₂²	1	1	yes	characteristic
16	#2169	C₄	1	1	yes	characteristic
17	#2168	C₄	1	2	yes	—
18	D₄×C₁₆	C₂²	1	1	yes	characteristic
19	D₄×C₁₆	C₂²	1	2	yes	—
20	#2164	C₄	1	2	yes	—
21	#2163	C₄	1	1	yes	characteristic
22	#2108	C₂²	1	2	yes	—
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	C₄⋊C₃₂	C₂²	1	2	yes	—
25	C₄⋊C₃₂	C₂²	1	2	yes	—
26	C₄⋊C₃₂	C₂²	1	2	yes	—
27	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
28	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
29	C₂²⋊C₃₂	C₂²	1	2	yes	—
30	C₂²⋊C₃₂	C₂²	1	2	yes	—
31	#1397	C₂²	1	2	yes	—
32	#1397	C₂²	1	2	yes	—
33	#1397	C₂²	1	2	yes	—
34	C₄×C₃₂	C₂²	1	1	yes	characteristic
35	C₄×C₃₂	C₂²	1	2	yes	—
36	C₄×C₃₂	C₂²	1	1	yes	characteristic
37	#529	C₈	1	1	yes	characteristic
38	C₂×M₃₂	C₂×C₄	1	2	yes	—
39	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₁₆	C₂³	1	1	yes	characteristic
41	C₂²×C₁₆	C₂³	1	2	yes	—
42	#446	C₈	1	1	yes	characteristic
43	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
44	#436	C₈	1	1	yes	characteristic
45	#435	C₈	1	2	yes	—
46	C₈×D₄	C₂×C₄	1	2	yes	—
47	C₈×D₄	C₂×C₄	1	1	yes	characteristic
48	#433	C₈	1	2	yes	—
49	#432	C₈	1	1	yes	characteristic
50	#405	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	M₆₄	—	2	4	no	—
53	M₆₄	—	2	4	no	—
54	C₂×C₃₂	—	2	4	no	—
55	C₂×C₃₂	C₂³	1	2	yes	—
56	C₂×C₃₂	C₂³	1	2	yes	—
57	C₂×C₃₂	C₂³	1	2	yes	—
58	C₂×C₃₂	C₂³	1	2	yes	—
59	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
62	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
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	#346	C₂×C₄	1	2	yes	—
67	#346	C₂×C₄	1	2	yes	—
68	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
69	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
70	C₄×C₁₆	C₂³	1	1	yes	characteristic
71	C₄×C₁₆	C₂³	1	1	yes	characteristic
72	C₄×C₁₆	C₂³	1	2	yes	—
73	C₂×M₁₆	C₂×C₈	1	2	yes	—
74	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₄	1	2	yes	—
77	#126	C₁₆	1	2	yes	—
78	#125	C₁₆	1	1	yes	characteristic
79	#124	C₁₆	1	1	yes	characteristic
80	#123	C₁₆	1	2	yes	—
81	C₂²⋊Q₈	C₁₆	1	1	yes	characteristic
82	#121	C₁₆	1	1	yes	characteristic
83	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
84	C₄×D₄	C₂×C₈	1	2	yes	—
85	C₄×D₄	C₂×C₈	1	1	yes	characteristic
86	#117	C₂×C₈	1	2	yes	—
87	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
88	M₃₂	—	2	4	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	2	2	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	2	yes	—
96	C₂×C₁₆	C₂²×C₄	1	2	yes	—
97	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
100	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₈	C₂×C₈	1	2	yes	—
102	C₄⋊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	#97	C₂×C₈	1	2	yes	—
107	#97	C₂×C₈	1	2	yes	—
108	C₄×C₈	C₂×C₈	1	1	yes	characteristic
109	C₄×C₈	C₂×C₈	1	1	yes	characteristic
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₄	1	2	yes	—
113	C₃₂	—	2	2	no	—
114	C₃₂	—	2	2	no	—
115	C₃₂	—	2	2	no	—
116	C₃₂	#41	1	4	yes	—
117	C₂×Q₈	C₂×C₁₆	1	1	yes	characteristic
118	C₂×D₄	C₂×C₁₆	1	2	yes	—
119	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
122	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₈	1	2	yes	—
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₈	C₂²×C₈	1	2	yes	—
133	C₂×C₈	C₂²×C₈	1	2	yes	—
134	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
136	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
137	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
138	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
139	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
140	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
141	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
142	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
143	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
144	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
145	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
146	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
147	C₄²	C₂×C₁₆	1	1	yes	characteristic
148	C₄²	C₂×C₁₆	1	1	yes	characteristic
149	C₄²	C₂²×C₈	1	1	yes	characteristic
150	C₄²	C₂²×C₈	1	1	yes	characteristic
151	C₄²	C₂²×C₈	1	2	yes	—
152	C₁₆	—	2	2	no	—
153	C₁₆	—	2	1	no	—
154	C₁₆	—	2	1	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	#131	1	2	yes	—
157	C₁₆	C₂×#41	1	2	yes	—
158	C₂³	C₂²×C₁₆	1	2	yes	—
159	C₂³	C₂²×C₁₆	1	1	yes	characteristic
160	Q₈	—	4	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	4	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
171	C₂×C₄	C₂²×C₁₆	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	C₂²×C₁₆	1	2	yes	—
175	C₂×C₄	C₂²×C₁₆	1	2	yes	—
176	C₂×C₄	C₂²×C₁₆	1	2	yes	—
177	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	2	no	—
182	C₈	#504	1	2	yes	—
183	C₈	C₄×#41	1	1	yes	characteristic
184	C₈	C₂×#131	1	1	yes	characteristic
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	C₂³×C₁₆	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	C₈×#41	1	1	yes	characteristic
201	C₄	C₂×#504	1	1	yes	characteristic
202	C₂	—	4	1	no	—
203	C₂	—	4	2	no	—
204	C₂	C₁₆×#41	1	1	yes	characteristic
205	C₂	#33974	1	1	yes	characteristic
206	C₂	#33974	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	#121	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. #13660	C₂	semidirect
3. #13646	C₂	semidirect
4. #13645	C₂	semidirect
5. #13645	C₂	semidirect
6. #13641	C₂	semidirect
7. #13640	C₂	semidirect
8. #13549	C₂	semidirect
9. C₄×M₆₄	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33480	2
#13660	2
#13646	2
#13645	2
#13641	2
#13640	2
#13549	2
C₄×M₆₄	2