Subgroups — #129071

Order 512.

320 conjugacy classes of subgroups.

The subgroup lattice has 217 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#12590	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	C₂×#1311	C₂	1	1	yes	maximal, characteristic
5	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#7136	C₂	1	1	yes	maximal, characteristic
7	#7136	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1374	C₄	1	2	yes	—
19	#1374	C₄	1	2	yes	—
20	#1374	C₄	1	2	yes	—
21	#1312	—	2	2	no	—
22	#1311	—	2	2	no	—
23	#1273	C₄	1	1	yes	characteristic
24	#1273	C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₈	C₈	1	2	yes	—
30	C₂×C₄⋊C₈	C₈	1	2	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	C₄³	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₁₆	—	2	4	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	—	2	1	no	—
45	C₄×C₁₆	—	2	1	no	—
46	C₄×C₁₆	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	D₄	1	2	yes	—
50	C₄×C₁₆	Q₈	1	2	yes	—
51	#336	C₈	1	2	yes	—
52	#336	C₈	1	2	yes	—
53	C₈²	—	2	1	no	—
54	C₈²	—	2	1	no	—
55	C₈²	—	2	1	no	—
56	C₈²	—	2	1	no	—
57	C₂²×C₈	C₄²	1	1	yes	characteristic
58	C₂²×C₈	C₄²	1	1	yes	characteristic
59	C₂²×C₈	C₂×C₈	1	2	yes	—
60	C₂²×C₈	C₂×C₈	1	2	yes	—
61	C₂²×C₈	C₂×C₈	1	2	yes	—
62	C₂²×C₈	C₂×C₈	1	2	yes	—
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄²	C₄²	1	1	yes	characteristic
66	C₂×C₄²	C₄²	1	1	yes	characteristic
67	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
83	C₂×C₁₆	M₁₆	1	4	yes	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	2	4	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
105	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
106	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
107	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
109	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	M₃₂	1	4	yes	—
129	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₈	#117	1	1	yes	characteristic
131	C₂×C₈	C₄×D₄	1	1	yes	characteristic
132	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
133	C₂×C₈	C₂×M₁₆	1	2	yes	—
134	C₂×C₈	#131	1	2	yes	—
135	C₂×C₈	#131	1	2	yes	—
136	C₂×C₈	#131	1	2	yes	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	M₃₂	1	4	yes	—
145	C₄²	C₄×D₄	1	1	yes	characteristic
146	C₄²	C₄×Q₈	1	1	yes	characteristic
147	C₄²	#131	1	1	yes	characteristic
148	C₄²	#131	1	1	yes	characteristic
149	C₁₆	—	2	4	no	—
150	C₁₆	—	2	4	no	—
151	C₁₆	—	4	2	no	—
152	C₁₆	—	4	2	no	—
153	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	#346	1	2	yes	—
165	C₂×C₄	#346	1	2	yes	—
166	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
168	C₂×C₄	#405	1	1	yes	characteristic
169	C₂×C₄	#405	1	1	yes	characteristic
170	C₂×C₄	#405	1	1	yes	characteristic
171	C₂×C₄	#405	1	1	yes	characteristic
172	C₂×C₄	C₈×D₄	1	1	yes	characteristic
173	C₂×C₄	C₈×D₄	1	1	yes	characteristic
174	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
175	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
176	C₂×C₄	C₂×M₃₂	1	2	yes	—
177	C₂×C₄	C₂×M₃₂	1	2	yes	—
178	C₂×C₄	#504	1	2	yes	—
179	C₂×C₄	#504	1	2	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	2	no	—
183	C₈	—	4	2	no	—
184	C₈	—	4	1	no	—
185	C₈	—	4	1	no	—
186	C₈	#424	1	2	yes	—
187	C₈	#433	1	2	yes	—
188	C₈	#435	1	2	yes	—
189	C₈	#446	1	2	yes	—
190	C₂²	C₈×M₁₆	1	1	yes	characteristic
191	C₂²	#1450	1	1	yes	characteristic
192	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
193	C₂²	C₂×#346	1	1	yes	characteristic
194	C₂²	C₄×M₃₂	1	1	yes	characteristic
195	C₂²	#2108	1	1	yes	characteristic
196	C₂²	#2108	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	4	1	no	—
201	C₄	#1453	1	1	yes	characteristic
202	C₄	#1454	1	1	yes	characteristic
203	C₄	#1768	1	1	yes	characteristic
204	C₄	#1771	1	1	yes	characteristic
205	C₄	#2168	1	2	yes	—
206	C₄	#2169	1	2	yes	—
207	C₄	#2183	1	2	yes	—
208	C₄	#2183	1	2	yes	—
209	C₂	#8264	1	1	yes	characteristic
210	C₂	#11408	1	1	yes	characteristic
211	C₂	#11409	1	1	yes	characteristic
212	C₂	#11411	1	1	yes	characteristic
213	C₂	#11412	1	1	yes	characteristic
214	C₂	#12662	1	1	yes	characteristic
215	C₂	#12663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator, socle, prime radical	C₂³	C₂×C₄×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₁₆	2
#12590	2
C₂×#1312	2
C₂×#1311	2
C₈×C₄⋊C₈	2
#7136	2