Subgroups — #140067

Order 512.

249 conjugacy classes of subgroups.

The subgroup lattice has 227 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13159	C₂	1	1	yes	maximal, characteristic
2	#12662	C₂	1	1	yes	maximal, characteristic
3	C₈×#107	C₂	1	1	yes	maximal, characteristic
4	#7167	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	1	yes	maximal, characteristic
6	#7151	C₂	1	1	yes	maximal, characteristic
7	#7136	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#346	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×#107	C₄	1	1	yes	characteristic
15	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×#335	C₄	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1374	—	2	1	no	—
20	#1374	—	2	1	no	—
21	#1279	C₄	1	1	yes	characteristic
22	#1279	C₄	1	1	yes	characteristic
23	C₂²×C₁₆	—	2	1	no	—
24	C₂²×C₁₆	—	2	1	no	—
25	C₂²×C₁₆	—	2	1	no	—
26	C₂²×C₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₁₆	D₄	1	1	yes	characteristic
28	C₂×#107	C₈	1	1	yes	characteristic
29	C₂×#107	C₈	1	1	yes	characteristic
30	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	—	2	1	no	—
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₁₆	—	2	1	no	—
40	C₄⋊C₁₆	—	2	1	no	—
41	C₄⋊C₁₆	—	2	1	no	—
42	C₄⋊C₁₆	—	4	2	no	—
43	C₄⋊C₁₆	Q₈	1	2	yes	—
44	#346	—	4	2	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	#340	C₈	1	1	yes	characteristic
50	#340	C₈	1	1	yes	characteristic
51	#336	—	2	1	no	—
52	#336	—	2	1	no	—
53	#335	—	2	2	no	—
54	C₈²	—	2	1	no	—
55	C₈²	—	2	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	#41	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄²	—	2	1	no	—
68	C₂×C₄²	—	2	1	no	—
69	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	4	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	—	4	1	no	—
83	C₂×C₁₆	—	4	1	no	—
84	C₂×C₁₆	D₈	1	2	yes	—
85	#107	—	2	2	no	—
86	#107	—	2	2	no	—
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	M₁₆	1	2	yes	—
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	—	4	1	no	—
96	C₄×C₈	—	4	1	no	—
97	C₄×C₈	—	4	1	no	—
98	C₄×C₈	—	4	1	no	—
99	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	#95	—	2	1	no	—
104	#95	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
111	C₂²×C₄	#131	1	1	yes	characteristic
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	M₃₂	1	2	yes	—
129	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₈	#123	1	1	yes	characteristic
131	C₂×C₈	#125	1	1	yes	characteristic
132	C₂×C₈	#126	1	1	yes	characteristic
133	C₂×C₈	C₂×D₈	1	1	yes	characteristic
134	C₂×C₈	#135	1	1	yes	characteristic
135	C₂×C₈	#137	1	1	yes	characteristic
136	C₂×C₈	#137	1	1	yes	characteristic
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	M₃₂	1	2	yes	—
145	C₄²	—	4	1	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	1	no	—
149	C₄²	#117	1	1	yes	characteristic
150	C₄²	C₄×Q₈	1	1	yes	characteristic
151	C₄²	C₂×M₁₆	1	1	yes	characteristic
152	C₄²	#131	1	1	yes	characteristic
153	C₁₆	—	2	2	no	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	4	2	no	—
156	C₁₆	—	4	1	no	—
157	C₁₆	—	8	1	no	—
158	C₂³	C₈×D₄	1	1	yes	characteristic
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	#387	1	1	yes	characteristic
174	C₂×C₄	#431	1	1	yes	characteristic
175	C₂×C₄	#433	1	1	yes	characteristic
176	C₂×C₄	#435	1	1	yes	characteristic
177	C₂×C₄	C₄×D₈	1	1	yes	characteristic
178	C₂×C₄	#441	1	1	yes	characteristic
179	C₂×C₄	#443	1	1	yes	characteristic
180	C₂×C₄	#444	1	1	yes	characteristic
181	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
182	C₂×C₄	#446	1	1	yes	characteristic
183	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
184	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
185	C₂×C₄	#504	1	1	yes	characteristic
186	C₂×C₄	#504	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	—	4	2	no	—
192	C₈	—	4	1	no	—
193	C₈	—	8	1	no	—
194	C₈	#474	1	1	yes	characteristic
195	C₈	#479	1	1	yes	characteristic
196	C₈	#480	1	1	yes	characteristic
197	C₈	#485	1	1	yes	characteristic
198	C₂²	C₈×D₈	1	1	yes	characteristic
199	C₂²	#1579	1	1	yes	characteristic
200	C₂²	#1916	1	1	yes	characteristic
201	C₂²	#2168	1	1	yes	characteristic
202	C₂²	#2169	1	1	yes	characteristic
203	C₂²	#2170	1	1	yes	characteristic
204	C₂²	#2171	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	2	no	—
208	C₄	—	8	1	no	—
209	C₄	—	8	1	no	—
210	C₄	—	8	1	no	—
211	C₄	#1581	1	1	yes	characteristic
212	C₄	#1587	1	1	yes	characteristic
213	C₄	#1918	1	1	yes	characteristic
214	C₄	#1922	1	1	yes	characteristic
215	C₄	#2163	1	1	yes	characteristic
216	C₄	#2164	1	1	yes	characteristic
217	C₄	#2183	1	1	yes	characteristic
218	C₄	#2183	1	1	yes	characteristic
219	C₂	#9533	1	1	yes	characteristic
220	C₂	#12038	1	1	yes	characteristic
221	C₂	#12039	1	1	yes	characteristic
222	C₂	#12060	1	1	yes	characteristic
223	C₂	#12061	1	1	yes	characteristic
224	C₂	#13105	1	1	yes	characteristic
225	C₂	#13106	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
#13159	2
#12662	2
C₈×#107	2
#7167	2
#7151	2
#7136	2