Finite Group Databasebeta

← #124562

Subgroups — #124562

Order 512.

298 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21339C211yesmaximal, characteristic
2#12894C211yesmaximal, characteristic
3#12879C211yesmaximal, characteristic
4#7128C212yesmaximal
5#7118C212yesmaximal
6C42×Q8C411yescharacteristic
7C2×C4⋊C16—21no—
8#1984C411yescharacteristic
9#1984C2211yescharacteristic
10#1939C411yescharacteristic
11#1939C2211yescharacteristic
12#1916—21no—
13#1916—21no—
14C4×C4⋊C8—21no—
15C4×C4⋊C8C411yescharacteristic
16#1744—21no—
17C42×C8C2211yescharacteristic
18#1374—21no—
19#1374—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C2×C4×Q8C811yescharacteristic
24C2×C4×Q8C811yescharacteristic
25C2×C4×Q8C2×C411yescharacteristic
26C22×C16—22no—
27C22×C16—22no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4×C8—21no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8C2311yescharacteristic
39C4×C4⋊C4—21no—
40C4×C4⋊C4—21no—
41C4×C4⋊C4—21no—
42C4×C4⋊C4C811yescharacteristic
43C4×C4⋊C4C811yescharacteristic
44C4×C4⋊C4C2×C411yescharacteristic
45C43—21no—
46C43C2×C411yescharacteristic
47C4⋊C16—42no—
48#336—21no—
49#336—21no—
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55C22×Q8C2×C811yescharacteristic
56C22×C8—21no—
57C22×C8—21no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C2×D411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—42no—
65C4×Q8—41no—
66C4×Q8—41no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2×C811yescharacteristic
71C2×C4⋊C4C2×C811yescharacteristic
72C2×C4⋊C4C2×C811yescharacteristic
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C2×C811yescharacteristic
82C2×C42C2×C811yescharacteristic
83C2×C42M1611yescharacteristic
84C2×C42M1611yescharacteristic
85C2×C42C22×C411yescharacteristic
86C2×C16—42no—
87C2×C16—42no—
88C2×C16—42no—
89C2×C16—42no—
90C2×C16—42no—
91C2×C16—42no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4×C8—21no—
98C4×C8—21no—
99C4×C8—22no—
100C4×C8—41no—
101C4×C8—41no—
102C4×C8—41no—
103C4×C8—41no—
104C4×C8—42no—
105#95—21no—
106#95—21no—
107#95—21no—
108C2×Q8—21no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8M3212yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4C22⋊C811yescharacteristic
117C22×C4C22⋊C811yescharacteristic
118C22×C4C22⋊C811yescharacteristic
119C22×C4C22⋊C811yescharacteristic
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C22×C811yescharacteristic
122C22×C4C2×M1611yescharacteristic
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8AGL(1,ℤ/8)11yescharacteristic
140C2×C8#13711yescharacteristic
141C2×C8#13712yes—
142C4⋊C4—24no—
143C4⋊C4—21no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4M3212yes—
149C42—24no—
150C42—22no—
151C42—22no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—42no—
162C42—42no—
163C42—42no—
164C42—41no—
165C42—41no—
166C42—41no—
167C42—41no—
168C16—82no—
169C16—82no—
170C23C2×C22⋊C811yescharacteristic
171Q8—42no—
172Q8—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4#34912yes—
200C2×C4#35012yes—
201C2×C4#35012yes—
202C2×C4#41011yescharacteristic
203C2×C4#41311yescharacteristic
204C2×C4#41711yescharacteristic
205C2×C4#41911yescharacteristic
206C2×C4#42111yescharacteristic
207C2×C4#42111yescharacteristic
208C2×C4C2×M3211yescharacteristic
209C2×C4C2×M3211yescharacteristic
210C8—21no—
211C8—21no—
212C8—22no—
213C8—41no—
214C8—41no—
215C8—81no—
216C8—81no—
217C22#145811yescharacteristic
218C22#147711yescharacteristic
219C22#147911yescharacteristic
220C22#211211yescharacteristic
221C22C2×#34911yescharacteristic
222C22C2×#35011yescharacteristic
223C22C2×#35011yescharacteristic
224C4—22no—
225C4—22no—
226C4—22no—
227C4—21no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—42no—
232C4—41no—
233C4—81no—
234C4—81no—
235C4#151211yescharacteristic
236C4#151511yescharacteristic
237C4#153511yescharacteristic
238C4#155911yescharacteristic
239C2#759611yescharacteristic
240C2#1146011yescharacteristic
241C2#1146211yescharacteristic
242C2#1162311yescharacteristic
243C2#1162611yescharacteristic
244C2#1163911yescharacteristic
245C2#1164211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socle, prime radicalC23C2×C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#213392
#128942
#128792
#71282
#71182