Finite Group Databasebeta

← #141900

Subgroups — #141900

Order 512.

296 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 114 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#21673C211yesmaximal, characteristic
2#12898C211yesmaximal, characteristic
3#12591C211yesmaximal, characteristic
4#7170C212yesmaximal
5#7170C212yesmaximal
6C4×#125C411yescharacteristic
7C2×C4⋊C16—21no—
8C2×C4⋊C16—21no—
9C2×C4⋊C16C2212yes—
10C2×#346C2212yes—
11#1939C411yescharacteristic
12#1939C411yescharacteristic
13#1916—21no—
14#1916—21no—
15#1831C411yescharacteristic
16#1771—21no—
17#1771—21no—
18#1770C2211yescharacteristic
19#1770C2211yescharacteristic
20C42×C8C2211yescharacteristic
21#1374—21no—
22#1374—21no—
23C2×#125C811yescharacteristic
24C22×C16—22no—
25C22×C16—22no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×#97—21no—
31C2×#97—21no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#389C811yescharacteristic
39#389C811yescharacteristic
40#387—21no—
41#387—21no—
42#382C811yescharacteristic
43C4×C4⋊C4—21no—
44C4×C4⋊C4—21no—
45C4×C4⋊C4C2×C411yescharacteristic
46C4×C4⋊C4C2×C411yescharacteristic
47C43C2×C411yescharacteristic
48C4⋊C16—24no—
49C4⋊C16—42no—
50C4⋊C16—42no—
51#346—22no—
52#346Q814yes—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22⋊C411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C22×C8C2×D411yescharacteristic
65#125—42no—
66C2×C4⋊C4—21no—
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×C4⋊C4C2×C811yescharacteristic
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22⋊C411yescharacteristic
78C2×C42C2×C811yescharacteristic
79C2×C42C2×C811yescharacteristic
80C2×C42M1611yescharacteristic
81C2×C42M1611yescharacteristic
82C2×C42C22×C411yescharacteristic
83C2×C16—24no—
84C2×C16—42no—
85C2×C16—42no—
86C2×C16—42no—
87C2×C16—42no—
88C2×C16—42no—
89C4⋊C8—22no—
90C4⋊C8—22no—
91C4⋊C8—42no—
92C4⋊C8—42no—
93#97—41no—
94#97—41no—
95#97—41no—
96#97—41no—
97C4×C8—21no—
98C4×C8—21no—
99C4×C8—21no—
100C4×C8—21no—
101C4×C8—22no—
102C4×C8—41no—
103C4×C8—41no—
104C4×C8C2×Q812yes—
105C4×C8#4112yes—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
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—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8C22⋊Q812yes—
138C2×C8#12411yescharacteristic
139C2×C8C4⋊Q811yescharacteristic
140C2×C8AGL(1,ℤ/8)11yescharacteristic
141C2×C8AGL(1,ℤ/8)11yescharacteristic
142C2×C8#13711yescharacteristic
143C2×C8#13711yescharacteristic
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—41no—
159C42—41no—
160C42—41no—
161C42—41no—
162C42—41no—
163C42—41no—
164C42C4×Q812yes—
165C42#13112yes—
166C16—44no—
167C16—82no—
168C23C2×C22⋊C811yescharacteristic
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
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—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4#34912yes—
196C2×C4#35012yes—
197C2×C4#39111yescharacteristic
198C2×C4#40811yescharacteristic
199C2×C4#41711yescharacteristic
200C2×C4#41711yescharacteristic
201C2×C4#42111yescharacteristic
202C2×C4#42111yescharacteristic
203C2×C4C8×Q812yes—
204C2×C4#44612yes—
205C8—22no—
206C8—41no—
207C8—41no—
208C8—81no—
209C8—81no—
210C8#49011yescharacteristic
211C8#49111yescharacteristic
212C8#50111yescharacteristic
213C8#50111yescharacteristic
214C22#147711yescharacteristic
215C22#147711yescharacteristic
216C22#198411yescharacteristic
217C22C2×#34911yescharacteristic
218C22#211511yescharacteristic
219C22C2×#35011yescharacteristic
220C22#211711yescharacteristic
221C4—22no—
222C4—22no—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—81no—
228C4—81no—
229C4—81no—
230C4—81no—
231C4#149511yescharacteristic
232C4#149511yescharacteristic
233C4#199211yescharacteristic
234C4#199511yescharacteristic
235C2#987211yescharacteristic
236C2#1162511yescharacteristic
237C2#1162611yescharacteristic
238C2#1162711yescharacteristic
239C2#1162811yescharacteristic
240C2#1326511yescharacteristic
241C2#1326811yescharacteristic
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 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#216732
#128982
#125912
#71702