Finite Group Databasebeta

← #150776

Subgroups — #150776

Order 512.

311 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 56 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#29112C211yesmaximal, characteristic
2#12860C211yesmaximal, characteristic
3C2×#1319C211yesmaximal, characteristic
4#7429C212yesmaximal, complemented
5#7413C212yesmaximal, complemented
6#2851C411yescharacteristic
7C2×#391C2211yescharacteristic
8#2074C411yescharacteristic
9#2074C2211yescharacteristic
10#2026C411yescharacteristic
11#2026C2211yescharacteristic
12#1935—21no—
13#1935—21no—
14#1824—21no—
15#1787—21no—
16#1784—21no—
17#1749C411yescharacteristic
18C2×#324—21no—
19#1394—21no—
20#1394C2212yes—
21#1319—21no—
22#1319C2212yes—
23C23×Q8D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C22×C42C2×C411yescharacteristic
27C2×#103—21no—
28C2×#103—21no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#101—22no—
32C2×#101—41no—
33#407—41no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—41no—
36C2×#97—21no—
37#391—41no—
38#391C2×C411yescharacteristic
39#391D411yescharacteristic
40#391D411yescharacteristic
41#391C2311yescharacteristic
42#389—21no—
43#387—41no—
44#385—41no—
45#382C2×C411yescharacteristic
46#380—21no—
47C2×#95—41no—
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336C2×C411yescharacteristic
53#336C2×C411yescharacteristic
54#324—22no—
55#324—41no—
56C22×Q8—21no—
57C22×Q8—22no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8—41no—
61C22×Q8—81no—
62C22×Q8C22⋊C411yescharacteristic
63C22×Q8C2×D411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C2×M16—42no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—81no—
72#125—81no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4C22⋊C411yescharacteristic
81C2×C4⋊C4C2×D411yescharacteristic
82C2×C22⋊C4—41no—
83C2×C42—21no—
84C2×C42—81no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22×C411yescharacteristic
87#103—42no—
88#103—42no—
89#103—81no—
90#103—81no—
91#101—42no—
92#101—42no—
93#101—82no—
94C22⋊C8—41no—
95C22⋊C8—41no—
96C22⋊C8—42no—
97C22⋊C8—81no—
98C22⋊C8—81no—
99C22⋊C8—81no—
100#97—81no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—41no—
108C24#9911yescharacteristic
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8—81no—
119C2×Q8—82no—
120C2×Q8—82no—
121C2×Q8—82no—
122C2×Q8—82no—
123C2×Q8#13512yes—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4—81no—
146C22×C4—82no—
147C22×C4#9911yescharacteristic
148C22×C4C2×C22⋊C411yescharacteristic
149M16—82no—
150M16—82no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—162no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—81no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C22⋊C4—82no—
170C42—41no—
171C42—41no—
172C42—81no—
173C23—21no—
174C23—22no—
175C23—21no—
176C23—41no—
177C23#35411yescharacteristic
178C23#35411yescharacteristic
179C23C2×#9911yescharacteristic
180Q8—42no—
181Q8—42no—
182Q8—81no—
183Q8—81no—
184Q8—84no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
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—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—84no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—161no—
221C2×C4#41611yescharacteristic
222C2×C4#42111yescharacteristic
223C8—81no—
224C8—81no—
225C8—162no—
226C8—162no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—42no—
235C22—42no—
236C22#141312yes—
237C22#151611yescharacteristic
238C22#212111yescharacteristic
239C22C2×#35411yescharacteristic
240C4—21no—
241C4—21no—
242C4—42no—
243C4—41no—
244C4—81no—
245C4—81no—
246C4—81no—
247C4—81no—
248C4—81no—
249C4—161no—
250C4—161no—
251C2—21no—
252C2—21no—
253C2—22no—
254C2—41no—
255C2#1184511yescharacteristic
256C2C2×#141311yescharacteristic
257C2#1360311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7429C2semidirect
5. #7413C2semidirect

Maximal subgroups

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

subgroupindex
#291122
#128602
C2×#13192
#74292
#74132