Finite Group Databasebeta

← #150772

Subgroups — #150772

Order 512.

314 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 54 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#29114C211yesmaximal, characteristic
2#12857C211yesmaximal, characteristic
3#11431C211yesmaximal, characteristic
4#7428C212yesmaximal, complemented
5#7412C212yesmaximal, complemented
6#2851C411yescharacteristic, complemented
7#2297C2211yescharacteristic
8#2074C411yescharacteristic, complemented
9#2073C2211yescharacteristic
10#2026C411yescharacteristic, complemented
11#2025C2211yescharacteristic
12#1936—21no—
13#1935—21no—
14#1825—21no—
15#1788—21no—
16#1782—21no—
17#1749C411yescharacteristic, complemented
18#1458—21no—
19#1393—21no—
20#1393C2212yes—
21#1319—21no—
22#1318C2212yes—
23C22×#41D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×#117D411yescharacteristic
26C22×C42C2×C411yescharacteristic
27#431—41no—
28C2×#103—21no—
29C2×#103—21no—
30C2×#102—21no—
31C2×#102—21no—
32C2×#99—22no—
33C2×#99—41no—
34C2×C22⋊C8—21no—
35C2×#97—21no—
36#391C2×C411yescharacteristic
37#391D411yescharacteristic
38#390D411yescharacteristic
39#390C2311yescharacteristic
40#389—21no—
41#387—41no—
42#386—41no—
43#385—41no—
44#382C2×C411yescharacteristic
45#381—41no—
46#380—21no—
47C4×C22⋊C4—41no—
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336C2×C411yescharacteristic
53#336C2×C411yescharacteristic
54#324—22no—
55#323—41no—
56C2×#41—22no—
57C2×#41—41no—
58C2×#41—41no—
59C2×#41—41no—
60C2×#41—81no—
61C22×Q8C22⋊C411yescharacteristic
62C22×D4—21no—
63C22×D4C2×D411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C22×C8—21no—
68C22×C8—21no—
69#125—81no—
70#117—41no—
71#117—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C22⋊C4—42no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—81no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22×C411yescharacteristic
88C4⋊C8—81no—
89#103—42no—
90#103—81no—
91#102—42no—
92#102—81no—
93#99—42no—
94#99—42no—
95#99—82no—
96C22⋊C8—41no—
97C22⋊C8—41no—
98C22⋊C8—42no—
99#97—81no—
100C4×C8—81no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—81no—
107#95—81no—
108C24—21no—
109C24#9911yescharacteristic
110#41—82no—
111#41—82no—
112#41—82no—
113#41—82no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×D4—21no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—81no—
124C2×D4#13512yes—
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—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4—82no—
145C22×C4#9911yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—82no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C22⋊C4—81no—
163C22⋊C4—82no—
164C22⋊C4—82no—
165C22⋊C4—82no—
166C22⋊C4—82no—
167C22⋊C4—82no—
168C42—41no—
169C42—41no—
170C42—41no—
171C42—41no—
172C42—81no—
173C23—21no—
174C23—21no—
175C23—22no—
176C23—21no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—81no—
182C23—81no—
183C23C2×#9911yescharacteristic
184Q8—42no—
185Q8—81no—
186D4—42no—
187D4—81no—
188D4—84no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
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—84no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—162no—
220C2×C4—162no—
221C2×C4—162no—
222C2×C4#35411yescharacteristic
223C2×C4#35411yescharacteristic
224C2×C4#41611yescharacteristic
225C2×C4#42111yescharacteristic
226C8—81no—
227C8—81no—
228C8—162no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—81no—
236C22—81no—
237C22—82no—
238C22—81no—
239C22—81no—
240C22—161no—
241C22#151611yescharacteristic
242C22#212111yescharacteristic
243C22C2×#35411yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4—161no—
254C4—161no—
255C4—322no—
256C2—21no—
257C2—21no—
258C2—82no—
259C2—81no—
260C2#1184511yescharacteristic
261C2#1359411yescharacteristic
262C2#1360411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7428C2semidirect
5. #7412C2semidirect
6. #2851C4semidirect
8. #2074C4semidirect
10. #2026C4semidirect
17. #1749C4semidirect

Maximal subgroups

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

subgroupindex
#291142
#128572
#114312
#74282
#74122