Finite Group Databasebeta

← #150690

Subgroups — #150690

Order 512.

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

normal subgroups only

This group has 325 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#23819C211yesmaximal, characteristic
2#12797C211yesmaximal, characteristic
3#11529C211yesmaximal, characteristic
4#7428C212yesmaximal, complemented
5#7412C212yesmaximal, complemented
6C4×#127C2211yescharacteristic
7C4×#125C411yescharacteristic, complemented
8C2×#353—21no—
9#2100—21no—
10#2073C411yescharacteristic, complemented
11#2073C2211yescharacteristic
12#2041—21no—
13#2025C411yescharacteristic, complemented
14#2025C2211yescharacteristic
15#1799—21no—
16C4×#102—21no—
17C4×#102—21no—
18#1744C411yescharacteristic, complemented
19#1465—21no—
20#1393C2212yes—
21#1325—21no—
22#1318C2212yes—
23C2×#127D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×C4×D4—41no—
26C2×C4×D4D411yescharacteristic
27C2×C4×D4D411yescharacteristic
28C2×#102—21no—
29C2×#102—21no—
30C2×#102—21no—
31C2×#102—21no—
32C2×#99—22no—
33#407—41no—
34C2×C4×C8—21no—
35#390—41no—
36#390C2×C411yescharacteristic
37#390C2311yescharacteristic
38#389—21no—
39#387—41no—
40#384D411yescharacteristic
41#382C2×C411yescharacteristic
42#380—41no—
43C4×C4⋊C4—21no—
44C4×C4⋊C4—41no—
45C43C2×C411yescharacteristic
46#353—42no—
47#342—41no—
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#323—41no—
57C22×D4—41no—
58C22×D4C22⋊C411yescharacteristic
59C22×D4C2×D411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C2×M16—81no—
63C22×C8—21no—
64C22×C8—21no—
65#127—21no—
66#127—21no—
67#127—41no—
68#125—42no—
69C4×D4—41no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—41no—
73C4×D4—84no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—81no—
80C2×C4⋊C4C22⋊C411yescharacteristic
81C2×C4⋊C4C2×D411yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—41no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22×C411yescharacteristic
93#102—42no—
94#102—42no—
95#102—42no—
96#102—41no—
97#102—41no—
98#99—42no—
99#99—42no—
100C22⋊C8—42no—
101C22⋊C8—81no—
102C4×C8—41no—
103C4×C8—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—81no—
108C24—21no—
109C24—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—82no—
120C2×D4—82no—
121C2×D4AGL(1,ℤ/8)11yescharacteristic
122C2×D4#13711yescharacteristic
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—82no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4#9911yescharacteristic
137C22×C4#9911yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139M16—162no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—82no—
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—162no—
158C22⋊C4—81no—
159C22⋊C4—81no—
160C22⋊C4—82no—
161C22⋊C4—82no—
162C22⋊C4—82no—
163C22⋊C4—82no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—41no—
169C42—42no—
170C42—42no—
171C42—82no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—81no—
180C23—81no—
181C23—81no—
182C23—81no—
183C23C2×#9911yescharacteristic
184D4—41no—
185D4—41no—
186D4—42no—
187D4—81no—
188D4—81no—
189D4—84no—
190D4—84no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—161no—
215C2×C4—161no—
216C2×C4—162no—
217C2×C4—162no—
218C2×C4—162no—
219C2×C4#35411yescharacteristic
220C2×C4#35411yescharacteristic
221C2×C4#41711yescharacteristic
222C2×C4#42111yescharacteristic
223C8—81no—
224C8—81no—
225C8—162no—
226C22—21no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—81no—
231C22—81no—
232C22—82no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—161no—
239C22—161no—
240C22#151311yescharacteristic
241C22#212011yescharacteristic
242C22C2×#35411yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—81no—
249C4—82no—
250C4—81no—
251C4—81no—
252C4—161no—
253C4—322no—
254C4#141212yes—
255C2—21no—
256C2—21no—
257C2—81no—
258C2—81no—
259C2—161no—
260C2#1182311yescharacteristic
261C2C2×#141211yescharacteristic
262C2#1360211yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ 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
7. C4×#125C4semidirect
10. #2073C4semidirect
13. #2025C4semidirect
18. #1744C4semidirect

Maximal subgroups

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

subgroupindex
#238192
#127972
#115292
#74282
#74122