Finite Group Databasebeta

← #150211

Subgroups — #150211

Order 512.

404 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 74 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#23631C211yesmaximal, characteristic
2#12581C211yesmaximal, characteristic
3#11431C211yesmaximal, characteristic
4#7412C214yesmaximal, complemented
5C2×#417—21no—
6C2×#410—21no—
7#2297C2211yescharacteristic
8C2×#382C411yescharacteristic, complemented
9#2074—21no—
10#2074C412yescomplemented
11#2073—21no—
12#2073C2212yes—
13#1736—21no—
14#1728C411yescharacteristic, complemented
15#1458—21no—
16#1393—21no—
17#1393C2212yes—
18#1319—21no—
19#1318C2212yes—
20C22×#41D411yescharacteristic
21C22×M16—21no—
22C2×#117—41no—
23C2×#117D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C42C2×C411yescharacteristic
26#431—41no—
27#417—42no—
28#417—42no—
29C2×#103—22no—
30C2×#102—22no—
31#410—24no—
32#410—42no—
33C2×#99—22no—
34C2×#99—41no—
35#391C2×C411yescharacteristic
36#391D411yescharacteristic
37#390D411yescharacteristic
38#390C2311yescharacteristic
39#386—41no—
40#382—21no—
41#382—41no—
42#382C2×C412yes—
43C4×C22⋊C4—41no—
44C2×#95—41no—
45#340—22no—
46#340—22no—
47#336—21no—
48#336C2×C412yes—
49#324—22no—
50#323—41no—
51C2×#41—22no—
52C2×#41—42no—
53C2×#41—41no—
54C2×#41—41no—
55C2×#41D812yes—
56C2×#41SD1612yes—
57C22×Q8C22⋊C411yescharacteristic
58C22×D4—21no—
59C22×D4C2×D411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—41no—
63C23×C4C22⋊C411yescharacteristic
64C2×M16—42no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—42no—
68C22×C8—22no—
69#117—41no—
70#117—41no—
71#117—44no—
72#117—82no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—81no—
78C2×C4⋊C4C22⋊C411yescharacteristic
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—42no—
87C2×C42—81no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22×C411yescharacteristic
90C4⋊C8—81no—
91#103—44no—
92#102—42no—
93#102—42no—
94#99—24no—
95#99—44no—
96#99—42no—
97C22⋊C8—42no—
98C4×C8—81no—
99#95—41no—
100#95—41no—
101#95—42no—
102#95—42no—
103C24—21no—
104C24#9911yescharacteristic
105#41—42no—
106#41—42no—
107#41—42no—
108#41—42no—
109#41—82no—
110#41—82no—
111C2×Q8—21no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×Q8#10212yes—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—21no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—81no—
121C2×D4—81no—
122C2×D4C2×D811yescharacteristic
123C2×D4C2×SD1611yescharacteristic
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4—82no—
144C22×C4—82no—
145C22×C4#9911yescharacteristic
146C22×C4#10212yes—
147C22×C4#10412yes—
148C22×C4C2×C22⋊C411yescharacteristic
149M16—82no—
150M16—82no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—82no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—84no—
160C4⋊C4—82no—
161C22⋊C4—44no—
162C22⋊C4—81no—
163C22⋊C4—82no—
164C22⋊C4—82no—
165C22⋊C4—82no—
166C42—41no—
167C42—41no—
168C42—42no—
169C42—41no—
170C42—41no—
171C42—84no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—81no—
181C23—81no—
182C23#32712yes—
183C23C2×#9911yescharacteristic
184Q8—42no—
185Q8—81no—
186D4—41no—
187D4—41no—
188D4—81no—
189D4—82no—
190D4—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—84no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4—164no—
220C2×C4—162no—
221C2×C4—162no—
222C2×C4#32712yes—
223C2×C4#35212yes—
224C2×C4#35411yescharacteristic
225C2×C4#35411yescharacteristic
226C2×C4C2×#10211yescharacteristic
227C2×C4C2×#10411yescharacteristic
228C8—82no—
229C8—162no—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—42no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—161no—
244C22C2×#32711yescharacteristic
245C22C2×#35211yescharacteristic
246C22C2×#35411yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4—81no—
254C4—162no—
255C4—164no—
256C4#134412yes—
257C4#134412yes—
258C2—21no—
259C2—21no—
260C2—42no—
261C2—81no—
262C2—81no—
263C2C2×#134411yescharacteristic
264C2#1359611yescharacteristic
265C2#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. #7412C2semidirect
8. C2×#382C4semidirect
10. #2074C4semidirect
14. #1728C4semidirect

Maximal subgroups

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

subgroupindex
#236312
#125812
#114312
#74122