Finite Group Databasebeta

← #125372

Subgroups — #125372

Order 512.

351 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 76 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#33053C211yesmaximal, characteristic
2#12605C211yesmaximal, characteristic
3C2×#1375C211yesmaximal, characteristic
4#7130C212yesmaximal, complemented
5#7117C212yesmaximal, complemented
6C22×M32—21no—
7C2×#491C411yescharacteristic
8C22×#97C2211yescharacteristic
9#2857C411yescharacteristic
10#2851C411yescharacteristic
11#2112—21no—
12C2×#347—21no—
13#1992C2211yescharacteristic
14#1960C2211yescharacteristic
15#1935—21no—
16#1935—21no—
17#1892—21no—
18#1892—21no—
19#1836C411yescharacteristic
20#1375—21no—
21#1375C2212yes—
22#1375C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×#125C2×C411yescharacteristic
27C2×C22⋊Q8—21no—
28C22×C42C2×C411yescharacteristic
29C2×M32—22no—
30C2×M32—22no—
31C2×M32—22no—
32C2×M32—42no—
33C2×M32—42no—
34C22×C16—22no—
35#491—42no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#97—42no—
41C2×#97C2×C411yescharacteristic
42C2×#97D411yescharacteristic
43C2×#97D411yescharacteristic
44C2×#97C2311yescharacteristic
45#391—21no—
46#389—21no—
47#389—41no—
48#387—41no—
49#385—41no—
50#384C2×C411yescharacteristic
51#382C2×C411yescharacteristic
52#380—21no—
53#380—41no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56#347—42no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61C22×Q8—21no—
62C23×C4—21no—
63C23×C4C22⋊C411yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—22no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—22no—
72C22×C8—41no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—41no—
79#125—22no—
80#125—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42—41no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C22×C411yescharacteristic
100M32—42no—
101M32—44no—
102M32—42no—
103M32—42no—
104M32—42no—
105C2×C16—24no—
106C2×C16—24no—
107C2×C16—44no—
108#103—42no—
109#103—42no—
110#103—42no—
111#103—41no—
112#103—41no—
113#97—22no—
114#97—22no—
115#97—41no—
116#97—41no—
117#95—41no—
118#95—42no—
119#95—41no—
120#95—41no—
121C24#10011yescharacteristic
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—81no—
141C22×C4#10011yescharacteristic
142C22×C4#10411yescharacteristic
143C22×C4#10411yescharacteristic
144C22×C4#10411yescharacteristic
145C22×C4#10411yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8#9912yes—
163C2×C8#13512yes—
164C2×C8AGL(1,ℤ/8)11yescharacteristic
165C2×C8#13711yescharacteristic
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C22⋊C4—82no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—42no—
185C42#13112yes—
186C16—44no—
187C16—44no—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—41no—
192C23#32911yescharacteristic
193C23#32911yescharacteristic
194C23C2×#10011yescharacteristic
195Q8—81no—
196Q8—81no—
197Q8—81no—
198Q8—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—84no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4#32911yescharacteristic
228C2×C4#32911yescharacteristic
229C2×C4#40811yescharacteristic
230C2×C4C2×#9911yescharacteristic
231C2×C4#41611yescharacteristic
232C2×C4#41711yescharacteristic
233C2×C4C2×#10411yescharacteristic
234C2×C4C2×#10411yescharacteristic
235C8—22no—
236C8—21no—
237C8—22no—
238C8—21no—
239C8—41no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22#135612yes—
250C22#147111yescharacteristic
251C22C2×#32911yescharacteristic
252C22#155111yescharacteristic
253C4—21no—
254C4—21no—
255C4—42no—
256C4—81no—
257C4—161no—
258C4—161no—
259C4—161no—
260C4—161no—
261C4#148311yescharacteristic
262C4#149411yescharacteristic
263C4#150111yescharacteristic
264C4#150411yescharacteristic
265C2—21no—
266C2—21no—
267C2—22no—
268C2—41no—
269C2#769011yescharacteristic
270C2C2×#135611yescharacteristic
271C2#1191111yescharacteristic
subgroupG/H
center, socleC22C2×#329
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7130C2semidirect
5. #7117C2semidirect

Maximal subgroups

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

subgroupindex
#330532
#126052
C2×#13752
#71302
#71172