Finite Group Databasebeta

← #124690

Subgroups — #124690

Order 512.

402 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 82 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#21330C211yesmaximal, characteristic
2#8730C211yesmaximal, characteristic
3#7868C211yesmaximal, characteristic
4#7134C214yesmaximal, complemented
5#3458C411yescharacteristic
6#2920—21no—
7#2905C411yescharacteristic
8C4×#117C2211yescharacteristic
9#1995—21no—
10#1995C412yes—
11#1994—21no—
12#1994C2212yes—
13#1733—21no—
14C2×#339—21no—
15#1458—21no—
16#1276—21no—
17#1276C2212yes—
18#1276C2212yes—
19#1274—21no—
20C22×M16—21no—
21#544—24no—
22#544—41no—
23#543—41no—
24#533—21no—
25#533—42no—
26#533C2×C412yes—
27C2×#117—21no—
28C2×#117C2×C411yescharacteristic
29C2×#117D411yescharacteristic
30C22×C42D411yescharacteristic
31#432—22no—
32#432—41no—
33#431—22no—
34#431—22no—
35#431—41no—
36#424—42no—
37#423—22no—
38#423—21no—
39#423C2×C412yes—
40#407—41no—
41C4×M16—41no—
42C4×C4⋊C4—41no—
43C4×C22⋊C4—41no—
44#376C2×C411yescharacteristic
45#376D411yescharacteristic
46C43D411yescharacteristic
47C43C2311yescharacteristic
48#339—22no—
49#339—22no—
50#339—22no—
51#339—22no—
52#324—41no—
53#323—41no—
54C23×C4—21no—
55C23×C4C22⋊C411yescharacteristic
56C2×M16—22no—
57C2×M16—41no—
58C2×M16—42no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C22×C8—41no—
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66C4⋊Q8—44no—
67#125—22no—
68#125—44no—
69#125M1614yescomplemented
70C22⋊Q8—44no—
71C22⋊Q8—44no—
72C22⋊Q8—42no—
73C22⋊Q8—42no—
74C4×Q8—42no—
75C4×Q8—42no—
76#117—22no—
77#117—22no—
78#117—22no—
79#117—22no—
80#117—22no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42—41no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42SD1611yescharacteristic
100C2×C42SD1611yescharacteristic
101C2×C42Q1611yescharacteristic
102C2×C42Q1611yescharacteristic
103C2×C42C22×C411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C4⋊C8—24no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#97—42no—
115C4×C8—42no—
116C4×C8—42no—
117#95—41no—
118#95—41no—
119C24#9911yescharacteristic
120C2×Q8—42no—
121C2×Q8—42no—
122C22×C4—21no—
123C22×C4—21no—
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—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—82no—
138C22×C4#9911yescharacteristic
139C22×C4#10311yescharacteristic
140C22×C4#10311yescharacteristic
141C22×C4#10411yescharacteristic
142C22×C4#10411yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144M16—42no—
145M16—42no—
146M16—81no—
147M16—81no—
148M16—84no—
149C2×C8—44no—
150C2×C8—42no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—82no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—42no—
178C42—42no—
179C42—82no—
180C42—82no—
181C42#10012yes—
182C42#10311yescharacteristic
183C42#10311yescharacteristic
184C42C2×M1612yes—
185C42C2×SD1611yescharacteristic
186C42C2×Q1611yescharacteristic
187C42#13712yes—
188C23—21no—
189C23—21no—
190C23—81no—
191C23#32811yescharacteristic
192C23#32811yescharacteristic
193C23C2×#9911yescharacteristic
194Q8—162no—
195Q8—162no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
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—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—162no—
226C2×C4#32811yescharacteristic
227C2×C4#32811yescharacteristic
228C2×C4#40711yescharacteristic
229C2×C4C2×#10011yescharacteristic
230C2×C4C2×#10311yescharacteristic
231C2×C4#41911yescharacteristic
232C2×C4C2×#10411yescharacteristic
233C2×C4#42111yescharacteristic
234C8—84no—
235C8—82no—
236C8—162no—
237C22—21no—
238C22—21no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—81no—
244C22#146211yescharacteristic
245C22C2×#32811yescharacteristic
246C22#151111yescharacteristic
247C4—22no—
248C4—21no—
249C4—21no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—42no—
255C4—42no—
256C4—81no—
257C4—84no—
258C4#148711yescharacteristic
259C4#149111yescharacteristic
260C4#152711yescharacteristic
261C4#154811yescharacteristic
262C2—21no—
263C2—21no—
264C2—21no—
265C2—41no—
266C2—41no—
267C2#760911yescharacteristic
subgroupG/H
centerC4#1548
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC2#7609
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7134C2semidirect
69. #125M16semidirect

Maximal subgroups

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

subgroupindex
#213302
#87302
#78682
#71342