Finite Group Databasebeta

← #128199

Subgroups — #128199

Order 512.

640 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 138 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#17365C212yesmaximal
2#13925C211yesmaximal, characteristic
3#7326C214yesmaximal
4C22×C4×Q8C2211yescharacteristic
5#2920—22no—
6C2×#382C411yescharacteristic
7C2×C4×C4⋊C4C411yescharacteristic
8#1467—22no—
9#1467C414yescomplemented
10#1463—22no—
11#1463C2214yes—
12#1461—22no—
13#1461C414yescomplemented
14#1446C412yes—
15#1445C2212yes—
16#1436—22no—
17C4⋊#95—22no—
18C23×Q8C2×C411yescharacteristic
19C2×C4×Q8—42no—
20C2×C4×Q8—41no—
21C2×C4×Q8—41no—
22C2×C4×Q8C2×C412yes—
23C2×C4×Q8C2×C412yes—
24C2×C4×Q8D412yes—
25C2×C4×Q8Q812yes—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4C2311yescharacteristic
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42D412yes—
33#432—24no—
34#432—42no—
35#431—24no—
36#431—42no—
37#423—42no—
38#407—24no—
39C4×M16—42no—
40#382—41no—
41#382—41no—
42C4×C4⋊C4—42no—
43C2×#95—24no—
44C2×#95—21no—
45C2×#95—24no—
46C2×#95C2×C412yes—
47C2×#95C2×C412yes—
48#324—24no—
49#324—24no—
50#324—24no—
51#324C2×C418yes—
52C22×Q8—41no—
53C22×Q8—41no—
54C22×Q8—44no—
55C22×Q8C4211yescharacteristic
56C22×Q8C4211yescharacteristic
57C22×Q8C4⋊C412yes—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D412yes—
65C2×M16—44no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×Q8—42no—
69C4×Q8—42no—
70C4×Q8—42no—
71C4×Q8—42no—
72C4×Q8—42no—
73C4×Q8—42no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4C22⋊C412yes—
85C2×C4⋊C4C22⋊C412yes—
86C2×C4⋊C4C4⋊C412yes—
87C2×C4⋊C4C4⋊C412yes—
88C2×C4⋊C4C22×C412yes—
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C4⋊C4C2×Q811yescharacteristic
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—42no—
99C2×C42—82no—
100C2×C42C4212yes—
101C2×C42C22⋊C412yes—
102C2×C42C22⋊C412yes—
103C2×C42C4⋊C412yes—
104C2×C42C4⋊C412yes—
105C2×C42C4⋊C412yes—
106C4⋊C8—44no—
107C4⋊C8—44no—
108C22⋊C8—28no—
109C22⋊C8—44no—
110#97—44no—
111C4×C8—44no—
112#95—42no—
113#95—42no—
114#95—48no—
115#95—42no—
116#95—84no—
117C24C2×C22⋊C411yescharacteristic
118C2×Q8—24no—
119C2×Q8—24no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—48no—
126C2×Q8—48no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4#9512yes—
155C22×C4#9512yes—
156C22×C4#9512yes—
157C22×C4#9512yes—
158C22×C4#9911yescharacteristic
159C22×C4#9912yes—
160C22×C4#9911yescharacteristic
161C22×C4C2×C4211yescharacteristic
162C22×C4C2×C22⋊C412yes—
163C22×C4C2×C4⋊C411yescharacteristic
164C22×C4C2×C4⋊C412yes—
165M16—84no—
166M16—84no—
167C2×C8—48no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—48no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C42—22no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—84no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23C2×#9511yescharacteristic
197C23C2×#9911yescharacteristic
198C23C2×#9911yescharacteristic
199Q8—44no—
200Q8—48no—
201Q8—44no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—48no—
232C2×C4—82no—
233C2×C4—84no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4#34212yes—
242C2×C4#34212yes—
243C2×C4#34414yes—
244C2×C4#35512yes—
245C2×C4#35612yes—
246C8—88no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22C2×#34211yescharacteristic
258C22C2×#34412yes—
259C22#212811yescharacteristic
260C22#212911yescharacteristic
261C22C2×#35511yescharacteristic
262C22C2×#35611yescharacteristic
263C4—22no—
264C4—24no—
265C4—42no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—84no—
272C4—162no—
273C4#139414yes—
274C2—21no—
275C2—21no—
276C2—21no—
277C2—21no—
278C2—22no—
279C2#811511yescharacteristic
280C2#1258311yescharacteristic
281C2C2×#139411yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1467C4semidirect
13. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#173652
#139252
#73262