Finite Group Databasebeta

← #128189

Subgroups — #128189

Order 512.

732 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 130 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#17357C212yesmaximal
2#13958C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C22×#117C2211yescharacteristic
5#2920—22no—
6C2×#381C411yescharacteristic
7C22×#95C411yescharacteristic
8#1464—22no—
9#1464C414yescomplemented
10#1463C2214yes—
11#1462—22no—
12#1458—22no—
13#1458C414yescomplemented
14#1445C412yes—
15#1445C2212yes—
16#1434—22no—
17C22⋊#95—22no—
18C24×C4—21no—
19C24×C4C2×C411yescharacteristic
20C2×#117—42no—
21C2×#117—41no—
22C2×#117—41no—
23C2×#117C2×C412yes—
24C2×#117C2×C412yes—
25C2×#117D412yes—
26C2×#117Q812yes—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2311yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4D412yes—
31C22×C42D412yes—
32#432—24no—
33#432—42no—
34#431—24no—
35#431—42no—
36#423—42no—
37#407—24no—
38C4×M16—42no—
39#381—41no—
40#381—41no—
41C2×#95—24no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—24no—
45C2×#95—42no—
46C2×#95C2×C412yes—
47C2×#95C2×C412yes—
48#324—24no—
49#324C2×C418yescomplemented
50#323—24no—
51#323—24no—
52C25C22⋊C411yescharacteristic
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—44no—
59C23×C4—44no—
60C23×C4—42no—
61C23×C4—44no—
62C23×C4—82no—
63C23×C4C4212yes—
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C4⋊C412yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C2×M16—44no—
70#117—42no—
71#117—42no—
72#117—42no—
73#117—42no—
74#117—42no—
75#117—42no—
76#117—42no—
77#117—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4C4211yescharacteristic
83C2×C4⋊C4C4211yescharacteristic
84C2×C4⋊C4C4⋊C412yes—
85C2×C4⋊C4C22×C412yes—
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C4⋊C4C2×Q811yescharacteristic
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4C22⋊C412yes—
93C2×C22⋊C4C22⋊C412yes—
94C2×C22⋊C4C4⋊C412yes—
95C2×C22⋊C4C4⋊C412yes—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42C22⋊C412yes—
101C2×C42C22⋊C412yes—
102C2×C42C4⋊C412yes—
103C2×C42C4⋊C412yes—
104C4⋊C8—44no—
105C4⋊C8—44no—
106C22⋊C8—28no—
107C22⋊C8—44no—
108#97—44no—
109C4×C8—44no—
110#95—42no—
111#95—42no—
112#95—48no—
113#95—42no—
114#95—41no—
115#95—41no—
116#95—84no—
117C24—24no—
118C24—21no—
119C24—21no—
120C24—42no—
121C24—41no—
122C24—41no—
123C24#9512yes—
124C24C2×C22⋊C411yescharacteristic
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—48no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—48no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—816no—
159C22×C4—84no—
160C22×C4—84no—
161C22×C4#9512yes—
162C22×C4#9512yes—
163C22×C4#9512yes—
164C22×C4#9912yes—
165C22×C4#9911yescharacteristic
166C22×C4#9911yescharacteristic
167C22×C4C2×C4211yescharacteristic
168C22×C4C2×C22⋊C412yes—
169C22×C4C2×C4⋊C411yescharacteristic
170C22×C4C2×C4⋊C412yes—
171M16—84no—
172M16—84no—
173C2×C8—48no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—84no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—22no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23—22no—
200C23—22no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—48no—
205C23—48no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23#34414yes—
211C23C2×#9511yescharacteristic
212C23C2×#9911yescharacteristic
213C23C2×#9911yescharacteristic
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—84no—
245C2×C4—82no—
246C2×C4—816no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—84no—
250C2×C4—84no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#34212yes—
254C2×C4#34212yes—
255C8—88no—
256C22—24no—
257C22—24no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—44no—
267C22—48no—
268C22—44no—
269C22—44no—
270C22—42no—
271C22—42no—
272C22C2×#34211yescharacteristic
273C22C2×#34412yes—
274C22#212811yescharacteristic
275C22#212911yescharacteristic
276C22#213212yes—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—41no—
284C4—41no—
285C4—84no—
286C4—162no—
287C2—21no—
288C2—22no—
289C2—21no—
290C2—22no—
291C2—44no—
292C2#811411yescharacteristic
293C2#1258311yescharacteristic
294C2#1258511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2semidirect
9. #1464C4semidirect
13. #1458C4semidirect
49. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#173572
#139582
#73262