Finite Group Databasebeta

← #128200

Subgroups — #128200

Order 512.

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

normal subgroups only

This group has 816 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#17362C212yesmaximal
2#13959C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#407—22no—
6C2×#383C411yescharacteristic
7C2×#376C411yescharacteristic
8#1465—22no—
9#1465C414yescomplemented
10#1463C2214yes—
11#1462—22no—
12#1459—22no—
13#1459C414yescomplemented
14#1445C2212yes—
15#1438C412yes—
16#1437—22no—
17C22⋊#95—22no—
18C23×D4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C2×C4×D4—42no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4C2×C412yes—
25C2×C4×D4C2×C412yes—
26C2×C4×D4D412yes—
27C2×C4×D4Q812yes—
28C22×C4⋊C4C2311yescharacteristic
29C22×C22⋊C4—24no—
30C22×C22⋊C4D412yes—
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33#407—24no—
34#407—24no—
35#407—42no—
36#407—42no—
37C2×C22⋊C8—24no—
38C2×C22⋊C8—42no—
39#383—41no—
40#383—41no—
41#376—42no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—24no—
46C2×#95C2×C412yes—
47C2×#95C2×C412yes—
48#324—24no—
49#324C2×C418yes—
50#323—24no—
51#323—24no—
52C25C22⋊C412yes—
53C22×D4—41no—
54C22×D4—41no—
55C22×D4—44no—
56C22×D4C4211yescharacteristic
57C22×D4C4211yescharacteristic
58C22×D4C4⋊C412yes—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—84no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C4⋊C412yes—
71C23×C4C4⋊C412yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D412yes—
74C2×M16—44no—
75C2×M16—44no—
76C2×M16—88no—
77C22×C8—44no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—42no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—42no—
85C4×D4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4C22×C412yes—
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C4⋊C4C2×Q811yescharacteristic
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—48no—
94C2×C22⋊C4—48no—
95C2×C22⋊C4—84no—
96C2×C22⋊C4C22⋊C412yes—
97C2×C22⋊C4C22⋊C412yes—
98C2×C22⋊C4C4⋊C412yes—
99C2×C22⋊C4C4⋊C412yes—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42—44no—
105C2×C42C4212yes—
106C2×C42C4⋊C412yes—
107C22⋊C8—28no—
108C22⋊C8—44no—
109C22⋊C8—44no—
110C22⋊C8—44no—
111#95—42no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—41no—
116#95—41no—
117#95—84no—
118C24—24no—
119C24—22no—
120C24—22no—
121C24—42no—
122C24—44no—
123C24—44no—
124C24#9512yes—
125C24#9512yes—
126C24C2×C22⋊C411yescharacteristic
127C2×D4—24no—
128C2×D4—24no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—48no—
135C2×D4—48no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—48no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—84no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—84no—
171C22×C4—84no—
172C22×C4#9512yes—
173C22×C4#9512yes—
174C22×C4#9912yes—
175C22×C4#9911yescharacteristic
176C22×C4#9911yescharacteristic
177C22×C4C2×C4211yescharacteristic
178C22×C4C2×C22⋊C412yes—
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4C2×C4⋊C412yes—
181M16—88no—
182M16—88no—
183C2×C8—48no—
184C2×C8—88no—
185C4⋊C4—42no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—816no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—48no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—44no—
207C23—44no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—48no—
213C23—48no—
214C23—44no—
215C23—44no—
216C23—82no—
217C23—82no—
218C23—84no—
219C23—84no—
220C23#34414yes—
221C23C2×#9511yescharacteristic
222C23C2×#9911yescharacteristic
223C23C2×#9911yescharacteristic
224D4—44no—
225D4—48no—
226D4—44no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—48no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—84no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#34212yes—
266C2×C4#34212yes—
267C8—88no—
268C22—24no—
269C22—24no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—24no—
277C22—44no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—82no—
285C22—88no—
286C22—88no—
287C22—82no—
288C22C2×#34211yescharacteristic
289C22C2×#34412yes—
290C22#212712yes—
291C22#213212yes—
292C4—22no—
293C4—22no—
294C4—42no—
295C4—42no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—84no—
300C4—162no—
301C2—22no—
302C2—22no—
303C2—22no—
304C2—44no—
305C2—82no—
306C2#811511yescharacteristic
307C2#1258211yescharacteristic
308C2#1258511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ 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. #1465C4semidirect
13. #1459C4semidirect

Maximal subgroups

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

subgroupindex
#173622
#139592
#73262