Finite Group Databasebeta

← #128179

Subgroups — #128179

Order 512.

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

normal subgroups only

This group has 796 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#14345C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#407—22no—
6C2×#383C411yescharacteristic
7C2×#382C411yescharacteristic
8#1465—22no—
9#1465C414yescomplemented
10#1463C2214yes—
11#1462—22no—
12#1459—22no—
13#1459C414yescomplemented
14#1445C2212yes—
15#1439—22no—
16#1438C412yes—
17C22⋊#95—22no—
18C23×D4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C2×C4×D4—42no—
22C2×C4×D4—42no—
23C2×C4×D4C2×C412yes—
24C2×C4×D4C2×C412yes—
25C2×C4×D4D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27C2×C4×D4Q811yescharacteristic
28C2×C4×D4Q811yescharacteristic
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2311yescharacteristic
31C22×C22⋊C4—24no—
32C22×C22⋊C4D412yes—
33C22×C42C2×C411yescharacteristic
34#407—24no—
35#407—24no—
36#407—42no—
37#407—42no—
38C2×C22⋊C8—24no—
39C2×C22⋊C8—42no—
40#383—42no—
41#382—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×D4C4212yes—
57C22×D4C4⋊C411yescharacteristic
58C22×D4C4⋊C411yescharacteristic
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⋊C4—42no—
89C2×C4⋊C4—82no—
90C2×C4⋊C4C22×C412yes—
91C2×C4⋊C4C2×D411yescharacteristic
92C2×C4⋊C4C2×Q811yescharacteristic
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—48no—
96C2×C22⋊C4—84no—
97C2×C22⋊C4—84no—
98C2×C22⋊C4C22⋊C412yes—
99C2×C22⋊C4C22⋊C412yes—
100C2×C22⋊C4C4⋊C412yes—
101C2×C22⋊C4C4⋊C412yes—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42C4211yescharacteristic
105C2×C42C4211yescharacteristic
106C2×C42C4⋊C411yescharacteristic
107C2×C42C4⋊C411yescharacteristic
108C22⋊C8—28no—
109C22⋊C8—44no—
110C22⋊C8—44no—
111C22⋊C8—44no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95—84no—
118C24—24no—
119C24—22no—
120C24—22no—
121C24—42no—
122C24—42no—
123C24—44no—
124C24—42no—
125C24#9512yes—
126C24#9512yes—
127C24C2×C22⋊C411yescharacteristic
128C2×D4—24no—
129C2×D4—24no—
130C2×D4—42no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—48no—
136C2×D4—48no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
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—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—48no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—84no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—84no—
173C22×C4—84no—
174C22×C4#9511yescharacteristic
175C22×C4#9511yescharacteristic
176C22×C4#9511yescharacteristic
177C22×C4#9511yescharacteristic
178C22×C4#9912yes—
179C22×C4#9912yes—
180C22×C4C2×C4211yescharacteristic
181C22×C4C2×C22⋊C412yes—
182C22×C4C2×C4⋊C411yescharacteristic
183C22×C4C2×C4⋊C412yes—
184M16—88no—
185M16—88no—
186C2×C8—48no—
187C2×C8—88no—
188C4⋊C4—42no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—84no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—816no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—22no—
201C23—22no—
202C23—21no—
203C23—21no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—44no—
211C23—44no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—48no—
217C23—48no—
218C23—44no—
219C23—42no—
220C23—42no—
221C23—82no—
222C23—82no—
223C23—84no—
224C23—84no—
225C23#34414yes—
226C23C2×#9511yescharacteristic
227C23C2×#9911yescharacteristic
228C23C2×#9911yescharacteristic
229D4—44no—
230D4—48no—
231D4—44no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4#34212yes—
274C2×C4#34212yes—
275C8—88no—
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—24no—
285C22—44no—
286C22—44no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—84no—
294C22—88no—
295C22—84no—
296C22—82no—
297C22C2×#34211yescharacteristic
298C22C2×#34412yes—
299C22#212712yes—
300C22#213411yescharacteristic
301C22#213511yescharacteristic
302C4—22no—
303C4—22no—
304C4—42no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—162no—
311C4—162no—
312C2—21no—
313C2—22no—
314C2—22no—
315C2—21no—
316C2—44no—
317C2—82no—
318C2#811211yescharacteristic
319C2#1258211yescharacteristic
320C2#1258711yescharacteristic
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
#143452
#73262