Finite Group Databasebeta

← #128198

Subgroups — #128198

Order 512.

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

normal subgroups only

This group has 866 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#17362C212yesmaximal
2#13905C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#407—22no—
6C2×#382C411yescharacteristic
7C2×#380—22no—
8C2×C4×C4⋊C4C411yescharacteristic
9#1465—22no—
10#1465C414yescomplemented
11#1463C2214yes—
12#1462—22no—
13#1459—22no—
14#1459C414yescomplemented
15#1445C2212yes—
16#1438C412yes—
17#1437—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⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2311yescharacteristic
31C22×C22⋊C4—24no—
32C22×C22⋊C4D412yes—
33C22×C42—21no—
34C22×C42C2×C411yescharacteristic
35#407—24no—
36#407—24no—
37#407—42no—
38#407—42no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—42no—
41#382—41no—
42#382—41no—
43#380—48no—
44C4×C4⋊C4—42no—
45C2×#95—21no—
46C2×#95—24no—
47C2×#95C2×C412yes—
48C2×#95C2×C412yes—
49#324—24no—
50#324C2×C418yes—
51#323—24no—
52#323—24no—
53C25C22⋊C412yes—
54C22×D4—41no—
55C22×D4—41no—
56C22×D4—44no—
57C22×D4C4211yescharacteristic
58C22×D4C4211yescharacteristic
59C22×D4C4⋊C412yes—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—44no—
67C23×C4—44no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C4⋊C412yes—
73C23×C4C4⋊C412yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D412yes—
76C2×M16—44no—
77C2×M16—44no—
78C2×M16—84no—
79C2×M16—84no—
80C22×C8—44no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—42no—
85C4×D4—42no—
86C4×D4—42no—
87C4×D4—42no—
88C4×D4—42no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4C22×C412yes—
98C2×C4⋊C4C2×D411yescharacteristic
99C2×C4⋊C4C2×Q811yescharacteristic
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—48no—
103C2×C22⋊C4—48no—
104C2×C22⋊C4—84no—
105C2×C22⋊C4C22⋊C412yes—
106C2×C22⋊C4C22⋊C412yes—
107C2×C22⋊C4C4⋊C412yes—
108C2×C22⋊C4C4⋊C412yes—
109C2×C42—22no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—42no—
113C2×C42—82no—
114C2×C42C4212yes—
115C2×C42C4⋊C412yes—
116C22⋊C8—28no—
117C22⋊C8—44no—
118C22⋊C8—44no—
119C22⋊C8—44no—
120#95—42no—
121#95—42no—
122#95—48no—
123#95—42no—
124C24—24no—
125C24—22no—
126C24—22no—
127C24—42no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—42no—
132C24#9512yes—
133C24#9512yes—
134C24C2×C22⋊C411yescharacteristic
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—48no—
143C2×D4—48no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4#9512yes—
186C22×C4#9512yes—
187C22×C4#9911yescharacteristic
188C22×C4#9912yes—
189C22×C4#9911yescharacteristic
190C22×C4C2×C4211yescharacteristic
191C22×C4C2×C22⋊C412yes—
192C22×C4C2×C4⋊C411yescharacteristic
193C22×C4C2×C4⋊C412yes—
194M16—84no—
195M16—84no—
196M16—88no—
197C2×C8—48no—
198C2×C8—88no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—48no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—816no—
210C42—22no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—84no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—44no—
226C23—44no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—44no—
233C23—44no—
234C23—44no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—82no—
240C23—82no—
241C23—82no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23#34414yes—
246C23C2×#9511yescharacteristic
247C23C2×#9911yescharacteristic
248C23C2×#9911yescharacteristic
249D4—44no—
250D4—48no—
251D4—44no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—84no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#34212yes—
293C2×C4#34212yes—
294C2×C4#35312yes—
295C2×C4#35412yes—
296C8—88no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—24no—
308C22—44no—
309C22—44no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—82no—
316C22—84no—
317C22—84no—
318C22—84no—
319C22—84no—
320C22—82no—
321C22C2×#34211yescharacteristic
322C22C2×#34412yes—
323C22C2×#35311yescharacteristic
324C22C2×#35411yescharacteristic
325C22#213411yescharacteristic
326C22#213511yescharacteristic
327C4—22no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—41no—
332C4—41no—
333C4—84no—
334C4—162no—
335C4#139314yes—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—21no—
340C2—21no—
341C2—44no—
342C2—82no—
343C2#811511yescharacteristic
344C2C2×#139311yescharacteristic
345C2#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
10. #1465C4semidirect
14. #1459C4semidirect

Maximal subgroups

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

subgroupindex
#173622
#139052
#73262