Finite Group Databasebeta

← #108243

Subgroups — #108243

Order 512.

1806 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 206 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#60064C211yesmaximal, characteristic
2#20584C211yesmaximal, characteristic
3#14337C211yesmaximal, characteristic
4#7983C214yesmaximal, complemented
5C23×#41C2211yescharacteristic
6C2×#568C412yescomplemented
7C2×#410—22no—
8#2880—28no—
9#2879—22no—
10#2879C412yescomplemented
11C2×#407—22no—
12C2×#407C2212yes—
13#2280C2212yes—
14C2×#380—22no—
15C2×#380C2212yes—
16#1782C412yescomplemented
17#1782C412yescomplemented
18#1781—22no—
19#1446C412yescomplemented
20#1438—22no—
21#1438C412yescomplemented
22#1289—28no—
23C22×#41—24no—
24C22×#41—24no—
25C22×#41—48no—
26C22×#41C2×C411yescharacteristic
27C22×#41C2×C411yescharacteristic
28C22×#41C2×C411yescharacteristic
29C22×#41C2×C411yescharacteristic
30C22×#41D412yes—
31C22×#41Q812yes—
32C23×Q8C2×C411yescharacteristic
33C23×D4C2×C411yescharacteristic
34C23×D4D412yes—
35C24×C4D412yes—
36C24×C4C2311yescharacteristic
37#568—48no—
38C2×#131—28no—
39C22×M16—24no—
40C22×M16C2×C414yes—
41C2×#117—22no—
42C22×C4⋊C4—22no—
43C22×C22⋊C4—24no—
44C22×C22⋊C4—22no—
45#410—48no—
46C2×#99—24no—
47C2×#99—24no—
48#408—216no—
49#408—28no—
50#407—28no—
51#407—24no—
52#407—22no—
53#407—22no—
54#407—24no—
55C2×C22⋊C8—28no—
56C2×C22⋊C8D418yes—
57#381C2×C414yes—
58#380—24no—
59#380—24no—
60#380—24no—
61#380—48no—
62#379—22no—
63C4×C22⋊C4—24no—
64C2×#95—24no—
65C2×#95C2×C414yes—
66C25C22⋊C412yes—
67C25C2×D411yescharacteristic
68C2×#41—28no—
69C2×#41—28no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—22no—
74C2×#41—24no—
75C2×#41—24no—
76C2×#41—24no—
77C2×#41—28no—
78C2×#41—24no—
79C2×#41—44no—
80C2×#41—44no—
81C2×#41—44no—
82C2×#41—44no—
83C2×#41—44no—
84C2×#41—44no—
85C2×#41—432no—
86C2×#41—432no—
87C22×Q8—24no—
88C22×Q8—44no—
89C22×Q8C4211yescharacteristic
90C22×Q8C4211yescharacteristic
91C22×Q8C4⋊C412yes—
92C22×D4—28no—
93C22×D4—24no—
94C22×D4—44no—
95C22×D4—44no—
96C22×D4—44no—
97C22×D4C4211yescharacteristic
98C22×D4C4211yescharacteristic
99C22×D4C22⋊C412yes—
100C22×D4C22⋊C412yes—
101C22×D4C4⋊C412yes—
102C22×D4C4⋊C412yes—
103C22×D4C4⋊C412yes—
104C23×C4—28no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—44no—
109C23×C4—44no—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C4⋊C412yes—
114C23×C4C4⋊C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D414yes—
121C23×C4C2×Q811yescharacteristic
122C23×C4#4114yes—
123#131—432no—
124C2×M16—28no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—48no—
128C2×M16—48no—
129C2×M16—48no—
130C22×C8—28no—
131C22×C8C22⋊C418yescomplemented
132#117—48no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—48no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—84no—
142C2×C22⋊C4—88no—
143C2×C42—42no—
144#99—48no—
145#99—48no—
146C22⋊C8—216no—
147C22⋊C8—28no—
148C22⋊C8—28no—
149#95—28no—
150#95—28no—
151#95—48no—
152C24—24no—
153C24—24no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—44no—
159C24—44no—
160C24#9512yes—
161C24#9512yes—
162C24C2×C22⋊C411yescharacteristic
163C24C2×C22⋊C411yescharacteristic
164C24C2×C4⋊C411yescharacteristic
165C24#12114yes—
166#41—216no—
167#41—216no—
168#41—416no—
169#41—416no—
170#41—48no—
171#41—48no—
172#41—416no—
173#41—432no—
174#41—416no—
175C2×Q8—24no—
176C2×Q8—24no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—24no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—416no—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—24no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—416no—
203C2×D4—432no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—48no—
221C22×C4—48no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—416no—
229C22×C4—416no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4#9512yes—
237C22×C4#9512yes—
238C22×C4#9918yes—
239C22×C4C2×C4211yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4#11712yes—
245C22×C4C4×D414yes—
246C22×C4C4×D414yes—
247C22×C4#12012yes—
248C22×C4#12012yes—
249C22×C4C22⋊Q814yes—
250C22×C4#12312yes—
251C22×C4#12312yes—
252M16—416no—
253M16—48no—
254M16—48no—
255C2×C8—28no—
256C2×C8—416no—
257C2×C8—48no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C4⋊C4—84no—
261C22⋊C4—48no—
262C22⋊C4—84no—
263C22⋊C4—84no—
264C22⋊C4—88no—
265C22⋊C4—816no—
266C42—84no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—24no—
272C23—24no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—44no—
280C23—42no—
281C23—42no—
282C23—44no—
283C23—48no—
284C23—44no—
285C23—44no—
286C23—44no—
287C23—48no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—84no—
292C23—82no—
293C23—82no—
294C23—84no—
295C23—84no—
296C23—82no—
297C23—82no—
298C23#34214yes—
299C23C2×#9511yescharacteristic
300C23#38012yes—
301C23#38612yes—
302C23#38612yes—
303Q8—28no—
304Q8—48no—
305Q8—44no—
306D4—216no—
307D4—28no—
308D4—48no—
309D4—48no—
310D4—44no—
311D4—48no—
312D4—48no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—84no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—84no—
341C2×C4—88no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4#34214yes—
345C2×C4C4×C22⋊C412yes—
346C2×C4#37911yescharacteristic
347C2×C4#38111yescharacteristic
348C2×C4#38512yes—
349C2×C4#38512yes—
350C2×C4C2×#9914yes—
351C8—48no—
352C22—22no—
353C22—24no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—24no—
360C22—22no—
361C22—24no—
362C22—24no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—44no—
368C22—84no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—84no—
373C22—88no—
374C22—88no—
375C22—82no—
376C22C22⋊#9511yescharacteristic
377C22C2×#34212yes—
378C22#190912yes—
379C22#191012yes—
380C4—22no—
381C4—24no—
382C4—22no—
383C4—22no—
384C4—42no—
385C4—164no—
386C4—164no—
387C4#178112yes—
388C4#181912yes—
389C2—22no—
390C2—22no—
391C2—22no—
392C2—42no—
393C2—42no—
394C2—44no—
395C2—82no—
396C2#1072511yescharacteristic
397C2#1074212yes—
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7983C2semidirect
6. C2×#568C4semidirect
10. #2879C4semidirect
16. #1782C4semidirect
17. #1782C4semidirect
19. #1446C4semidirect
21. #1438C4semidirect
131. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#600642
#205842
#143372
#79832