Finite Group Databasebeta

← #204635

Subgroups — #204635

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35103C211yesmaximal, characteristic
2#20536C211yesmaximal, characteristic, complemented
3C2×#1521C212yesmaximal, complemented
4C2×#1500C212yesmaximal, complemented
5#13964C211yesmaximal, characteristic, complemented
6#9684C212yesmaximal
7#9683C212yesmaximal, complemented
8#9682C214yesmaximal, complemented
9C2×#529C2212yes—
10C2×#431C2212yes—
11#2857C2211yescharacteristic
12#2856C2211yescharacteristic
13#2850C2212yes—
14#2618—22no—
15#2601—24no—
16#2566C412yescomplemented
17#2565C412yescomplemented
18#2545C412yescomplemented
19#2544C412yescomplemented
20#2325—22no—
21C2×#383C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1770—24no—
25#1749C2212yes—
26#1749C2212yes—
27#1521—28no—
28#1500—28no—
29C2×#328C2214yes—
30C2×#328C2214yes—
31C2×#327C2214yes—
32C2×#327C2214yes—
33#1441—22no—
34#1436—22no—
35C4⋊#95C2212yes—
36#1434C2212yes—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×#127C2×C411yescharacteristic
39#529—24no—
40#529—24no—
41#529—24no—
42#529—24no—
43C2×#126—22no—
44C2×#125C2×C412yes—
45C2×#124C2×C412yes—
46C2×#124C2×C412yes—
47C2×#123—22no—
48C2×C22⋊Q8C2312yes—
49C2×#121—22no—
50C2×#121C2312yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×#117—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D412yes—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#431—24no—
62#431D418yescomplemented
63C2×C4⋊C8—22no—
64C2×C22⋊C8C2314yes—
65C2×#97—22no—
66C2×#97—22no—
67C2×C4×C8—22no—
68#398—24no—
69#396—24no—
70#396—24no—
71#394—24no—
72#391C2×C412yes—
73#391C2×C412yes—
74#390—22no—
75#390C2×C412yes—
76#390C2×C412yes—
77#389—22no—
78#388—24no—
79#388—24no—
80#387—24no—
81#386—24no—
82#386—24no—
83#385—24no—
84#384C2×C411yescharacteristic
85#384C2×C411yescharacteristic
86#383—24no—
87#382C2×C412yes—
88#380—22no—
89C4×C4⋊C4C2×C412yes—
90C4×C4⋊C4C2×C412yes—
91C4×C4⋊C4D414yes—
92#376—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95D412yes—
100C2×#95C2312yes—
101C2×#95C2312yes—
102#336—22no—
103#336—22no—
104#328—28no—
105#328—28no—
106#327—28no—
107#327—28no—
108C22×Q8C22×C412yes—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4C22×C412yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—24no—
121C4⋊Q8—81no—
122#127—81no—
123#126—44no—
124#126—44no—
125#125—24no—
126#125—24no—
127#124—24no—
128#124—24no—
129#124—82no—
130#123—44no—
131#123—44no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136#121—22no—
137#121—22no—
138#121—22no—
139#121—22no—
140#121—44no—
141#121—44no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—44no—
147C4×D4—44no—
148#117—44no—
149#117—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22⋊C414yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C42—22no—
175C2×C42—44no—
176C2×C42—42no—
177C2×C42—81no—
178C2×C42—81no—
179C2×C42C22×C412yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D414yes—
186C2×C42C2×D412yes—
187C2×C42#4114yes—
188C4⋊C8—28no—
189C22⋊C8—24no—
190C22⋊C8C2×D418yes—
191#97—82no—
192#97—82no—
193C4×C8—28no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—44no—
200#95—44no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—82no—
205#95C22⋊C414yes—
206#95C22⋊C414yes—
207C24—22no—
208C24C22×D411yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—82no—
223C2×D4—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—84no—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4#12412yes—
262C22×C4#12412yes—
263C22×C4#12712yes—
264C22×C4#12712yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C2×#4112yes—
269C2×C8—28no—
270C2×C8—44no—
271C2×C8—44no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—88no—
287C22⋊C4—48no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—84no—
296C22⋊C4—84no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—41no—
301C42—41no—
302C42—48no—
303C42—44no—
304C42C4×D414yes—
305C42C4×D414yes—
306C42#12114yes—
307C42#12114yes—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—42no—
312C23—42no—
313C23—44no—
314C23—44no—
315C23—42no—
316C23C22×C22⋊C411yescharacteristic
317C23C2×#12411yescharacteristic
318C23C2×#12711yescharacteristic
319C23#57512yes—
320C23#57512yes—
321Q8—82no—
322Q8—82no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—84no—
328D4—164no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—88no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#39014yes—
378C2×C4#39014yes—
379C2×C4C2×C4×D412yes—
380C2×C4C2×#12112yes—
381C8—48no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—82no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22—82no—
395C22C2×#39011yescharacteristic
396C22#288311yescharacteristic
397C22#288411yescharacteristic
398C22#289612yes—
399C22#290112yes—
400C22#313511yescharacteristic
401C22#313511yescharacteristic
402C22#315012yes—
403C4—24no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—44no—
408C4—42no—
409C4—42no—
410C4—84no—
411C4—82no—
412C4—82no—
413C4—82no—
414C4—162no—
415C2—22no—
416C2—21no—
417C2—21no—
418C2—162no—
419C2#1762312yes—
420C2#1762412yes—
421C2#2119911yescharacteristic
422C2#2121411yescharacteristic
423C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20536C2semidirect
3. C2×#1521C2semidirect
4. C2×#1500C2semidirect
5. #13964C2semidirect
7. #9683C2semidirect
8. #9682C2semidirect
16. #2566C4semidirect
17. #2565C4semidirect
18. #2545C4semidirect
19. #2544C4semidirect
62. #431D4semidirect

Maximal subgroups

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

subgroupindex
#351032
#205362
C2×#15212
C2×#15002
#139642
#96842
#96832
#96822