Finite Group Databasebeta

← #495421

Subgroups — #495421

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35532C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#20393C211yesmaximal, characteristic
4#13300C214yesmaximal, complemented
5#13298C214yesmaximal, complemented
6#12583C212yesmaximal
7#12581C212yesmaximal
8C2×#544C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10C2×#410C2212yes—
11C2×#410C2212yes—
12#2567C412yescomplemented
13#2566C2212yes—
14#2565C412yescomplemented
15#2542C2212yes—
16#2449—22no—
17#2449—22no—
18#2441—21no—
19#2441—21no—
20#2440—21no—
21#2440—21no—
22#2317C411yescharacteristic, complemented
23#2316—21no—
24#2311C2211yescharacteristic
25#2297C2211yescharacteristic
26#2296C411yescharacteristic, complemented
27C2×#384C411yescharacteristic, complemented
28#2274—21no—
29#1799—22no—
30#1799C2214yes—
31C4×#99—22no—
32C4×#99C2214yes—
33#1740—21no—
34#1740—21no—
35#1393C2214yes—
36#1393C2214yes—
37#1393C2218yes—
38C22×#41C2311yescharacteristic
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yes—
42C2×#125—21no—
43C2×#125C2×C412yes—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47#520—44no—
48C2×C4×Q8C2×C412yes—
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4C2312yes—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42C2×C411yescharacteristic
65#410—24no—
66#410—28no—
67#410Q818yescomplemented
68C2×#99—22no—
69C2×#99—22no—
70C2×#99C2314yes—
71C2×#99C2314yes—
72#399—22no—
73#399—22no—
74#398—22no—
75#398—22no—
76#396—22no—
77#396—22no—
78#395—22no—
79#395—22no—
80#391C2×C411yescharacteristic
81#391C2×C412yes—
82#391C2×C411yescharacteristic
83#390C2311yescharacteristic
84#390C2312yes—
85#390C2311yescharacteristic
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—22no—
92#389—44no—
93#387—22no—
94#387—22no—
95#386—21no—
96#386—21no—
97#386—21no—
98#386—21no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#384—21no—
104#384—21no—
105#384C2×C412yes—
106#384C2×C412yes—
107#384C2×C412yes—
108#382C2×C412yes—
109#380—21no—
110#380—21no—
111#380—24no—
112C4×C4⋊C4—44no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—24no—
118#342—24no—
119#342—24no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41C2×Q814yes—
123C2×#41#4114yes—
124C22×Q8—22no—
125C22×Q8C22×C411yescharacteristic
126C22×D4C2×D411yescharacteristic
127C22×D4C2×D411yescharacteristic
128C22×D4C2411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4C22⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139#125—44no—
140#125—44no—
141C22⋊Q8—48no—
142C22⋊Q8—84no—
143C22⋊Q8—84no—
144C4×Q8—44no—
145C4×D4—44no—
146C4×D4—44no—
147C4×D4—44no—
148#117—24no—
149#117—28no—
150#117—42no—
151#117—42no—
152#117—44no—
153#117—42no—
154#117—42no—
155#117—44no—
156#117—44no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—84no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C2×D412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—44no—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213#99—28no—
214#99—24no—
215#99—28no—
216#99—44no—
217#99C2×Q818yes—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—44no—
229#95—44no—
230C24C2×C22⋊C411yescharacteristic
231C24C22×D411yescharacteristic
232C24C22×D411yescharacteristic
233#41—44no—
234#41—44no—
235#41—44no—
236#41—44no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—44no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8C4×Q814yes—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4C22⋊Q814yes—
250C2×D4#12412yes—
251C2×D4C4⋊Q812yes—
252C2×D4C22×Q812yes—
253C2×D4C2×#4112yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C4×Q814yes—
287C22×C4C22⋊Q814yes—
288C22×C4#12412yes—
289C22×C4C4⋊Q812yes—
290C22×C4C23×C411yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—84no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—164no—
316C4⋊C4#9914yes—
317C22⋊C4—24no—
318C22⋊C4—28no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C42—24no—
333C42—42no—
334C42—42no—
335C42—41no—
336C42—41no—
337C42—42no—
338C42—48no—
339C42—42no—
340C42—42no—
341C42—84no—
342C42—164no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23#39112yes—
356C23C22×C22⋊C411yescharacteristic
357C23C2×C22⋊Q812yes—
358C23C2×#12411yescharacteristic
359C23C2×C4⋊Q811yescharacteristic
360Q8—44no—
361Q8—164no—
362D4—44no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—42no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—162no—
414C2×C4—162no—
415C2×C4#39112yes—
416C2×C4#39112yes—
417C2×C4#39112yes—
418C2×C4C2×#9914yes—
419C2×C4C2×#9912yes—
420C2×C4#41012yes—
421C2×C4C2×C4×Q812yes—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—22no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—81no—
440C22—81no—
441C22C2×#39111yescharacteristic
442C22C22×#9911yescharacteristic
443C22C2×#41011yescharacteristic
444C4—24no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—82no—
454C4—88no—
455C4—82no—
456C4—81no—
457C4—81no—
458C4—164no—
459C4#199614yes—
460C2—21no—
461C2—21no—
462C2—42no—
463C2—41no—
464C2—42no—
465C2—41no—
466C2C2×#199611yescharacteristic
467C2#3335911yescharacteristic
468C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13300C2semidirect
5. #13298C2semidirect
8. C2×#544C4semidirect
12. #2567C4semidirect
14. #2565C4semidirect
22. #2317C4semidirect
26. #2296C4semidirect
27. C2×#384C4semidirect
67. #410Q8semidirect

Maximal subgroups

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

subgroupindex
#355322
#206342
#203932
#133002
#132982
#125832
#125812