Finite Group Databasebeta

← #205031

Subgroups — #205031

Order 512.

1373 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35238C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1505C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6C2×#1500C211yesmaximal, characteristic, complemented
7#13750C211yesmaximal, characteristic, complemented
8#9713C214yesmaximal, complemented
9#9712C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#529C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2856C2212yes—
15#2850C2212yes—
16#2662—22no—
17#2567C411yescharacteristic, complemented
18#2566C412yescomplemented
19#2563C411yescharacteristic, complemented
20#2548C411yescharacteristic, complemented
21#2544C412yescomplemented
22#2541C411yescharacteristic, complemented
23#2430—22no—
24#2416—22no—
25#2324—22no—
26C2×C4×C4⋊C4—22no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1940—22no—
31#1939—22no—
32#1797C2214yes—
33#1509—24no—
34#1505—24no—
35#1502—24no—
36#1500—24no—
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41C2×#327C2212yes—
42C2×#327C2214yes—
43C4×#95—22no—
44C4×#95C2214yes—
45C22×#41—21no—
46#532—22no—
47#532—22no—
48#532—22no—
49#532—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#127C2×C412yes—
53#529—22no—
54#529—22no—
55#529—22no—
56#529—22no—
57C2×#126—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124—21no—
60C2×#124C2×C411yescharacteristic
61C2×#124C2×C412yes—
62C2×#123—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121—21no—
65C2×#121—24no—
66C2×#121—22no—
67C2×#121C2312yes—
68C2×#121C2311yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C412yes—
74C2×#117—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42—24no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#432—22no—
83#432—22no—
84#431—22no—
85#431—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—24no—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×#97—21no—
92C2×C4×C8—21no—
93#396—24no—
94#395—22no—
95#394—24no—
96#394—22no—
97#393—22no—
98#392—22no—
99#391C2×C412yes—
100#390—22no—
101#390—24no—
102#390C2×C412yes—
103#390C2×C412yes—
104#390C2×C412yes—
105#389—22no—
106#388—24no—
107#386—24no—
108#386—24no—
109#385—24no—
110#384C2×C412yes—
111#382C2×C412yes—
112#380—22no—
113C4×C4⋊C4—216no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D414yescomplemented
121#376—22no—
122#376D414yescomplemented
123C2×#95—21no—
124C2×#95—24no—
125C2×#95D412yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128#336—24no—
129#328—24no—
130#328—24no—
131#327—24no—
132#327—28no—
133#327—28no—
134#327—24no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41—82no—
138C2×#41—82no—
139C22×Q8—21no—
140C22×Q8C22×C411yescharacteristic
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—24no—
145C22×D4—21no—
146C22×D4C22×C411yescharacteristic
147C22×D4C22×C412yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4C2×D414yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160#127—22no—
161#127—22no—
162#127—82no—
163#126—42no—
164#126—42no—
165#125—22no—
166#125—22no—
167#124—22no—
168#124—22no—
169#124—42no—
170#124—42no—
171#124—82no—
172#123—42no—
173#123—42no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178#121—21no—
179#121—21no—
180#121—22no—
181#121—22no—
182#121—21no—
183#121—21no—
184#121—22no—
185#121—22no—
186#121—42no—
187#121—42no—
188#121—48no—
189#121—84no—
190#121—84no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C414yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—216no—
233C2×C42—24no—
234C2×C42—82no—
235C2×C42—82no—
236C2×C42—84no—
237C2×C42—84no—
238C2×C42C22×C412yes—
239C2×C42C22×C412yes—
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D414yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4112yes—
248C2×C42#4112yes—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—84no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256#97—42no—
257C4×C8—42no—
258#95—216no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95C22⋊C414yes—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C2×D414yes—
268#95C2×D414yes—
269C24—22no—
270C24—21no—
271C24C22×D411yescharacteristic
272#41—84no—
273#41—84no—
274#41—88no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—82no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—44no—
300C2×D4—48no—
301C2×D4—82no—
302C2×D4—84no—
303C2×D4—84no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—44no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—84no—
333C22×C4C2×C22⋊C414yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C414yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—216no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—84no—
371C4⋊C4#10418yes—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—48no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C42—216no—
387C42—44no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—84no—
392C42—816no—
393C42C4×D412yes—
394C42C4×D412yes—
395C42C4×D412yes—
396C42C4×D412yes—
397C42#12412yes—
398C42#12412yes—
399C42#12712yes—
400C42#12712yes—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—41no—
413C23—42no—
414C23C22×C22⋊C411yescharacteristic
415C23C2×#12111yescharacteristic
416C23C2×#12111yescharacteristic
417C23#57612yes—
418C23#57712yes—
419Q8—81no—
420Q8—81no—
421Q8—84no—
422D4—81no—
423D4—81no—
424D4—82no—
425D4—82no—
426D4—84no—
427D4—81no—
428D4—82no—
429D4—81no—
430D4—82no—
431D4—82no—
432D4—84no—
433D4—84no—
434D4—168no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—48no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—88no—
480C2×C4—84no—
481C2×C4#39012yes—
482C2×C4#39012yes—
483C2×C4#39012yes—
484C2×C4#39012yes—
485C2×C4C2×#10418yes—
486C2×C4C2×#10414yes—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×#12411yescharacteristic
490C2×C4C2×#12711yescharacteristic
491C8—82no—
492C8—82no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—82no—
502C22—81no—
503C22—82no—
504C22—84no—
505C22—82no—
506C22—84no—
507C22—81no—
508C22—82no—
509C22—82no—
510C22—81no—
511C22C2×#39011yescharacteristic
512C22#288311yescharacteristic
513C22#288411yescharacteristic
514C22#289712yes—
515C22C22×#10412yes—
516C22#304111yescharacteristic
517C22#304311yescharacteristic
518C22#304512yes—
519C4—22no—
520C4—22no—
521C4—28no—
522C4—22no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—48no—
528C4—48no—
529C4—44no—
530C4—82no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4#196214yes—
535C2—21no—
536C2—22no—
537C2—21no—
538C2—162no—
539C2—161no—
540C2#1754712yes—
541C2#1755012yes—
542C2C2×#196211yescharacteristic
543C2#2121411yescharacteristic
544C2#2126311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35238C2
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. #21953C2semidirect
3. C2×#1505C2semidirect
4. C2×#1509C2semidirect
5. C2×#1502C2semidirect
6. C2×#1500C2semidirect
7. #13750C2semidirect
8. #9713C2semidirect
9. #9712C2semidirect
17. #2567C4semidirect
18. #2566C4semidirect
19. #2563C4semidirect
20. #2548C4semidirect
21. #2544C4semidirect
22. #2541C4semidirect
120. C4×C4⋊C4D4semidirect
122. #376D4semidirect

Maximal subgroups

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

subgroupindex
#352382
#219532
C2×#15052
C2×#15092
C2×#15022
C2×#15002
#137502
#97132
#97122