Finite Group Databasebeta

← #204992

Subgroups — #204992

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35103C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3C2×#1521C212yesmaximal, complemented
4C2×#1500C212yesmaximal, complemented
5#13806C211yesmaximal, characteristic, complemented
6#9716C212yesmaximal, complemented
7#9715C211yesmaximal, characteristic
8#9714C211yesmaximal, characteristic, complemented
9#9713C212yesmaximal, complemented
10#9712C211yesmaximal, characteristic
11#9711C211yesmaximal, characteristic, complemented
12C2×#529C2212yes—
13C2×#431C2212yes—
14#2857C2211yescharacteristic
15#2856C2211yescharacteristic
16#2850C2212yes—
17#2618—21no—
18#2618—21no—
19#2601—22no—
20#2601—22no—
21#2566C412yescomplemented
22#2565C412yescomplemented
23#2545C412yescomplemented
24#2544C412yescomplemented
25#2325—21no—
26#2325—21no—
27C2×#383C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1939—22no—
31#1939—22no—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1521—28no—
36#1500—28no—
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1436—21no—
46#1436—21no—
47#1434—21no—
48#1434C2211yescharacteristic
49#1434C2211yescharacteristic
50C4×#95—21no—
51C4×#95C2211yescharacteristic
52C4×#95C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55#529—24no—
56#529—24no—
57#529—24no—
58#529—24no—
59C2×#126—22no—
60C2×#125C2×C412yes—
61C2×#124C2×C412yes—
62C2×#124C2×C412yes—
63C2×#123—22no—
64C2×C22⋊Q8C2312yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121C2312yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×D4—22no—
70C2×C4×D4C2×C412yes—
71C2×#117—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#431—24no—
81#431—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88C2×C4×C8—22no—
89#398—22no—
90#398—22no—
91#396—22no—
92#396—22no—
93#396—22no—
94#396—22no—
95#394—22no—
96#394—22no—
97#391C2×C412yes—
98#391C2×C412yes—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2×C412yes—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#385—22no—
116#385—22no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#383—22no—
120#383D414yescomplemented
121#382C2×C412yes—
122#380—21no—
123#380—21no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128#376—22no—
129#376D414yescomplemented
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145#336—22no—
146#336—21no—
147#336—21no—
148#328—24no—
149#328—24no—
150#328—24no—
151#328—24no—
152#327—24no—
153#327—24no—
154#327—24no—
155#327—24no—
156C22×Q8C22×C412yes—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—81no—
176#127—81no—
177#126—44no—
178#126—44no—
179#125—24no—
180#125—24no—
181#124—24no—
182#124—24no—
183#124—82no—
184#123—44no—
185#123—44no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190#121—22no—
191#121—22no—
192#121—22no—
193#121—22no—
194#121—42no—
195#121—42no—
196#121—42no—
197#121—42no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—44no—
203C4×D4—44no—
204#117—44no—
205#117—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—81no—
246C2×C42—81no—
247C2×C42—81no—
248C2×C42—81no—
249C2×C42C22×C412yes—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C4⋊C8—44no—
262C4⋊C8—81no—
263C4⋊C8—81no—
264C4⋊C8—82no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C4×C8—44no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—81no—
288#95—81no—
289#95C22⋊C412yes—
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293#95C2×D412yes—
294#95C2×D412yes—
295#95C2×D412yes—
296#95C2×D412yes—
297C24—22no—
298C24C22×D411yescharacteristic
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4—81no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—44no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4#12112yes—
384C22×C4#12112yes—
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4#13512yes—
388C22×C4#13512yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C42—41no—
434C42—41no—
435C42—42no—
436C42—42no—
437C42—41no—
438C42—41no—
439C42—48no—
440C42—44no—
441C42—84no—
442C42—84no—
443C42C4×D414yes—
444C42C4×D414yes—
445C42#12414yes—
446C42#12714yes—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23C22×C22⋊C411yescharacteristic
459C23C2×#12111yescharacteristic
460C23C2×#12111yescharacteristic
461C23C2×#13511yescharacteristic
462C23C2×#13511yescharacteristic
463C23#57511yescharacteristic
464C23#57511yescharacteristic
465Q8—82no—
466Q8—82no—
467D4—82no—
468D4—82no—
469D4—82no—
470D4—82no—
471D4—82no—
472D4—82no—
473D4—162no—
474D4—162no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—88no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4#39014yes—
556C2×C4#39014yes—
557C2×C4#41612yes—
558C2×C4#41612yes—
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×#12411yescharacteristic
562C2×C4C2×#12711yescharacteristic
563C8—82no—
564C8—82no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22C2×#39011yescharacteristic
585C22#288311yescharacteristic
586C22#288411yescharacteristic
587C22C2×#41611yescharacteristic
588C22#289611yescharacteristic
589C22#290111yescharacteristic
590C22#290111yescharacteristic
591C22#303511yescharacteristic
592C22#303511yescharacteristic
593C22#303511yescharacteristic
594C22#303511yescharacteristic
595C4—24no—
596C4—22no—
597C4—22no—
598C4—24no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—44no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—162no—
619C2#1761511yescharacteristic
620C2#1761611yescharacteristic
621C2#1762311yescharacteristic
622C2#1762411yescharacteristic
623C2#2121011yescharacteristic
624C2#2121411yescharacteristic
625C2#2125711yescharacteristic
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. #21951C2semidirect
3. C2×#1521C2semidirect
4. C2×#1500C2semidirect
5. #13806C2semidirect
6. #9716C2semidirect
8. #9714C2semidirect
9. #9713C2semidirect
11. #9711C2semidirect
21. #2566C4semidirect
22. #2565C4semidirect
23. #2545C4semidirect
24. #2544C4semidirect
120. #383D4semidirect
129. #376D4semidirect

Maximal subgroups

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

subgroupindex
#351032
#219512
C2×#15212
C2×#15002
#138062
#97162
#97152
#97142
#97132
#97122
#97112