Finite Group Databasebeta

← #205020

Subgroups — #205020

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35032C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3C2×#1513C212yesmaximal, complemented
4C2×#1504C212yesmaximal, complemented
5#13802C211yesmaximal, characteristic, complemented
6#9716C212yesmaximal, complemented
7#9715C211yesmaximal, characteristic, complemented
8#9714C211yesmaximal, characteristic, complemented
9#9713C212yesmaximal, complemented
10#9712C211yesmaximal, characteristic, complemented
11#9711C211yesmaximal, characteristic, complemented
12C2×#525C2212yes—
13C2×#407C2212yes—
14#2857C2211yescharacteristic
15#2856C2211yescharacteristic
16#2850C2212yes—
17#2617—21no—
18#2617—21no—
19#2602—22no—
20#2602—22no—
21#2500C412yescomplemented
22#2499C412yescomplemented
23#2432C412yescomplemented
24#2431C412yescomplemented
25#2325—21no—
26#2325—21no—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4C2211yescharacteristic
30C22×#95C2211yescharacteristic
31#1851—22no—
32#1851—22no—
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1797C2212yes—
36#1513—28no—
37#1504—28no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45C2×#327C2212yes—
46#1436—21no—
47#1436—21no—
48#1434C2211yescharacteristic
49#1434C2211yescharacteristic
50C22⋊#95—21no—
51C4×#95C2211yescharacteristic
52C4×#95C2211yescharacteristic
53C22×#41C2×C412yes—
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C22×M16—22no—
57C2×C4⋊Q8C2×C411yescharacteristic, complemented
58C2×#127C2×C411yescharacteristic, complemented
59C2×#124C2×C412yescomplemented
60#525—24no—
61#525—24no—
62#525—24no—
63#525—24no—
64C2×#123—22no—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2312yes—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121C2×C412yes—
70C2×#121C2312yes—
71C2×#120—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×#117C2×C412yes—
75C22×C4⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86#407—24no—
87#407—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91#397—22no—
92#397—22no—
93#394—22no—
94#394—22no—
95#394—22no—
96#394—22no—
97#392—22no—
98#392—22no—
99#391C2×C412yescomplemented
100#391C2×C412yescomplemented
101#390—21no—
102#390—21no—
103#390C2×C412yescomplemented
104#390C2×C412yescomplemented
105#389—21no—
106#389—21no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#385—22no—
118#385—22no—
119#384C2×C411yescharacteristic, complemented
120#384C2×C411yescharacteristic, complemented
121#382C2×C412yescomplemented
122#380—21no—
123#380—21no—
124#380—22no—
125#380D414yescomplemented
126C4×C22⋊C4—44no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141C2×#95D414yescomplemented
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146#336—22no—
147#336—21no—
148#336—21no—
149#328—24no—
150#328—24no—
151#328—24no—
152#328—24no—
153#327—24no—
154#327—24no—
155#327—24no—
156#327—24no—
157C25C2×D411yescharacteristic
158C2×#41—24no—
159C2×#41—24no—
160C2×#41—82no—
161C2×#41—82no—
162C22×Q8C22×C412yes—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C22×C412yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—81no—
180C23×C4—81no—
181C23×C4—81no—
182C23×C4—81no—
183C23×C4C22×C412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C2×M16—44no—
196C2×M16—82no—
197C2×M16—82no—
198C22×C8—22no—
199C22×C8—22no—
200C4⋊Q8—81no—
201#127—81no—
202#124—82no—
203#123—44no—
204#123—44no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211#121—24no—
212#121—24no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—22no—
217#121—42no—
218#121—42no—
219#121—42no—
220#121—42no—
221#120—44no—
222#120—44no—
223C4×D4—44no—
224C4×D4—44no—
225C4×D4—44no—
226C4×D4—44no—
227#117—24no—
228#117—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—81no—
239C2×C4⋊C4—81no—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4C22⋊C412yes—
261C2×C22⋊C4C22⋊C412yes—
262C2×C22⋊C4C22⋊C412yes—
263C2×C22⋊C4C22⋊C412yes—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—81no—
270C2×C42—81no—
271C2×C42—81no—
272C2×C42—81no—
273C2×C42C22×C412yes—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C4⋊C8—81no—
280C4⋊C8—81no—
281C4⋊C8—82no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—44no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—81no—
303#95—81no—
304#95C22⋊C412yes—
305#95C22⋊C412yes—
306#95C22⋊C412yes—
307#95C22⋊C412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24C2×C22⋊C412yes—
322C24#12112yes—
323C24#12112yes—
324C24C22×D411yescharacteristic
325#41—88no—
326#41—88no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—42no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—48no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—48no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—44no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C4×D414yes—
428C22×C4C4×D414yes—
429C22×C4#12112yes—
430C22×C4#12112yes—
431C22×C4#12414yes—
432C22×C4#12714yes—
433C22×C4AGL(1,ℤ/8)12yes—
434C22×C4#13712yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441M16—88no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—84no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C42—41no—
481C42—41no—
482C42—42no—
483C42—42no—
484C42—41no—
485C42—41no—
486C42—44no—
487C42—84no—
488C42—84no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23#39014yes—
523C23C22×C22⋊C411yescharacteristic
524C23C2×#12111yescharacteristic
525C23C2×#12111yescharacteristic
526C23C2×AGL(1,ℤ/8)11yescharacteristic
527C23C2×#13711yescharacteristic
528C23#57511yescharacteristic
529C23#57511yescharacteristic
530Q8—82no—
531Q8—82no—
532D4—82no—
533D4—82no—
534D4—88no—
535D4—88no—
536D4—82no—
537D4—82no—
538D4—82no—
539D4—82no—
540D4—162no—
541D4—162no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—88no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—816no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4#39014yes—
622C2×C4#41712yes—
623C2×C4#41712yes—
624C2×C4C2×C4×D411yescharacteristic
625C2×C4C2×C4×D411yescharacteristic
626C2×C4C2×#12411yescharacteristic
627C2×C4C2×#12711yescharacteristic
628C8—82no—
629C8—82no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C2×#39011yescharacteristic
661C22#288311yescharacteristic
662C22#288311yescharacteristic
663C22C2×#41711yescharacteristic
664C22#289611yescharacteristic
665C22#290111yescharacteristic
666C22#290111yescharacteristic
667C22#303611yescharacteristic
668C22#303611yescharacteristic
669C22#303711yescharacteristic
670C22#303711yescharacteristic
671C4—22no—
672C4—22no—
673C4—21no—
674C4—24no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—44no—
679C4—41no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C2—21no—
690C2—21no—
691C2—21no—
692C2—21no—
693C2—44no—
694C2—162no—
695C2#1762211yescharacteristic
696C2#1762311yescharacteristic
697C2#1762911yescharacteristic
698C2#1763011yescharacteristic
699C2#2121011yescharacteristic
700C2#2121311yescharacteristic
701C2#2125811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21934C2semidirect
3. C2×#1513C2semidirect
4. C2×#1504C2semidirect
5. #13802C2semidirect
6. #9716C2semidirect
7. #9715C2semidirect
8. #9714C2semidirect
9. #9713C2semidirect
10. #9712C2semidirect
11. #9711C2semidirect
21. #2500C4semidirect
22. #2499C4semidirect
23. #2432C4semidirect
24. #2431C4semidirect
57. C2×C4⋊Q8C2×C4semidirect
58. C2×#127C2×C4semidirect
59. C2×#124C2×C4semidirect
99. #391C2×C4semidirect
100. #391C2×C4semidirect
103. #390C2×C4semidirect
104. #390C2×C4semidirect
119. #384C2×C4semidirect
120. #384C2×C4semidirect
121. #382C2×C4semidirect
125. #380D4semidirect
141. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
#350322
#219342
C2×#15132
C2×#15042
#138022
#97162
#97152
#97142
#97132
#97122
#97112