Finite Group Databasebeta

← #497491

Subgroups — #497491

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36386C211yesmaximal, characteristic
2#22405C212yesmaximal
3#13300C212yesmaximal, complemented
4#13091C212yesmaximal, complemented
5#13090C212yesmaximal, complemented
6#12857C211yesmaximal, characteristic, complemented
7#12856C211yesmaximal, characteristic, complemented
8#12798C211yesmaximal, characteristic, complemented
9C4×#354C211yesmaximal, characteristic, complemented
10#12581C211yesmaximal, characteristic, complemented
11C2×#1393C211yesmaximal, characteristic, complemented
12C2×#552C411yescharacteristic, complemented
13C2×#546C2211yescharacteristic
14C2×#410C2212yes—
15C22×#99C2212yes—
16#2861C411yescharacteristic, complemented
17#2857C411yescharacteristic, complemented
18#2850C411yescharacteristic, complemented
19#2566C2211yescharacteristic
20#2559—22no—
21#2558—22no—
22#2545C411yescharacteristic, complemented
23#2534—22no—
24#2503C411yescharacteristic, complemented
25#2500C411yescharacteristic, complemented
26#2435C2211yescharacteristic
27#2432C2211yescharacteristic
28#2342—22no—
29#2342—22no—
30C4×#124C2211yescharacteristic
31#2297C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C42×D4C411yescharacteristic, complemented
34C2×#354C2212yes—
35C2×#354C2212yes—
36#1964—22no—
37#1899—22no—
38#1898—22no—
39#1799C2212yes—
40#1782C2212yes—
41#1781C2212yes—
42C4×#99C2212yes—
43#1393C2212yes—
44#1393C2212yes—
45#1393C2212yes—
46#1393C2212yes—
47#1393C2211yescharacteristic
48#1393C2211yescharacteristic
49#1393C2211yescharacteristic
50#1393C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C23×D4C2311yescharacteristic
53#552—41no—
54#552—41no—
55#552—41no—
56#552—41no—
57#546—41no—
58#546—41no—
59#546—41no—
60#546—41no—
61C2×C4⋊Q8C2×C411yescharacteristic, complemented
62C2×#126—21no—
63C2×#125—21no—
64C2×#125C2×C411yescharacteristic, complemented
65C2×#124C2×C411yescharacteristic
66C2×#124C2×C411yescharacteristic
67C2×#124C2311yescharacteristic
68C2×#123—21no—
69C2×#123D412yes—
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C22⋊Q8C2×C411yescharacteristic, complemented
73C2×#121C2×C411yescharacteristic
74C2×#121D412yes—
75C2×#120D412yes—
76C2×C4×Q8C2×C411yescharacteristic, complemented
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C2×#117—22no—
82C2×#117C2×C411yescharacteristic, complemented
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D412yes—
87C22×C42C2×C411yescharacteristic
88C22×C42C2×C411yescharacteristic, complemented
89#410—24no—
90#410—42no—
91C2×#99—24no—
92C2×#99—22no—
93C2×#99—22no—
94C2×#99—22no—
95C2×#99—22no—
96C2×#99—42no—
97C2×#99C2312yes—
98C2×#99C2312yes—
99C2×#99C2312yes—
100C2×#99C2312yes—
101#400—22no—
102#396—22no—
103#396—22no—
104#394—22no—
105#394—22no—
106#394—22no—
107#391C2×C411yescharacteristic, complemented
108#391C2×C411yescharacteristic, complemented
109#391C2×C411yescharacteristic, complemented
110#391C2×C411yescharacteristic, complemented
111#390—22no—
112#390C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#390C2311yescharacteristic
116#390C2311yescharacteristic
117#390C2311yescharacteristic
118#389—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—42no—
126#388—42no—
127#386—22no—
128#386—22no—
129#385—22no—
130#385—22no—
131#385—42no—
132#384C2×C411yescharacteristic, complemented
133#382C2×C411yescharacteristic, complemented
134#382C2×C411yescharacteristic, complemented
135#381—22no—
136#381—22no—
137#380—22no—
138#380—22no—
139#379—22no—
140#379—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4C2×C411yescharacteristic, complemented
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—42no—
149C4×C22⋊C4C2×C411yescharacteristic
150C4×C22⋊C4C2×C411yescharacteristic, complemented
151C43C2×C411yescharacteristic, complemented
152#354—24no—
153#354—42no—
154#342—22no—
155#342—22no—
156C25C2×D412yes—
157C2×#41—22no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—41no—
161C2×#41—41no—
162C2×#41—81no—
163C22×Q8C22×C411yescharacteristic
164C22×Q8C22×C411yescharacteristic
165C22×D4—22no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—81no—
171C22×D4C22⋊C411yescharacteristic
172C22×D4C22⋊C412yes—
173C22×D4C22×C411yescharacteristic
174C22×D4C2×D412yes—
175C22×D4C2×D412yes—
176C22×D4C2411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—42no—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C4⋊Q8—81no—
186#126—42no—
187#126—42no—
188#125—41no—
189#125—41no—
190#125—42no—
191#125—42no—
192#124—22no—
193#124—21no—
194#124—21no—
195#124—41no—
196#124—41no—
197#124—81no—
198#123—42no—
199#123—42no—
200#123—42no—
201#123—42no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208#121—41no—
209#121—41no—
210#121—42no—
211#121—42no—
212#120—42no—
213#120—42no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—44no—
223#117—41no—
224#117—41no—
225#117—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—44no—
230#117—82no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
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—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4C22⋊C412yes—
278C2×C22⋊C4C22⋊C412yes—
279C2×C22⋊C4C22⋊C412yes—
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C412yes—
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C2×D412yes—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—81no—
293C2×C42—81no—
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299#99—22no—
300#99—22no—
301#99—22no—
302#99—22no—
303#99—22no—
304#99—22no—
305#99—22no—
306#99—44no—
307#99—44no—
308#99—82no—
309#99—82no—
310#99#4114yes—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—42no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—42no—
329C24—44no—
330C24—82no—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C412yes—
333C24C2×C22⋊C411yescharacteristic
334C24C22×D412yes—
335#41—81no—
336#41—82no—
337#41—81no—
338#41—82no—
339#41—81no—
340#41—81no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—82no—
374C2×D4C2×#4112yes—
375C2×D4H(5,𝔽2)11yescharacteristic
376C2×D42−1+411yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—162no—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C23×C411yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—84no—
445C4⋊C4—162no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—24no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—162no—
482C22⋊C4#9911yescharacteristic
483C22⋊C4#9911yescharacteristic
484C22⋊C4#9911yescharacteristic
485C22⋊C4#9911yescharacteristic
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—81no—
499C42—81no—
500C42—81no—
501C42—84no—
502C42—162no—
503C23—21no—
504C23—22no—
505C23—21no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—21no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—44no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—84no—
530C23—84no—
531C23—164no—
532C23C2×#9911yescharacteristic
533C23C2×#9911yescharacteristic
534C23C22×C22⋊C411yescharacteristic
535C23#54612yes—
536C23#54712yes—
537Q8—81no—
538Q8—81no—
539Q8—81no—
540Q8—81no—
541D4—42no—
542D4—41no—
543D4—41no—
544D4—42no—
545D4—42no—
546D4—81no—
547D4—81no—
548D4—82no—
549D4—82no—
550D4—82no—
551D4—82no—
552D4—162no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—168no—
620C2×C4C2×#9911yescharacteristic
621C2×C4C2×#9911yescharacteristic
622C2×C4C2×#9911yescharacteristic
623C2×C4C2×#9911yescharacteristic
624C2×C4C4×#4111yescharacteristic
625C2×C4#52011yescharacteristic
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—22no—
633C22—22no—
634C22—21no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—41no—
644C22—42no—
645C22—41no—
646C22—42no—
647C22—82no—
648C22—84no—
649C22—84no—
650C22—81no—
651C22—81no—
652C22—82no—
653C22—82no—
654C22—162no—
655C22—162no—
656C22#212612yes—
657C22#233911yescharacteristic
658C22C22×#9911yescharacteristic
659C22#288311yescharacteristic
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—42no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—84no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—162no—
680C4—162no—
681C4—162no—
682C2—21no—
683C2—21no—
684C2—22no—
685C2—42no—
686C2—42no—
687C2—41no—
688C2—81no—
689C2—82no—
690C2#2247611yescharacteristic
691C2C2×#212611yescharacteristic
692C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#546C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#546 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13300C2semidirect
4. #13091C2semidirect
5. #13090C2semidirect
6. #12857C2semidirect
7. #12856C2semidirect
8. #12798C2semidirect
9. C4×#354C2semidirect
10. #12581C2semidirect
11. C2×#1393C2semidirect
12. C2×#552C4semidirect
16. #2861C4semidirect
17. #2857C4semidirect
18. #2850C4semidirect
22. #2545C4semidirect
24. #2503C4semidirect
25. #2500C4semidirect
33. C42×D4C4semidirect
61. C2×C4⋊Q8C2×C4semidirect
64. C2×#125C2×C4semidirect
72. C2×C22⋊Q8C2×C4semidirect
76. C2×C4×Q8C2×C4semidirect
82. C2×#117C2×C4semidirect
88. C22×C42C2×C4semidirect
107. #391C2×C4semidirect
108. #391C2×C4semidirect
109. #391C2×C4semidirect
110. #391C2×C4semidirect
132. #384C2×C4semidirect
133. #382C2×C4semidirect
134. #382C2×C4semidirect
142. C4×C4⋊C4C2×C4semidirect
150. C4×C22⋊C4C2×C4semidirect
151. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#363862
#224052
#133002
#130912
#130902
#128572
#128562
#127982
C4×#3542
#125812
C2×#13932