Finite Group Databasebeta

← #481781

Subgroups — #481781

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35730C211yesmaximal, characteristic
2#20392C211yesmaximal, characteristic, complemented
3#17369C211yesmaximal, characteristic, complemented
4#11787C212yesmaximal, complemented
5#11774C212yesmaximal, complemented
6#11759C212yesmaximal, complemented
7#11753C212yesmaximal, complemented
8#11731C212yesmaximal, complemented
9#11725C212yesmaximal, complemented
10C2×#547C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2975—21no—
13C22×#99—21no—
14#2638C411yescharacteristic, complemented
15#2638C2211yescharacteristic
16#2628C411yescharacteristic, complemented
17#2563C2211yescharacteristic
18#2561C411yescharacteristic, complemented
19#2561C2211yescharacteristic
20#2535C2211yescharacteristic
21#2457—21no—
22#2457—21no—
23#2420—21no—
24#2418—21no—
25#2418—21no—
26#2408—21no—
27#2404C411yescharacteristic, complemented
28#2404C2211yescharacteristic
29#2392C411yescharacteristic, complemented
30#2388C411yescharacteristic, complemented
31#2388C2211yescharacteristic
32#2361—21no—
33#2357C411yescharacteristic, complemented
34#2352—21no—
35#2273—21no—
36C4×#99—21no—
37C4×#99—21no—
38C4×#99C2212yes—
39C4×#99C2212yes—
40C2×#342C2212yes—
41#1465—21no—
42#1465C2212yes—
43#1462—21no—
44#1462C2212yes—
45#1460C2212yes—
46#1458—21no—
47#1351C2214yescomplemented
48#1351C2214yescomplemented
49#1344C2214yescomplemented
50#1343C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic, complemented
53C23×D4D411yescharacteristic
54#547—44no—
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×#123—21no—
57C2×#123—21no—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×C22⋊Q8C2×C411yescharacteristic, complemented
60C2×#121—21no—
61C2×#121—21no—
62C2×#120—21no—
63C2×#120—21no—
64#518—44no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic, complemented
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4C2311yescharacteristic
85#436—44no—
86#431—22no—
87C2×#99—28no—
88C2×#99—22no—
89C2×#99—22no—
90#407—22no—
91#407—22no—
92#400—21no—
93#399—21no—
94#398C2×C411yescharacteristic, complemented
95#397—21no—
96#397—21no—
97#397C2×C411yescharacteristic
98#397C2×C411yescharacteristic, complemented
99#397C2311yescharacteristic
100#396—21no—
101#395C2×C411yescharacteristic, complemented
102#395C2×C411yescharacteristic, complemented
103#394—21no—
104#394—21no—
105#394—21no—
106#394—21no—
107#394—21no—
108#394C2×C411yescharacteristic, complemented
109#393C2×C411yescharacteristic
110#393C2×C411yescharacteristic
111#393C2×C411yescharacteristic, complemented
112#393C2×C411yescharacteristic, complemented
113#393C2311yescharacteristic
114#393C2311yescharacteristic
115#392—21no—
116#392—21no—
117#392C2×C411yescharacteristic
118#392C2×C411yescharacteristic, complemented
119#392C2311yescharacteristic
120#390—21no—
121#390—21no—
122#389—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#387—21no—
130#387—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#385—21no—
140#385—21no—
141#382—21no—
142#382C2×C411yescharacteristic, complemented
143#381—21no—
144#381—21no—
145#381C2×C411yescharacteristic
146#381C2×C411yescharacteristic, complemented
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#379—21no—
152#379—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2×C411yescharacteristic, complemented
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—44no—
161C4×C22⋊C4C2×C411yescharacteristic
162C4×C22⋊C4C2×C411yescharacteristic, complemented
163#342—22no—
164#342C2314yes—
165#324—22no—
166#323—22no—
167#323—22no—
168#323C2314yes—
169C25C22⋊C411yescharacteristic
170C25C2×D411yescharacteristic
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—82no—
174C2×#41—82no—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—24no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4C22⋊C411yescharacteristic
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22×C411yescharacteristic
188C22×D4C2×D411yescharacteristic
189C22×D4C2×D411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—42no—
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C2×M16—42no—
204C2×M16—42no—
205C4⋊Q8—42no—
206#123—44no—
207#123—44no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210#121—44no—
211#121—44no—
212#120—82no—
213#120—82no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—44no—
224C4×D4—82no—
225C4×D4—82no—
226#117—42no—
227#117—42no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
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—44no—
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22×C411yescharacteristic
279C2×C22⋊C4C22×C411yescharacteristic
280C2×C22⋊C4C2×D411yescharacteristic
281C2×C22⋊C4C2×D411yescharacteristic
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2411yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C4⋊C8—42no—
294#99—28no—
295#99—28no—
296C22⋊C8—24no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C4×C8—42no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—42no—
316#95—42no—
317#95C22×C411yescharacteristic
318#95C22×C411yescharacteristic
319#95C22×C411yescharacteristic
320#95C22×C411yescharacteristic
321C24—22no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—22no—
326C24—41no—
327C24—41no—
328C24—81no—
329C24—81no—
330C24—81no—
331C24—81no—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C22×D411yescharacteristic
337#41—84no—
338#41—84no—
339#41—84no—
340#41—84no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4—81no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—162no—
413C22×C4—162no—
414C22×C4#10414yes—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425M16—162no—
426M16—162no—
427C2×C8—44no—
428C4⋊C4—22no—
429C4⋊C4—22no—
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—42no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—28no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—48no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4#9914yes—
478C42—22no—
479C42—22no—
480C42—41no—
481C42—41no—
482C42—42no—
483C42—81no—
484C42—81no—
485C42—162no—
486C42—162no—
487C23—24no—
488C23—24no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—44no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—82no—
520C23—82no—
521C23—161no—
522C23—161no—
523C23—162no—
524C23—162no—
525C23—162no—
526C23—162no—
527C23C2×#9912yes—
528C23C2×#10412yes—
529C23C22×C22⋊C411yescharacteristic
530C23#57512yes—
531Q8—82no—
532Q8—82no—
533D4—44no—
534D4—44no—
535D4—44no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—84no—
540D4—84no—
541D4—84no—
542D4—84no—
543D4—84no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—161no—
597C2×C4—161no—
598C2×C4—168no—
599C2×C4C2×#9912yes—
600C2×C4C2×#9912yes—
601C2×C4C2×#10412yes—
602C2×C4C2×#10412yes—
603C8—84no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—84no—
637C22—84no—
638C22—81no—
639C22—81no—
640C22—164no—
641C22C22×#9911yescharacteristic
642C22#289611yescharacteristic
643C22C22×#10411yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—88no—
659C4—81no—
660C4—81no—
661C4—81no—
662C2—21no—
663C2—21no—
664C2—42no—
665C2—42no—
666C2—42no—
667C2—81no—
668C2—82no—
669C2—81no—
670C2#1754211yescharacteristic
671C2#3334811yescharacteristic
672C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35730C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20392C2semidirect
3. #17369C2semidirect
4. #11787C2semidirect
5. #11774C2semidirect
6. #11759C2semidirect
7. #11753C2semidirect
8. #11731C2semidirect
9. #11725C2semidirect
10. C2×#547C4semidirect
14. #2638C4semidirect
16. #2628C4semidirect
18. #2561C4semidirect
27. #2404C4semidirect
29. #2392C4semidirect
30. #2388C4semidirect
33. #2357C4semidirect
47. #1351C22semidirect
48. #1351C22semidirect
49. #1344C22semidirect
50. #1343C22semidirect
52. C22×#41C2×C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
59. C2×C22⋊Q8C2×C4semidirect
70. C2×C4×D4C2×C4semidirect
94. #398C2×C4semidirect
98. #397C2×C4semidirect
101. #395C2×C4semidirect
102. #395C2×C4semidirect
108. #394C2×C4semidirect
111. #393C2×C4semidirect
112. #393C2×C4semidirect
118. #392C2×C4semidirect
142. #382C2×C4semidirect
146. #381C2×C4semidirect
154. C4×C4⋊C4C2×C4semidirect
162. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#357302
#203922
#173692
#117872
#117742
#117592
#117532
#117312
#117252