Finite Group Databasebeta

← #482681

Subgroups — #482681

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36022C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic, complemented
3#17367C211yesmaximal, characteristic, complemented
4#11901C212yesmaximal, complemented
5#11875C212yesmaximal, complemented
6#11824C212yesmaximal, complemented
7#11822C212yesmaximal, complemented
8#11757C212yesmaximal, complemented
9#11755C212yesmaximal, complemented
10C2×#555C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2973—21no—
13C22×#99—21no—
14#2635C411yescharacteristic, complemented
15#2635C2211yescharacteristic
16#2628C411yescharacteristic, complemented
17#2542C2211yescharacteristic
18#2539C411yescharacteristic, complemented
19#2539C2211yescharacteristic
20#2535C2211yescharacteristic
21#2482C411yescharacteristic, complemented
22#2482C2211yescharacteristic
23#2472—21no—
24#2472—21no—
25#2461—21no—
26#2461—21no—
27#2422—21no—
28#2408—21no—
29#2395C411yescharacteristic, complemented
30#2395C2211yescharacteristic
31#2392C411yescharacteristic, complemented
32#2363—21no—
33#2357C411yescharacteristic, complemented
34#2354—21no—
35#2315—21no—
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40C2×#342C2212yes—
41#1466C2212yes—
42#1465—21no—
43#1464—21no—
44#1462C2212yes—
45#1459C2212yes—
46C2×#323—21no—
47#1352C2214yescomplemented
48#1352C2214yescomplemented
49#1344C2214yescomplemented
50#1343C2214yescomplemented
51C23×D4D411yescharacteristic
52#555—44no—
53C2×C4⋊Q8C2×C411yescharacteristic, complemented
54C2×#124—21no—
55C2×#124—21no—
56C2×#123—21no—
57C2×#123—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C411yescharacteristic
63C2×#121C2×C411yescharacteristic, complemented
64C2×#120—21no—
65C2×#120—21no—
66#518—44no—
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×#117—21no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic, complemented
81C2×#117D411yescharacteristic
82C22×C22⋊C4—22no—
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4C2311yescharacteristic
87#435—44no—
88#432—22no—
89C2×#99—22no—
90C2×#99—22no—
91C2×#99—44no—
92#407—22no—
93C2×C22⋊C8—22no—
94#400C2×C411yescharacteristic, complemented
95#400C2×C411yescharacteristic, complemented
96#399C2×C411yescharacteristic
97#399C2×C411yescharacteristic
98#399C2×C411yescharacteristic, complemented
99#399C2×C411yescharacteristic, complemented
100#399C2311yescharacteristic
101#399C2311yescharacteristic
102#398C2×C411yescharacteristic, complemented
103#397—21no—
104#397—21no—
105#397C2×C411yescharacteristic
106#397C2×C411yescharacteristic, complemented
107#397C2311yescharacteristic
108#396—21no—
109#395—21no—
110#394—21no—
111#394—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#394C2×C411yescharacteristic, complemented
116#393—21no—
117#392—21no—
118#392—21no—
119#392C2×C411yescharacteristic
120#392C2×C411yescharacteristic, complemented
121#392C2311yescharacteristic
122#390—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#390—21no—
127#390—21no—
128#389—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#385—21no—
142#385—21no—
143#385—44no—
144#384—21no—
145#382C2×C411yescharacteristic, complemented
146#381C2×C411yescharacteristic
147#381C2×C411yescharacteristic, complemented
148#380—21no—
149#380—21no—
150#380—22no—
151#380—22no—
152#379—21no—
153#379—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4C2×C411yescharacteristic, complemented
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
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
171C22×Q8—21no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22⋊C411yescharacteristic
183C22×D4C22×C411yescharacteristic
184C22×D4C2×D411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C23×C4—21no—
187C23×C4—42no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C2×M16—42no—
198C22×C8—42no—
199C4⋊Q8—42no—
200#124—44no—
201#124—44no—
202#123—82no—
203#123—82no—
204C22⋊Q8—82no—
205C22⋊Q8—82no—
206#121—42no—
207#121—44no—
208#121—42no—
209#121—42no—
210#121—42no—
211#120—82no—
212#120—82no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—44no—
223#117—42no—
224#117—42no—
225#117—44no—
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—21no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C22⋊C4—24no—
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—24no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22×C411yescharacteristic
285C2×C22⋊C4C22×C411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C22⋊C4C2×D411yescharacteristic
289C2×C22⋊C4C2411yescharacteristic
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C4⋊C8—42no—
300#99—44no—
301#99—44no—
302C22⋊C8—24no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305#97—42no—
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—21no—
316#95—21no—
317#95—42no—
318#95—42no—
319#95C22×C411yescharacteristic
320#95C22×C411yescharacteristic
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323C24—22no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—41no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24—81no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C22×D411yescharacteristic
340C2×Q8—42no—
341C2×Q8—81no—
342C2×Q8—81no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4AGL(1,ℤ/8)12yes—
376C2×D4#13712yes—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—84no—
408C22×C4—162no—
409C22×C4—162no—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420M16—162no—
421C2×C8—44no—
422C2×C8—162no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—162no—
441C4⋊C4—162no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—84no—
480C22⋊C4#9914yes—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—81no—
489C42—81no—
490C23—24no—
491C23—24no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—41no—
504C23—41no—
505C23—44no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—161no—
524C23—161no—
525C23—162no—
526C23—161no—
527C23—161no—
528C23—162no—
529C23—162no—
530C23C2×#9912yes—
531C23#41712yes—
532C23#42112yes—
533C23C22×C22⋊C411yescharacteristic
534C23C2×AGL(1,ℤ/8)11yescharacteristic
535C23C2×#13711yescharacteristic
536Q8—162no—
537D4—42no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—42no—
542D4—42no—
543D4—84no—
544D4—82no—
545D4—162no—
546D4—161no—
547D4—161no—
548D4—162no—
549D4—162no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
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—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—161no—
602C2×C4—161no—
603C2×C4—168no—
604C2×C4C2×#9912yes—
605C2×C4C2×#9912yes—
606C2×C4#41712yes—
607C8—84no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—84no—
642C22—84no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—164no—
647C22C22×#9911yescharacteristic
648C22C2×#41711yescharacteristic
649C22C2×#42111yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—164no—
668C2—21no—
669C2—21no—
670C2—41no—
671C2—42no—
672C2—41no—
673C2—42no—
674C2—42no—
675C2—81no—
676C2—161no—
677C2#1759211yescharacteristic
678C2#3335011yescharacteristic
679C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36022C2
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. #20633C2semidirect
3. #17367C2semidirect
4. #11901C2semidirect
5. #11875C2semidirect
6. #11824C2semidirect
7. #11822C2semidirect
8. #11757C2semidirect
9. #11755C2semidirect
10. C2×#555C4semidirect
14. #2635C4semidirect
16. #2628C4semidirect
18. #2539C4semidirect
21. #2482C4semidirect
29. #2395C4semidirect
31. #2392C4semidirect
33. #2357C4semidirect
47. #1352C22semidirect
48. #1352C22semidirect
49. #1344C22semidirect
50. #1343C22semidirect
53. C2×C4⋊Q8C2×C4semidirect
63. C2×#121C2×C4semidirect
70. C2×C4×D4C2×C4semidirect
80. C2×#117C2×C4semidirect
94. #400C2×C4semidirect
95. #400C2×C4semidirect
98. #399C2×C4semidirect
99. #399C2×C4semidirect
102. #398C2×C4semidirect
106. #397C2×C4semidirect
115. #394C2×C4semidirect
120. #392C2×C4semidirect
145. #382C2×C4semidirect
147. #381C2×C4semidirect
157. 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
#360222
#206332
#173672
#119012
#118752
#118242
#118222
#117572
#117552