Finite Group Databasebeta

← #924620

Subgroups — #924620

Order 512.

2832 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17234C212yesmaximal, complemented
2#17224C212yesmaximal, complemented
3#16660C214yesmaximal, complemented
4#16357C214yesmaximal, complemented
5#15952C212yesmaximal, complemented
6#15506C212yesmaximal, complemented
7#15198C212yesmaximal, complemented
8#15144C212yesmaximal, complemented
9#15125C211yesmaximal, characteristic, complemented
10#15124C211yesmaximal, characteristic
11#14257C212yesmaximal, complemented
12#14025C212yesmaximal, complemented
13#13970C212yesmaximal, complemented
14#13916C212yesmaximal, complemented
15#13853C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#121C2212yes—
19C22×C4×D4C2212yes—
20C22×#117C2211yescharacteristic
21C2×#393C2212yes—
22C2×#393C2214yes—
23C2×#393C2214yes—
24C2×#392C2212yes—
25C2×#392C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#381C2214yes—
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2212yes—
38#1445C2211yescharacteristic
39#1445C2211yescharacteristic
40#1445C2211yescharacteristic
41#1445C2211yescharacteristic
42#1443C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2212yes—
47#1440C2212yes—
48#1440C2214yes—
49#1440C2212yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C23×Q8C2312yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C23×D4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2314yes—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—44no—
84C2×#127—28no—
85C2×#127—42no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—44no—
89C2×C4×D4—28no—
90C2×C4×D4—44no—
91C2×C4×D4—44no—
92C2×#117—24no—
93C2×#117—42no—
94C2×#117—42no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#393—44no—
121#393—48no—
122#393—48no—
123#392—44no—
124#392—44no—
125#390—44no—
126#390—44no—
127#390—44no—
128#390—44no—
129#386—48no—
130#386—48no—
131#386—48no—
132#381—48no—
133#380—44no—
134C4×C22⋊C4—42no—
135C4×C22⋊C4—42no—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C25C2414yes—
164C25C2411yescharacteristic
165C22×Q8—24no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8—42no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—44no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4#4118yes—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4#4114yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—416no—
220#127—416no—
221#121—432no—
222C4×D4—432no—
223#117—416no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—42no—
306#95—42no—
307#95—44no—
308#95—41no—
309#95—41no—
310#95—41no—
311#95—41no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24—21no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—21no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24C2×#4118yes—
348C24C2×#4112yes—
349C24C2511yescharacteristic
350C2×Q8—48no—
351C2×Q8—84no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×Q8—84no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—416no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—88no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—416no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C42−1+412yes—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C42—48no—
492C42—48no—
493C42—44no—
494C42—44no—
495C23—22no—
496C23—28no—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—28no—
501C23—22no—
502C23—22no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—41no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—41no—
525C23—44no—
526C23—41no—
527C23—44no—
528C23—41no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—41no—
541C23—44no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—44no—
546C23—41no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23C22×#4111yescharacteristic
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×2−1+411yescharacteristic
563C23#58512yes—
564Q8—816no—
565D4—432no—
566D4—816no—
567D4—816no—
568D4—832no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—416no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—832no—
645C2×C4#53814yes—
646C2×C4#53818yes—
647C2×C4#55114yes—
648C2×C4#55118yes—
649C2×C4#56014yes—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—44no—
654C22—48no—
655C22—42no—
656C22—44no—
657C22—48no—
658C22—48no—
659C22—42no—
660C22—42no—
661C22—48no—
662C22—81no—
663C22—82no—
664C22—84no—
665C22—82no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—81no—
671C22—84no—
672C22—81no—
673C22—84no—
674C22—81no—
675C22—84no—
676C22C2×#53812yes—
677C22C2×#53811yescharacteristic
678C22#346012yes—
679C22C2×#55111yescharacteristic
680C22C2×#55112yes—
681C22#350811yescharacteristic
682C22#351111yescharacteristic
683C22#351312yes—
684C22#351412yes—
685C22#351912yes—
686C22#352012yes—
687C22C2×#56011yescharacteristic
688C22#353611yescharacteristic
689C22#353711yescharacteristic
690C22#354212yes—
691C22#354312yes—
692C22#354312yes—
693C22#354811yescharacteristic
694C22#354911yescharacteristic
695C22#354911yescharacteristic
696C22#355111yescharacteristic
697C22#355412yes—
698C22#355812yes—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—84no—
706C4—82no—
707C4—84no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4#276018yes—
712C2—88no—
713C2—82no—
714C2#3731611yescharacteristic
715C2#3751311yescharacteristic
716C2#3825012yes—
717C2#3906412yes—
718C2#4253212yes—
719C2#4254712yes—
720C2#4508811yescharacteristic
721C2#4521411yescharacteristic
722C2C2×#276011yescharacteristic
723C2#5296011yescharacteristic
724C2#5296911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17234C2semidirect
2. #17224C2semidirect
3. #16660C2semidirect
4. #16357C2semidirect
5. #15952C2semidirect
6. #15506C2semidirect
7. #15198C2semidirect
8. #15144C2semidirect
9. #15125C2semidirect
11. #14257C2semidirect
12. #14025C2semidirect
13. #13970C2semidirect
14. #13916C2semidirect
15. #13853C2semidirect

Maximal subgroups

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

subgroupindex
#172342
#172242
#166602
#163572
#159522
#155062
#151982
#151442
#151252
#151242
#142572
#140252
#139702
#139162
#138532