Finite Group Databasebeta

← #7642039

Subgroups — #7642039

Order 512.

8314 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#537C211yesmaximal, characteristic, complemented
2C22×#539C211yesmaximal, characteristic, complemented
3#37185C218yesmaximal, complemented
4#37118C218yesmaximal, complemented
5#36961C214yesmaximal, complemented
6#36958C214yesmaximal, complemented
7#36741C2116yesmaximal, complemented
8#36710C214yesmaximal, complemented
9#36707C214yesmaximal, complemented
10#35088C214yesmaximal, complemented
11#35019C212yesmaximal, complemented
12#35001C212yesmaximal, complemented
13#35000C214yesmaximal, complemented
14C2×#2313C211yesmaximal, characteristic, complemented
15C23×#41C2212yes—
16C2×#539—216no—
17C2×#539C2218yes—
18C2×#539C2218yescomplemented
19C2×#537—216no—
20C2×#537C2214yes—
21C2×#537C2214yes—
22C2×#537C2218yescomplemented
23C2×#534C2218yes—
24C22×#126C2212yescomplemented
25C22×#125C2212yescomplemented
26C22×#123C2212yes—
27C22×#123C2212yescomplemented
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2212yescomplemented
30C22×C22⋊Q8C2214yescomplemented
31C22×#121C2212yes—
32C22×#121C2211yescharacteristic
33C22×#121C2212yes—
34C22×#121C2214yescomplemented
35C22×C4×D4C2212yes—
36C23×C4⋊C4C2211yescharacteristic
37#2856C2218yescomplemented
38#2854C2218yescomplemented
39#2850C2218yescomplemented
40#2849C22116yescomplemented
41#2847C2218yescomplemented
42#2723C22132yescomplemented
43#2720C22132yescomplemented
44#2714C22132yescomplemented
45#2711C22132yescomplemented
46#2696C22132yescomplemented
47#2692C22132yescomplemented
48#2676C22132yescomplemented
49#2673C22132yescomplemented
50#2656C2218yescomplemented
51#2639C2218yescomplemented
52#2618C2218yescomplemented
53#2617C2218yescomplemented
54#2614C2218yescomplemented
55#2613C2218yescomplemented
56#2591C2218yescomplemented
57#2561C2218yescomplemented
58#2543C22116yescomplemented
59#2539C2218yes—
60#2539C2218yescomplemented
61#2534C22116yescomplemented
62#2504C2214yescomplemented
63#2503C2214yescomplemented
64#2501C2214yes—
65#2501C2214yescomplemented
66#2486C2218yescomplemented
67#2436C2214yescomplemented
68#2435C2214yescomplemented
69#2433C2214yes—
70#2433C2214yescomplemented
71#2430C22116yescomplemented
72#2430C22116yescomplemented
73#2416C22116yescomplemented
74#2408C2218yescomplemented
75#2406C2218yescomplemented
76#2406C2218yescomplemented
77#2404C2218yescomplemented
78#2404C2218yescomplemented
79#2313C2218yescomplemented
80#2313C2214yescomplemented
81#2313C2214yescomplemented
82#2308C2218yes—
83C2×#387C2218yescomplemented
84C2×#381C2212yes—
85C2×#380C2212yes—
86C2×C4×C22⋊C4C2212yes—
87C22×#41—216no—
88C22×#41—416no—
89C22×#41C2318yes—
90C22×#41C2318yes—
91C23×Q8C2312yes—
92C23×Q8C2311yescharacteristic
93C23×D4C2314yes—
94C23×D4C2312yes—
95C23×D4C2312yes—
96C23×D4C2312yes—
97C23×D4C2312yes—
98C23×D4C2311yescharacteristic
99C24×C4C2311yescharacteristic
100C24×C4C2314yes—
101C24×C4C2312yes—
102C24×C4C2311yescharacteristic
103C24×C4C2311yescharacteristic
104#539—264no—
105#539—264no—
106#537—232no—
107#537—232no—
108#537—264no—
109#534—432no—
110C2×#127C2318yes—
111C2×#126—216no—
112C2×#126C23116yes—
113C2×#125—216no—
114C2×#125C23116yes—
115C2×#124C2318yes—
116C2×#123—216no—
117C2×#123—216no—
118C2×#123C2318yes—
119C2×#123C2318yes—
120C2×#123C23116yes—
121C2×#123C23116yes—
122C2×#123C23116yes—
123C2×C22⋊Q8—28no—
124C2×C22⋊Q8—232no—
125C2×C22⋊Q8—216no—
126C2×C22⋊Q8C2314yes—
127C2×C22⋊Q8C2314yes—
128C2×C22⋊Q8C23116yes—
129C2×C22⋊Q8C23116yes—
130C2×C22⋊Q8C23116yes—
131C2×#121—216no—
132C2×#121—216no—
133C2×#121—28no—
134C2×#121—232no—
135C2×#121C2314yes—
136C2×#121C2318yes—
137C2×#121C2314yes—
138C2×#121C2314yes—
139C2×#121C2314yes—
140C2×#121C2318yes—
141C2×#121C2314yes—
142C2×#121C2314yes—
143C2×#121C23116yes—
144C2×#121C23116yes—
145C2×#121C23132yes—
146C2×#121C23116yes—
147C2×#120C23116yes—
148C2×#120C23116yes—
149C2×C4×D4—216no—
150C2×C4×D4C2318yes—
151C2×C4×D4C2318yes—
152C22×C4⋊C4D4116yescomplemented
153C22×C4⋊C4C2312yes—
154C22×C4⋊C4C2312yes—
155C22×C4⋊C4C2312yes—
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2312yes—
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2314yes—
162C22×C4⋊C4C2314yes—
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2314yes—
166C22×C22⋊C4C2314yes—
167C22×C22⋊C4C2312yes—
168C22×C22⋊C4C2314yes—
169C22×C22⋊C4C2312yes—
170C22×C22⋊C4C2312yes—
171C22×C22⋊C4C2312yes—
172C22×C22⋊C4C2314yes—
173C22×C22⋊C4C2312yes—
174C22×C42C2312yes—
175C22×C42C2314yes—
176#400C23116yes—
177#399C23116yes—
178#399C23116yes—
179#398C23116yes—
180#397C23116yes—
181#397C23116yes—
182#396C23132yes—
183#395C23116yes—
184#395C23116yes—
185#394C23116yes—
186#394C23116yes—
187#394C23116yes—
188#394C23132yes—
189#394C23116yes—
190#393C23116yes—
191#393C23116yes—
192#393C23116yes—
193#392C23116yes—
194#392C23116yes—
195#392C23116yes—
196#392C23116yes—
197#392C23116yes—
198#391C2318yes—
199#390C2318yes—
200#390C23116yes—
201#390C23132yes—
202#390C2318yes—
203#390C2318yes—
204#389C23116yes—
205#388C23132yes—
206#388C23132yes—
207#387C23132yes—
208#387C23132yes—
209#386C23116yes—
210#386C23116yes—
211#386C23132yes—
212#386C23132yes—
213#386C23132yes—
214#386C23132yes—
215#385C23132yes—
216#385C23132yes—
217#384C2318yes—
218#382C2318yes—
219#381C23116yes—
220#381C2318yes—
221#381C2318yes—
222#380C2318yes—
223#380C2318yes—
224#380C23116yes—
225#379C23116yes—
226C4×C22⋊C4C2314yes—
227C4×C22⋊C4C2314yes—
228C4×C22⋊C4C2314yes—
229C4×C22⋊C4C2314yes—
230C2×#95C2314yes—
231C2×#95C2314yes—
232C2×#95C2314yes—
233C2×#95C2314yes—
234C25C2412yes—
235C25C2414yes—
236C25C2411yescharacteristic
237C2×#41—232no—
238C2×#41—232no—
239C2×#41—432no—
240C2×#41—432no—
241C2×#41—4128no—
242C22×Q8—28no—
243C22×Q8—24no—
244C22×Q8—48no—
245C22×Q8—82no—
246C22×Q8C2414yes—
247C22×Q8C2418yes—
248C22×D4—28no—
249C22×D4—216no—
250C22×D4—216no—
251C22×D4—232no—
252C22×D4—28no—
253C22×D4—28no—
254C22×D4—28no—
255C22×D4—24no—
256C22×D4—48no—
257C22×D4—84no—
258C22×D4—84no—
259C22×D4—82no—
260C22×D4C2414yes—
261C22×D4C2414yes—
262C22×D4C2418yes—
263C22×D4C2418yes—
264C22×D4C2418yes—
265C22×D4C2414yes—
266C22×D4C2414yes—
267C22×D4C2418yes—
268C22×D4C2414yes—
269C22×D4C24132yes—
270C23×C4—28no—
271C23×C4—216no—
272C23×C4—28no—
273C23×C4—416no—
274C23×C4—82no—
275C23×C4—82no—
276C23×C4—84no—
277C23×C4C2×D418yes—
278C23×C4C2×D418yes—
279C23×C4C2×D418yes—
280C23×C4C2414yes—
281C23×C4C2412yes—
282C23×C4C2412yes—
283C23×C4C2412yes—
284C23×C4C2412yes—
285C23×C4C2412yes—
286C23×C4C2412yes—
287C23×C4C2418yes—
288C23×C4C2418yes—
289C23×C4C24116yes—
290C23×C4C2412yes—
291C23×C4C2411yescharacteristic
292C23×C4C2412yes—
293C23×C4C2412yes—
294C23×C4C2411yescharacteristic
295C23×C4C2412yes—
296C23×C4C2412yes—
297#127—88no—
298#126—264no—
299#125—264no—
300#124—88no—
301#123—232no—
302#123—232no—
303#123—264no—
304#123—432no—
305#123—432no—
306C22⋊Q8—216no—
307C22⋊Q8—216no—
308C22⋊Q8—264no—
309C22⋊Q8—264no—
310C22⋊Q8—264no—
311#121—232no—
312#121—216no—
313#121—216no—
314#121—232no—
315#121—216no—
316#121—216no—
317#121—216no—
318#121—216no—
319#121—264no—
320#121—264no—
321#121—464no—
322#121—816no—
323#120—432no—
324#120—432no—
325C4×D4—232no—
326C4×D4—232no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4C2×D4132yes—
337C2×C4⋊C4C2×D4116yes—
338C2×C4⋊C4C2×D4116yes—
339C2×C4⋊C4C2×D4132yes—
340C2×C4⋊C4C2418yes—
341C2×C4⋊C4C2418yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2418yes—
346C2×C4⋊C4C2418yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2418yes—
349C2×C4⋊C4C24116yes—
350C2×C4⋊C4C24132yes—
351C2×C4⋊C4C2418yes—
352C2×C4⋊C4C2418yes—
353C2×C22⋊C4—28no—
354C2×C22⋊C4—216no—
355C2×C22⋊C4—28no—
356C2×C22⋊C4—216no—
357C2×C22⋊C4—216no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4C2414yes—
362C2×C22⋊C4C2414yes—
363C2×C22⋊C4C2418yes—
364C2×C22⋊C4C2418yes—
365C2×C22⋊C4C2418yes—
366C2×C22⋊C4C24116yes—
367C2×C22⋊C4C2418yes—
368C2×C22⋊C4C2418yes—
369C2×C22⋊C4C2418yes—
370C2×C22⋊C4C2418yes—
371C2×C22⋊C4C2418yes—
372C2×C22⋊C4C24132yes—
373C2×C22⋊C4C24132yes—
374C2×C22⋊C4C24132yes—
375C2×C22⋊C4C24132yes—
376C2×C42—28no—
377C2×C42—84no—
378C2×C42C2418yes—
379C2×C42C2418yes—
380C2×C42C2418yes—
381#95C2418yes—
382#95C2418yes—
383#95C24116yes—
384#95C24116yes—
385#95C24116yes—
386#95C24132yes—
387#95C24132yes—
388C24—24no—
389C24—28no—
390C24—28no—
391C24—216no—
392C24—42no—
393C24—42no—
394C24—44no—
395C24—48no—
396C24—44no—
397C24—48no—
398C24C22×D414yes—
399C24H(5,𝔽2)18yes—
400C24C2514yes—
401C24C2518yes—
402C24C2511yescharacteristic
403C24C2511yescharacteristic
404C24C2511yescharacteristic
405#41—4128no—
406#41—4128no—
407C2×Q8—216no—
408C2×Q8—28no—
409C2×Q8—416no—
410C2×Q8—48no—
411C2×Q8—432no—
412C2×Q8—88no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—232no—
419C2×D4—216no—
420C2×D4—232no—
421C2×D4—216no—
422C2×D4—216no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—416no—
427C2×D4—432no—
428C2×D4—416no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—432no—
433C2×D4—432no—
434C2×D4—432no—
435C2×D4—432no—
436C2×D4—464no—
437C2×D4—88no—
438C2×D4—816no—
439C2×D4—816no—
440C22×C4—232no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—216no—
450C22×C4—216no—
451C22×C4—232no—
452C22×C4—216no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—432no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—464no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—816no—
470C22×C4C22×D418yes—
471C22×C4C22×D418yes—
472C22×C4C22×D418yes—
473C22×C4C22×D418yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C22×D418yes—
477C22×C4C22×D414yes—
478C22×C4C22×D418yes—
479C22×C4C22×D418yes—
480C22×C4C25116yes—
481C22×C4C2514yes—
482C22×C4C2514yes—
483C22×C4C2518yes—
484C22×C4C2514yes—
485C22×C4C2514yes—
486C22×C4C2512yes—
487C22×C4C2512yes—
488C22×C4C2514yes—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—216no—
498C4⋊C4—416no—
499C4⋊C4—432no—
500C4⋊C4C22×D4132yes—
501C4⋊C4C22×D4132yes—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—232no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—432no—
514C22⋊C4—432no—
515C22⋊C4—432no—
516C22⋊C4—432no—
517C42—216no—
518C42—48no—
519C42—48no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—216no—
530C23—216no—
531C23—28no—
532C23—216no—
533C23—28no—
534C23—44no—
535C23—44no—
536C23—416no—
537C23—48no—
538C23—416no—
539C23—48no—
540C23—416no—
541C23—416no—
542C23—48no—
543C23—832no—
544C23—816no—
545C23#535116yes—
546C23C23×D412yes—
547C23C23×D412yes—
548C23C23×D412yes—
549C23C2×H(5,𝔽2)14yes—
550C23C2×H(5,𝔽2)14yes—
551C23C2×H(5,𝔽2)14yes—
552C23C2611yescharacteristic
553Q8—432no—
554Q8—88no—
555D4—432no—
556D4—416no—
557D4—416no—
558D4—432no—
559D4—432no—
560D4—816no—
561D4—88no—
562D4—88no—
563D4—88no—
564D4—1632no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—216no—
570C2×C4—216no—
571C2×C4—216no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—216no—
577C2×C4—28no—
578C2×C4—216no—
579C2×C4—216no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—216no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—464no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—416no—
594C2×C4—48no—
595C2×C4—816no—
596C2×C4—816no—
597C2×C4C23×D418yes—
598C2×C4C23×D414yes—
599C2×C4C23×D414yes—
600C2×C4C23×D418yes—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—416no—
611C22—44no—
612C22—48no—
613C22—44no—
614C22—48no—
615C22—416no—
616C22—432no—
617C22—48no—
618C22—44no—
619C22—832no—
620C22—816no—
621C22C2×#53514yes—
622C22C2×#53514yes—
623C22C2×#53514yes—
624C22C24×D411yescharacteristic
625C22C22×H(5,𝔽2)12yes—
626C22H(7,𝔽2)12yes—
627C222−1+612yes—
628C4—28no—
629C4—24no—
630C4—24no—
631C4—28no—
632C4—44no—
633C4—44no—
634C4—416no—
635C4—48no—
636C4—44no—
637C2—24no—
638C2—24no—
639C2—24no—
640C2—84no—
641C2—88no—
642C2C22×#53511yescharacteristic
643C2#6284211yescharacteristic
644C2#6284811yescharacteristic
645C2#6286614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#537C2semidirect
2. C22×#539C2semidirect
3. #37185C2semidirect
4. #37118C2semidirect
5. #36961C2semidirect
6. #36958C2semidirect
7. #36741C2semidirect
8. #36710C2semidirect
9. #36707C2semidirect
10. #35088C2semidirect
11. #35019C2semidirect
12. #35001C2semidirect
13. #35000C2semidirect
14. C2×#2313C2semidirect
18. C2×#539C22semidirect
22. C2×#537C22semidirect
24. C22×#126C22semidirect
25. C22×#125C22semidirect
27. C22×#123C22semidirect
29. C22×C22⋊Q8C22semidirect
30. C22×C22⋊Q8C22semidirect
34. C22×#121C22semidirect
37. #2856C22semidirect
38. #2854C22semidirect
39. #2850C22semidirect
40. #2849C22semidirect
41. #2847C22semidirect
42. #2723C22semidirect
43. #2720C22semidirect
44. #2714C22semidirect
45. #2711C22semidirect
46. #2696C22semidirect
47. #2692C22semidirect
48. #2676C22semidirect
49. #2673C22semidirect
50. #2656C22semidirect
51. #2639C22semidirect
52. #2618C22semidirect
53. #2617C22semidirect
54. #2614C22semidirect
55. #2613C22semidirect
56. #2591C22semidirect
57. #2561C22semidirect
58. #2543C22semidirect
60. #2539C22semidirect
61. #2534C22semidirect
62. #2504C22semidirect
63. #2503C22semidirect
65. #2501C22semidirect
66. #2486C22semidirect
67. #2436C22semidirect
68. #2435C22semidirect
70. #2433C22semidirect
71. #2430C22semidirect
72. #2430C22semidirect
73. #2416C22semidirect
74. #2408C22semidirect
75. #2406C22semidirect
76. #2406C22semidirect
77. #2404C22semidirect
78. #2404C22semidirect
79. #2313C22semidirect
80. #2313C22semidirect
81. #2313C22semidirect
83. C2×#387C22semidirect
152. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5372
C22×#5392
#371852
#371182
#369612
#369582
#367412
#367102
#367072
#350882
#350192
#350012
#350002
C2×#23132