Finite Group Databasebeta

← #620759

Subgroups — #620759

Order 512.

2616 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2639C211yesmaximal, characteristic
2C22×#383C211yesmaximal, characteristic, complemented
3#15936C214yesmaximal, complemented
4#15931C214yesmaximal, complemented
5#15924C214yesmaximal, complemented
6#15917C214yesmaximal, complemented
7#14934C211yesmaximal, characteristic, complemented
8#14926C211yesmaximal, characteristic
9#14914C211yesmaximal, characteristic, complemented
10#14913C211yesmaximal, characteristic
11#14706C212yesmaximal, complemented
12#14259C212yesmaximal, complemented
13#14020C212yesmaximal, complemented
14#13996C212yesmaximal, complemented
15#13796C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2639—28no—
21C2×#400C2212yes—
22C2×#397C2212yes—
23C2×#394C2212yes—
24C2×#393C2212yes—
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#383—28no—
32C2×#383C2214yes—
33C2×#383C2214yes—
34C2×#381C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1446C2212yes—
43#1446C2212yes—
44#1445C2214yes—
45#1444C2212yes—
46#1444C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—44no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C2×#121—24no—
93C2×#121—44no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42C2314yes—
115C22×C42C2314yes—
116#400—28no—
117#397—28no—
118#394—28no—
119#393—28no—
120#386—28no—
121#386—28no—
122#386—28no—
123#386—28no—
124#386—44no—
125#385—44no—
126#383—216no—
127#383—216no—
128#381—44no—
129C4×C22⋊C4—28no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95—24no—
133C2×#95—24no—
134C2×#95—24no—
135C2×#95D418yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×Q8—22no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227#123—432no—
228C22⋊Q8—416no—
229#121—416no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—42no—
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#4118yes—
275C2×C42—232no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—28no—
279C2×C42—28no—
280C2×C42—44no—
281C2×C42—44no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—42no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C2×D418yes—
311#95C2×D418yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24C22×D412yes—
328C24C2×#4112yes—
329C24C2×#4112yes—
330C24C2×#4114yes—
331C24H(5,𝔽2)14yes—
332C24C2511yescharacteristic
333C2×Q8—44no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8—88no—
337C2×D4—44no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4—88no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—216no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4114yes—
448C4⋊C4—416no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—88no—
453C22⋊C4—216no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C42—232no—
466C42—88no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23#52514yes—
503C23#53214yes—
504C23#53514yes—
505C23#55118yes—
506C23#55518yes—
507C23C23×D411yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4112yes—
511C23C2×H(5,𝔽2)12yes—
512C23#58512yes—
513C23#58512yes—
514C23#58512yes—
515C23#58512yes—
516Q8—88no—
517D4—88no—
518D4—816no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4—88no—
595C22—24no—
596C22—24no—
597C22—216no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—216no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—84no—
613C22—84no—
614C22#261018yes—
615C22C2×#52511yescharacteristic
616C22C2×#53211yescharacteristic
617C22C2×#53511yescharacteristic
618C22#345512yes—
619C22#345512yes—
620C22C2×#55112yes—
621C22C2×#55512yes—
622C22#348511yescharacteristic
623C22#348511yescharacteristic
624C22#348511yescharacteristic
625C22#348711yescharacteristic
626C22#349112yes—
627C22#349312yes—
628C22#352114yes—
629C22#352114yes—
630C22#356014yes—
631C22#356014yes—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—48no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C2—28no—
646C2—162no—
647C2C2×#261011yescharacteristic
648C2#3509311yescharacteristic
649C2#3509411yescharacteristic
650C2#3546711yescharacteristic
651C2#3546911yescharacteristic
652C2#3696111yescharacteristic
653C2#3696911yescharacteristic
654C2#3805914yes—
655C2#3893514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#383C2semidirect
3. #15936C2semidirect
4. #15931C2semidirect
5. #15924C2semidirect
6. #15917C2semidirect
7. #14934C2semidirect
9. #14914C2semidirect
11. #14706C2semidirect
12. #14259C2semidirect
13. #14020C2semidirect
14. #13996C2semidirect
15. #13796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26392
C22×#3832
#159362
#159312
#159242
#159172
#149342
#149262
#149142
#149132
#147062
#142592
#140202
#139962
#137962