Finite Group Databasebeta

← #740203

Subgroups — #740203

Order 512.

3078 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3C2×#2639C211yesmaximal, characteristic
4C2×#2364C211yesmaximal, characteristic
5#16803C212yesmaximal, complemented
6#16701C212yesmaximal, complemented
7#15789C214yesmaximal, complemented
8#15495C214yesmaximal, complemented
9#15161C212yesmaximal, complemented
10#15016C212yesmaximal, complemented
11#14912C214yesmaximal, complemented
12#14159C211yesmaximal, characteristic, complemented
13#13968C214yesmaximal, complemented
14C2×#1445C211yesmaximal, characteristic, complemented
15#13782C211yesmaximal, characteristic, complemented
16C22×#123C2214yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#2639—28no—
22#2503—28no—
23#2501—28no—
24C2×#400C2212yes—
25C2×#400C2214yes—
26C2×#399C2212yes—
27C2×#399C2212yes—
28C2×#397C2212yes—
29C2×#394C2212yes—
30C2×#393C2212yes—
31#2364—28no—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#384C2212yes—
41C2×#382C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2214yes—
50#1446C2211yescharacteristic
51#1446C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1445C2211yescharacteristic
56#1445C2211yescharacteristic
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74#1434C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C2×#123—216no—
88C2×#123—216no—
89C2×#123—48no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121D418yes—
96C2×C4×D4—28no—
97C2×C4×D4—48no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#400—28no—
126#400—216no—
127#399—28no—
128#399—28no—
129#397—28no—
130#394—28no—
131#393—28no—
132#391—28no—
133#390—28no—
134#388—28no—
135#387—28no—
136#386—216no—
137#386—28no—
138#386—28no—
139#385—28no—
140#384—28no—
141#382—28no—
142#381—28no—
143#381—28no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4D418yes—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—48no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—44no—
190C22×D4C2×D418yes—
191C22×D4C2×D414yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4#4114yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#123—464no—
230C22⋊Q8—416no—
231#121—232no—
232C4×D4—432no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—42no—
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×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4#4118yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D418yes—
298#95—22no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D418yes—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24C22×D414yes—
339C24C22×D412yes—
340C24C22×D412yes—
341C24C2×#4112yes—
342C24C2×#4114yes—
343C24H(5,𝔽2)14yes—
344C24C2511yescharacteristic
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×D4—216no—
350C2×D4—232no—
351C2×D4—28no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—416no—
356C2×D4—84no—
357C2×D4—84no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
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—24no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—416no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4#121116yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C42−1+414yes—
449C4⋊C4—28no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—88no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—216no—
461C22⋊C4—216no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—84no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C42—48no—
471C42—48no—
472C42—48no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23C2×#12114yes—
508C23#53214yes—
509C23#53514yes—
510C23#54618yes—
511C23#55918yes—
512C23C23×D411yescharacteristic
513C23C23×D411yescharacteristic
514C23C22×#4111yescharacteristic
515C23C22×#4112yes—
516C23C2×H(5,𝔽2)12yes—
517C23C2×2−1+412yes—
518C23#58512yes—
519C23#58514yes—
520Q8—88no—
521D4—432no—
522D4—88no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—816no—
590C2×C4—816no—
591C2×C4—816no—
592C2×C4—816no—
593C2×C4C2×#12114yes—
594C2×C4C2×#12114yes—
595C22—216no—
596C22—24no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—24no—
602C22—44no—
603C22—44no—
604C22—48no—
605C22—48no—
606C22—44no—
607C22—84no—
608C22—84no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22#254418yes—
614C22C22×#12111yescharacteristic
615C22C2×#53211yescharacteristic
616C22C2×#53511yescharacteristic
617C22#346112yes—
618C22C2×#54612yes—
619C22C2×#55912yes—
620C22#348512yes—
621C22#348512yes—
622C22#348912yes—
623C22#349112yes—
624C22#349911yescharacteristic
625C22#350011yescharacteristic
626C22#351514yes—
627C22#352114yes—
628C22#353314yes—
629C22#353514yes—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—84no—
639C4—82no—
640C4—84no—
641C4—82no—
642C4—82no—
643C2—28no—
644C2—84no—
645C2C2×#254411yescharacteristic
646C2#3530811yescharacteristic
647C2#3544711yescharacteristic
648C2#3604512yes—
649C2#3926614yes—
650C2#4140712yes—
651C2#4349612yes—
652C2#4445012yes—
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
1. C2×#2503C2semidirect
5. #16803C2semidirect
6. #16701C2semidirect
7. #15789C2semidirect
8. #15495C2semidirect
9. #15161C2semidirect
10. #15016C2semidirect
11. #14912C2semidirect
12. #14159C2semidirect
13. #13968C2semidirect
14. C2×#1445C2semidirect
15. #13782C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#25012
C2×#26392
C2×#23642
#168032
#167012
#157892
#154952
#151612
#150162
#149122
#141592
#139682
C2×#14452
#137822