Finite Group Databasebeta

← #725981

Subgroups — #725981

Order 512.

3038 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic, complemented
2C2×#2626C211yesmaximal, characteristic
3C22×#397C211yesmaximal, characteristic
4C2×#2368C211yesmaximal, characteristic, complemented
5#16472C214yesmaximal, complemented
6#15886C214yesmaximal, complemented
7#15504C214yesmaximal, complemented
8#14935C212yesmaximal, complemented
9#14918C212yesmaximal
10#14854C214yesmaximal, complemented
11#13992C212yesmaximal, complemented
12#13908C211yesmaximal, characteristic, complemented
13#13904C211yesmaximal, characteristic, complemented
14#13903C212yesmaximal
15#13775C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20#2626—28no—
21#2503—28no—
22C2×#400C2214yescomplemented
23C2×#399C2212yes—
24C2×#397—28no—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#393C2212yes—
30#2368—28no—
31C2×#391C2212yescomplemented
32C2×#391C2212yescomplemented
33C2×#387C2212yescomplemented
34C2×#387C2212yescomplemented
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#382C2212yescomplemented
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#379C2211yescharacteristic
45C2×#379C2212yes—
46C2×#379C2211yescharacteristic
47C2×C4×C4⋊C4C2212yescomplemented
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2212yes—
52C22×#95C2211yescharacteristic
53#1445C2214yescomplemented
54#1444C2212yescomplemented
55#1443C2214yescomplemented
56#1443C2214yescomplemented
57#1442C2212yescomplemented
58#1442C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yescomplemented
73#1436C2214yescomplemented
74C4⋊#95C2214yescomplemented
75#1434C2214yescomplemented
76C22⋊#95C2212yes—
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2211yescharacteristic
79C26C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8D418yes—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#400—216no—
125#399—28no—
126#397—28no—
127#397—28no—
128#397—216no—
129#396—216no—
130#393—28no—
131#391—28no—
132#391—28no—
133#387—28no—
134#387—28no—
135#385—28no—
136#385—28no—
137#382—28no—
138#381—28no—
139#381—28no—
140#381—28no—
141#381—44no—
142#380—24no—
143#380—44no—
144#379—24no—
145#379—28no—
146#379—24no—
147C4×C4⋊C4—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C2×#95—28no—
152C2×#95D418yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C25—22no—
180C25—22no—
181C25—22no—
182C25—22no—
183C25—22no—
184C25—22no—
185C25—44no—
186C25—44no—
187C25#4114yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—44no—
193C22×Q8C2×D414yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4#4114yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C22⋊Q8—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D418yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—432no—
286C2×C22⋊C4—416no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4#4114yes—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95C2×D414yes—
329#95C2×D414yes—
330#95C2×D418yes—
331C24—24no—
332C24—24no—
333C24—28no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—28no—
345C24—28no—
346C24—416no—
347C24—48no—
348C24—48no—
349C24—416no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—44no—
354C24—84no—
355C24—84no—
356C24—84no—
357C24—84no—
358C24—88no—
359C24—88no—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C2×#4114yes—
363C24C2×#4112yes—
364C24C2×#4112yes—
365C24C2×#4112yes—
366C24H(5,𝔽2)12yes—
367C242−1+412yes—
368C24C2511yescharacteristic
369C2×Q8—28no—
370C2×Q8—48no—
371C2×Q8—84no—
372C2×Q8—84no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4#121116yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—28no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—48no—
484C42—48no—
485C42—84no—
486C42—84no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—432no—
503C23—42no—
504C23—44no—
505C23—416no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—816no—
523C23—816no—
524C23—88no—
525C23—816no—
526C23—816no—
527C23—816no—
528C23—88no—
529C23C2×#12114yes—
530C23#52914yes—
531C23#54214yes—
532C23#54614yes—
533C23#54814yes—
534C23#54814yes—
535C23#55014yes—
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23#58514yes—
544C23#58514yes—
545Q8—88no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—816no—
613C2×C4—88no—
614C2×C4—816no—
615C2×C4C2×#12114yes—
616C2×C4C2×#12114yes—
617C22—24no—
618C22—28no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—24no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—832no—
631C22—82no—
632C22—84no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—82no—
637C22#242618yes—
638C22C22×#12111yescharacteristic
639C22C2×#52911yescharacteristic
640C22#344912yes—
641C22C2×#54211yescharacteristic
642C22C2×#54611yescharacteristic
643C22C2×#54811yescharacteristic
644C22C2×#54811yescharacteristic
645C22C2×#55011yescharacteristic
646C22#348512yes—
647C22#348512yes—
648C22#349112yes—
649C22#349114yes—
650C22#350012yes—
651C22#351012yes—
652C22#352114yes—
653C22#353314yes—
654C22#356414yes—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—84no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—82no—
668C2—28no—
669C2—84no—
670C2C2×#242611yescharacteristic
671C2#3532711yescharacteristic
672C2#3544711yescharacteristic
673C2#3601312yes—
674C2#3784212yes—
675C2#3911712yes—
676C2#4139112yes—
677C2#4349312yes—
678C2#4445112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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
4. C2×#2368C2semidirect
5. #16472C2semidirect
6. #15886C2semidirect
7. #15504C2semidirect
8. #14935C2semidirect
10. #14854C2semidirect
11. #13992C2semidirect
12. #13908C2semidirect
13. #13904C2semidirect
22. C2×#400C22semidirect
31. C2×#391C22semidirect
32. C2×#391C22semidirect
33. C2×#387C22semidirect
34. C2×#387C22semidirect
37. C2×#382C22semidirect
47. C2×C4×C4⋊C4C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1443C22semidirect
56. #1443C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
72. #1436C22semidirect
73. #1436C22semidirect
74. C4⋊#95C22semidirect
75. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#26262
C22×#3972
C2×#23682
#164722
#158862
#155042
#149352
#149182
#148542
#139922
#139082
#139042
#139032
#137752