Finite Group Databasebeta

← #4416075

Subgroups — #4416075

Order 512.

2660 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic
3#17041C214yesmaximal, complemented
4#17008C211yesmaximal, characteristic, complemented
5#16989C211yesmaximal, characteristic, complemented
6#16978C212yesmaximal, complemented
7#16887C211yesmaximal, characteristic, complemented
8#16811C211yesmaximal, characteristic, complemented
9#16803C212yesmaximal, complemented
10#14927C214yesmaximal, complemented
11#14836C212yesmaximal, complemented
12#14766C212yesmaximal, complemented
13#14759C212yesmaximal, complemented
14#14747C212yesmaximal
15#14631C212yesmaximal, complemented
16#14624C212yesmaximal, complemented
17C2×#1434C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2855—28no—
23#2854—28no—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#394C2212yes—
27C2×#393C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#385C2212yes—
44C2×#381C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2214yes—
48C22×#95C2212yes—
49C22×#95C2212yes—
50C22×#95C2212yes—
51#1446C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1438C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2211yescharacteristic
82#1434C2211yescharacteristic
83#1434C2212yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#123—28no—
93C2×#123—48no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—44no—
96C2×#121—24no—
97C2×#121—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42—44no—
124C22×C42D418yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#399—44no—
130#398—28no—
131#394—28no—
132#393—44no—
133#391—24no—
134#391—24no—
135#390—24no—
136#390—24no—
137#389—24no—
138#389—24no—
139#389—24no—
140#389—24no—
141#388—28no—
142#388—28no—
143#387—28no—
144#387—28no—
145#386—28no—
146#386—48no—
147#386—48no—
148#385—28no—
149#381—28no—
150#380—24no—
151#380—24no—
152#380—48no—
153C2×#95—28no—
154C2×#95—44no—
155C2×#95—44no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×D4—24no—
193C22×D4—28no—
194C22×D4—28no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—42no—
200C22×D4—42no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#123—432no—
237C22⋊Q8—416no—
238#121—416no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
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⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C42—416no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95#4118yes—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24C22×D412yes—
354C24H(5,𝔽2)14yes—
355C24H(5,𝔽2)14yes—
356C242−1+414yes—
357C24C2511yescharacteristic
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—84no—
362C2×Q8—84no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—44no—
371C2×D4—88no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
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—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+412yes—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—44no—
495C42—44no—
496C42—44no—
497C42—44no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#53414yes—
535C23#53514yes—
536C23#53614yes—
537C23#562116yes—
538C23C23×D411yescharacteristic
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×2−1+412yes—
544C23C2×2−1+411yescharacteristic
545C23#58512yes—
546C23#58512yes—
547Q8—88no—
548D4—88no—
549D4—816no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—416no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—832no—
629C2×C4#53214yes—
630C2×C4#53714yes—
631C22—216no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—216no—
637C22—24no—
638C22—24no—
639C22—84no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—84no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22#281618yes—
651C22C2×#53211yescharacteristic
652C22C2×#53411yescharacteristic
653C22C2×#53511yescharacteristic
654C22C2×#53611yescharacteristic
655C22C2×#53711yescharacteristic
656C22#344911yescharacteristic
657C22#344911yescharacteristic
658C22#349514yes—
659C22#351314yes—
660C22#351912yes—
661C22#352212yes—
662C22C2×#56214yes—
663C22#353412yes—
664C22#353512yes—
665C22#356014yes—
666C22#356112yes—
667C22#356312yes—
668C4—24no—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—48no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—82no—
683C2—28no—
684C2—162no—
685C2#3506411yescharacteristic
686C2#3523811yescharacteristic
687C2#3532611yescharacteristic
688C2C2×#281611yescharacteristic
689C2#3745211yescharacteristic
690C2#3755511yescharacteristic
691C2#3755611yescharacteristic
692C2#4554814yes—
693C2#4558014yes—
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×#2855C2semidirect
3. #17041C2semidirect
4. #17008C2semidirect
5. #16989C2semidirect
6. #16978C2semidirect
7. #16887C2semidirect
8. #16811C2semidirect
9. #16803C2semidirect
10. #14927C2semidirect
11. #14836C2semidirect
12. #14766C2semidirect
13. #14759C2semidirect
15. #14631C2semidirect
16. #14624C2semidirect
17. C2×#1434C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#28542
#170412
#170082
#169892
#169782
#168872
#168112
#168032
#149272
#148362
#147662
#147592
#147472
#146312
#146242
C2×#14342