Finite Group Databasebeta

← #4417814

Subgroups — #4417814

Order 512.

2440 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3#16980C212yesmaximal, complemented
4#16891C211yesmaximal, characteristic, complemented
5#16890C211yesmaximal, characteristic, complemented
6#16888C211yesmaximal, characteristic, complemented
7#16887C211yesmaximal, characteristic, complemented
8#16885C214yesmaximal, complemented
9#16805C212yesmaximal, complemented
10#15252C214yesmaximal, complemented
11#15024C212yesmaximal, complemented
12#14932C212yesmaximal, complemented
13#14842C212yesmaximal, complemented
14#14765C212yesmaximal, complemented
15#14660C212yesmaximal, complemented
16#14659C212yesmaximal
17C4×#380C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2855—28no—
24#2850—28no—
25C2×#397C2212yes—
26C2×#393C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#388C2214yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#382C2211yescharacteristic
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1446C2212yes—
48#1446C2212yes—
49#1446C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1438C2212yes—
74#1438C2212yes—
75#1438C2214yes—
76#1438C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2211yescharacteristic
82#1434C2211yescharacteristic
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#124—24no—
93C2×#124—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—44no—
98C2×#121—24no—
99C2×#121—28no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42—44no—
124C22×C42D418yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#397—44no—
131#393—44no—
132#391—24no—
133#391—24no—
134#391—24no—
135#390—24no—
136#389—24no—
137#389—24no—
138#388—28no—
139#388—28no—
140#388—48no—
141#387—28no—
142#387—28no—
143#386—28no—
144#386—28no—
145#385—28no—
146#382—24no—
147#381—28no—
148#380—24no—
149#380—28no—
150C4×C4⋊C4—44no—
151C4×C22⋊C4—44no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—44no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4#4118yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#124—416no—
231C22⋊Q8—416no—
232C22⋊Q8—416no—
233#121—416no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
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⋊C4—44no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—416no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
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—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24C22×D412yes—
350C24C2×#4114yes—
351C24C2511yescharacteristic
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—88no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—44no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
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—28no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
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—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4#124116yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C4⋊C4—44no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C42—28no—
482C42—28no—
483C42—44no—
484C42—44no—
485C42—88no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23C2×#12414yes—
526C23C23×D411yescharacteristic
527C23C22×#4112yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23#58512yes—
532C23#58512yes—
533C23#58512yes—
534C23#58512yes—
535Q8—88no—
536Q8—88no—
537D4—88no—
538D4—816no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
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—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
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—44no—
613C2×C4C2×#12414yes—
614C2×C4C2×#12414yes—
615C2×C4#53414yes—
616C2×C4#53714yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—48no—
625C22—48no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22C22×#12411yescharacteristic
638C22C2×#53411yescharacteristic
639C22C2×#53711yescharacteristic
640C22#344911yescharacteristic
641C22#344911yescharacteristic
642C22#344911yescharacteristic
643C22#344911yescharacteristic
644C22#350514yes—
645C22#350612yes—
646C22#350712yes—
647C22#351014yes—
648C22#352914yes—
649C22#353412yes—
650C22#353512yes—
651C22#356212yes—
652C22#356512yes—
653C22#356614yes—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—84no—
668C4—84no—
669C2—44no—
670C2—162no—
671C2#3501911yescharacteristic
672C2#3518011yescharacteristic
673C2#3533811yescharacteristic
674C2#3665211yescharacteristic
675C2#3745211yescharacteristic
676C2#3755311yescharacteristic
677C2#3755811yescharacteristic
678C2#4552014yes—
679C2#4552114yes—
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. #16980C2semidirect
4. #16891C2semidirect
5. #16890C2semidirect
6. #16888C2semidirect
7. #16887C2semidirect
8. #16885C2semidirect
9. #16805C2semidirect
10. #15252C2semidirect
11. #15024C2semidirect
12. #14932C2semidirect
13. #14842C2semidirect
14. #14765C2semidirect
15. #14660C2semidirect
17. C4×#380C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#28502
#169802
#168912
#168902
#168882
#168872
#168852
#168052
#152522
#150242
#149322
#148422
#147652
#146602
#146592
C4×#3802