Finite Group Databasebeta

← #4868766

Subgroups — #4868766

Order 512.

2606 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic, complemented
2C2×#2543C211yesmaximal, characteristic
3#17229C212yesmaximal, complemented
4#17147C212yesmaximal, complemented
5#16652C212yesmaximal, complemented
6#16629C212yesmaximal, complemented
7#16511C212yesmaximal, complemented
8#16465C214yesmaximal, complemented
9#16398C212yesmaximal, complemented
10#16392C212yesmaximal, complemented
11#16363C212yesmaximal, complemented
12#15673C214yesmaximal, complemented
13#15110C212yesmaximal, complemented
14#15015C212yesmaximal, complemented
15#14344C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×#121C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2547—28no—
22#2543—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2211yescharacteristic
26C2×#397C2211yescharacteristic
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#395C2214yes—
30C2×#394C2211yescharacteristic
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#393C2214yes—
34C2×#392C2211yescharacteristic
35C2×#391C2212yes—
36C2×#388C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#381C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1445C2212yes—
46#1445C2212yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1440C2214yes—
58#1440C2212yes—
59#1440C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—42no—
88C2×#124—28no—
89C2×#124—44no—
90C2×#124—44no—
91C2×#121—28no—
92C2×#121—24no—
93C2×#121—42no—
94C2×#121—42no—
95C2×#121—48no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4D418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#399—44no—
127#399—44no—
128#398—24no—
129#397—24no—
130#397—28no—
131#397—28no—
132#395—48no—
133#394—24no—
134#394—48no—
135#394—48no—
136#393—48no—
137#392—24no—
138#391—28no—
139#388—28no—
140#386—28no—
141#386—28no—
142#381—42no—
143#380—28no—
144#380—28no—
145#380—42no—
146C4×C4⋊C4—24no—
147C2×#95—24no—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C25C2412yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—22no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—44no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4C2×D414yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—416no—
227#124—432no—
228#121—416no—
229#121—432no—
230C4×D4—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
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—42no—
287C2×C22⋊C4—42no—
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⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42C2×D414yes—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—24no—
352C24C22×D412yes—
353C24C22×D412yes—
354C24H(5,𝔽2)14yes—
355C24C2511yescharacteristic
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×Q8—84no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—84no—
375C2×D4—88no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—88no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
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—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
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—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—88no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—28no—
486C42—48no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23#53514yes—
542C23C23×D411yescharacteristic
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×2−1+412yes—
548C23#58514yes—
549C23#58512yes—
550Q8—816no—
551D4—416no—
552D4—816no—
553D4—88no—
554D4—816no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
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×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4—816no—
621C2×C4#53414yes—
622C2×C4#53514yes—
623C22—24no—
624C22—28no—
625C22—42no—
626C22—48no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—84no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22C2×#53411yescharacteristic
651C22C2×#53511yescharacteristic
652C22C2×#53511yescharacteristic
653C22#348512yes—
654C22#348912yes—
655C22#353712yes—
656C22#353712yes—
657C22#354512yes—
658C22#354512yes—
659C22#354712yes—
660C22#355011yescharacteristic
661C22#355111yescharacteristic
662C22#355914yes—
663C22#356014yes—
664C22#356114yes—
665C22#356214yes—
666C4—24no—
667C4—24no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—84no—
674C4—84no—
675C4—84no—
676C4—82no—
677C4—84no—
678C2—44no—
679C2—82no—
680C2—162no—
681C2#3711911yescharacteristic
682C2#3737611yescharacteristic
683C2#3745011yescharacteristic
684C2#4141312yes—
685C2#4268012yes—
686C2#5364212yes—
687C2#5385614yes—
688C2#5389312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2543C2
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×#2547C2semidirect
3. #17229C2semidirect
4. #17147C2semidirect
5. #16652C2semidirect
6. #16629C2semidirect
7. #16511C2semidirect
8. #16465C2semidirect
9. #16398C2semidirect
10. #16392C2semidirect
11. #16363C2semidirect
12. #15673C2semidirect
13. #15110C2semidirect
14. #15015C2semidirect
15. #14344C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25432
#172292
#171472
#166522
#166292
#165112
#164652
#163982
#163922
#163632
#156732
#151102
#150152
#143442