Finite Group Databasebeta

← #4866520

Subgroups — #4866520

Order 512.

2596 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic, complemented
2C2×#2619C211yesmaximal, characteristic
3#17090C214yesmaximal, complemented
4#17048C212yesmaximal, complemented
5#16684C212yesmaximal, complemented
6#16664C212yesmaximal, complemented
7#16652C212yesmaximal, complemented
8#16535C214yesmaximal, complemented
9#16503C212yesmaximal, complemented
10#16253C212yesmaximal, complemented
11#15833C212yesmaximal, complemented
12#15643C212yesmaximal, complemented
13#14996C212yesmaximal, complemented
14#14813C212yesmaximal, complemented
15#13938C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2212yes—
19C22×#117C2211yescharacteristic
20#2619—28no—
21#2499—28no—
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#393C2214yes—
31C2×#392C2212yes—
32C2×#392C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#390C2212yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#384C2211yescharacteristic
41C2×#383C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1445C2214yes—
47#1444C2212yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2212yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1438C2214yes—
64#1438C2212yes—
65#1438C2214yes—
66#1438C2212yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2212yes—
73#1437C2214yes—
74#1436C2212yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2314yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C24×C4C2311yescharacteristic
83C2×#127—24no—
84C2×#127—42no—
85C2×#127—42no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—42no—
88C2×C22⋊Q8—42no—
89C2×#121—28no—
90C2×#121—48no—
91C2×#117—42no—
92C2×#117—42no—
93C2×#117D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#397—28no—
125#397—28no—
126#394—28no—
127#394—28no—
128#394—48no—
129#394—44no—
130#394—44no—
131#394—48no—
132#393—48no—
133#392—28no—
134#392—28no—
135#391—24no—
136#391—24no—
137#390—28no—
138#388—48no—
139#388—48no—
140#387—28no—
141#387—28no—
142#384—24no—
143#383—42no—
144#382—24no—
145#382—24no—
146C4×C4⋊C4—42no—
147C4×C22⋊C4—24no—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C25C2414yes—
172C25C2411yescharacteristic
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—42no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4C2×D414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218#127—416no—
219C22⋊Q8—416no—
220#121—432no—
221#117—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
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×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×D414yes—
294C2×C22⋊C4C2×D414yes—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
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—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24C22×D412yes—
353C24C2511yescharacteristic
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—84no—
358C2×Q8—84no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—416no—
364C2×D4—48no—
365C2×D4—88no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—88no—
374C2×D4—84no—
375C2×D4—88no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
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—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4H(5,𝔽2)14yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)14yes—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—28no—
483C42—28no—
484C42—44no—
485C42—44no—
486C42—84no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
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—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23C23×D411yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)12yes—
543C23#58512yes—
544C23#58512yes—
545C23#58514yes—
546Q8—88no—
547D4—816no—
548D4—816no—
549D4—816no—
550D4—832no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—416no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
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—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4#53414yes—
622C2×C4#53514yes—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22C2×#53411yescharacteristic
647C22C2×#53511yescharacteristic
648C22#344911yescharacteristic
649C22#348812yes—
650C22#350012yes—
651C22#353314yes—
652C22#353412yes—
653C22#353412yes—
654C22#354012yes—
655C22#354014yes—
656C22#354614yes—
657C22#354612yes—
658C22#354911yescharacteristic
659C22#355111yescharacteristic
660C22#355112yes—
661C22#356714yes—
662C4—24no—
663C4—24no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—84no—
675C2—82no—
676C2—164no—
677C2#3727311yescharacteristic
678C2#3741911yescharacteristic
679C2#3744011yescharacteristic
680C2#4194412yes—
681C2#4298712yes—
682C2#5560412yes—
683C2#5658612yes—
684C2#5673614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2619C2
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×#2499C2semidirect
3. #17090C2semidirect
4. #17048C2semidirect
5. #16684C2semidirect
6. #16664C2semidirect
7. #16652C2semidirect
8. #16535C2semidirect
9. #16503C2semidirect
10. #16253C2semidirect
11. #15833C2semidirect
12. #15643C2semidirect
13. #14996C2semidirect
14. #14813C2semidirect
15. #13938C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#26192
#170902
#170482
#166842
#166642
#166522
#165352
#165032
#162532
#158332
#156432
#149962
#148132
#139382