Finite Group Databasebeta

← #766203

Subgroups — #766203

Order 512.

3500 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic, complemented
2C2×#2321C211yesmaximal, characteristic
3#15828C214yesmaximal, complemented
4#15432C214yesmaximal, complemented
5#15334C214yesmaximal, complemented
6#15327C214yesmaximal, complemented
7#15092C211yesmaximal, characteristic, complemented
8#15091C211yesmaximal, characteristic, complemented
9#15026C211yesmaximal, characteristic, complemented
10#14946C211yesmaximal, characteristic, complemented
11#14134C212yesmaximal
12#14039C212yesmaximal, complemented
13#14015C212yesmaximal, complemented
14#13877C212yesmaximal, complemented
15#13794C211yesmaximal, characteristic, complemented
16C22×#120C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C2×#399—28no—
24C2×#399C2214yes—
25C2×#399C2214yes—
26#2321—28no—
27C2×#390C2212yes—
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#387C2214yescomplemented
31C2×#386C2211yescharacteristic
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#385C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yescomplemented
40C2×#380C2214yes—
41C2×#379C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1444C2214yescomplemented
47#1443C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1438C2214yes—
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C26C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2314yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#120—24no—
83C2×#120—24no—
84C2×#120Q818yes—
85C2×C4×D4—24no—
86C2×C4×D4—44no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—48no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4D418yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#399—216no—
125#399—216no—
126#390—28no—
127#388—28no—
128#388—28no—
129#387—48no—
130#386—24no—
131#386—28no—
132#386—28no—
133#386—48no—
134#386—44no—
135#385—24no—
136#385—28no—
137#385—48no—
138#385—44no—
139#380—48no—
140#379—24no—
141C4×C22⋊C4—24no—
142C2×#95—24no—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25—24no—
164C25—24no—
165C25—44no—
166C25—44no—
167C25C2×D414yes—
168C25C2×Q814yes—
169C25#4114yes—
170C25C2411yescharacteristic
171C25C2412yes—
172C25C2411yescharacteristic
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4C2×Q814yes—
182C22×D4C2×Q818yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—28no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221#120—232no—
222C4×D4—416no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—432no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—432no—
270C2×C22⋊C4—416no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—432no—
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⋊C4C2×D418yes—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4C2×Q818yes—
290C2×C22⋊C4C2×Q814yes—
291C2×C22⋊C4#4118yes—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—44no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—28no—
327C24—28no—
328C24—216no—
329C24—216no—
330C24—21no—
331C24—22no—
332C24—21no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—48no—
337C24—48no—
338C24—416no—
339C24—416no—
340C24—88no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24C22⋊Q8116yes—
345C24C22×D414yes—
346C24C22×D412yes—
347C24C22×Q812yes—
348C24C22×Q814yes—
349C24C22×Q812yes—
350C24C2×#4114yes—
351C24C2×#4112yes—
352C24C2×#4114yes—
353C24H(5,𝔽2)14yes—
354C24C2511yescharacteristic
355C2×D4—28no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—416no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—88no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
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—28no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q814yes—
449C22×C4C2×#4118yes—
450C22×C4H(5,𝔽2)18yes—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—44no—
456C4⋊C4—84no—
457C4⋊C4—88no—
458C22⋊C4—216no—
459C22⋊C4—28no—
460C22⋊C4—216no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C22⋊C4—88no—
475C42—44no—
476C42—84no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—232no—
500C23—22no—
501C23—24no—
502C23—28no—
503C23—432no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—816no—
523C23—88no—
524C23—88no—
525C23—832no—
526C23—832no—
527C23C2×C22⋊Q818yes—
528C23C2×C22⋊Q814yes—
529C23#53214yes—
530C23#53414yes—
531C23#55118yes—
532C23#55418yes—
533C23C23×D411yescharacteristic
534C23C23×Q811yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4112yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)14yes—
539C23#58514yes—
540D4—416no—
541D4—816no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—432no—
560C2×C4—416no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
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—88no—
605C2×C4—832no—
606C2×C4—88no—
607C2×C4—816no—
608C22—28no—
609C22—28no—
610C22—24no—
611C22—24no—
612C22—216no—
613C22—216no—
614C22—28no—
615C22—28no—
616C22—28no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—48no—
622C22—48no—
623C22—48no—
624C22—48no—
625C22—42no—
626C22—44no—
627C22—48no—
628C22—832no—
629C22—82no—
630C22—81no—
631C22—81no—
632C22—84no—
633C22#256418yes—
634C22C22×C22⋊Q811yescharacteristic
635C22C2×#53211yescharacteristic
636C22C2×#53411yescharacteristic
637C22C2×#55112yes—
638C22C2×#55412yes—
639C22#348412yes—
640C22#348512yes—
641C22#350114yes—
642C22#350514yes—
643C22#351314yes—
644C22#351314yes—
645C22#353314yes—
646C22#354314yes—
647C4—42no—
648C4—44no—
649C4—48no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—48no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4—84no—
658C4—84no—
659C2—28no—
660C2—48no—
661C2—82no—
662C2C2×#256411yescharacteristic
663C2#3534912yes—
664C2#3539412yes—
665C2#3695511yescharacteristic
666C2#3723211yescharacteristic
667C2#4312114yes—
668C2#4314714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#399C2semidirect
3. #15828C2semidirect
4. #15432C2semidirect
5. #15334C2semidirect
6. #15327C2semidirect
7. #15092C2semidirect
8. #15091C2semidirect
9. #15026C2semidirect
10. #14946C2semidirect
12. #14039C2semidirect
13. #14015C2semidirect
14. #13877C2semidirect
15. #13794C2semidirect
30. C2×#387C22semidirect
39. C2×#385C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C2×#23212
#158282
#154322
#153342
#153272
#150922
#150912
#150262
#149462
#141342
#140392
#140152
#138772
#137942