Finite Group Databasebeta

← #559981

Subgroups — #559981

Order 512.

2960 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2#14683C212yesmaximal, complemented
3#14666C212yesmaximal, complemented
4#14516C212yesmaximal, complemented
5#14513C212yesmaximal, complemented
6#14432C214yesmaximal, complemented
7#14431C214yesmaximal, complemented
8#13956C211yesmaximal, characteristic
9#13944C212yesmaximal, complemented
10#13937C211yesmaximal, characteristic
11#13919C212yesmaximal, complemented
12#13913C212yesmaximal, complemented
13#13901C212yesmaximal, complemented
14#13892C212yesmaximal, complemented
15#13832C212yesmaximal, complemented
16C23×#41C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2297C4116yescomplemented
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#387C2214yes—
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#386C2211yescharacteristic
28C2×#386C2211yescharacteristic
29C2×#386C2211yescharacteristic
30C2×#386C2211yescharacteristic
31C2×#383C2212yes—
32C2×#383C2212yes—
33C2×#381C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C22×#95C2211yescharacteristic
39#1446C2212yes—
40#1445C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2212yes—
51#1438C2212yes—
52#1438C2212yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59#1434C2214yes—
60#1434C2212yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2214yes—
65#1434C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C22×#41—28no—
77C22×#41—42no—
78C22×#41—42no—
79C22×#41—44no—
80C22×#41C2×C418yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×#117—44no—
92C2×#117C2×C418yescomplemented
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C42—42no—
110C22×C42—42no—
111C22×C42C2×C418yescomplemented
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117#391—44no—
118#391C2×C4116yescomplemented
119#390—28no—
120#390C2×C4116yes—
121#387—48no—
122#386—28no—
123#386—44no—
124#386C2×C418yes—
125#386C2×C418yes—
126#386C2×C418yescomplemented
127#386C2×C418yescomplemented
128#383—44no—
129#383—44no—
130#381—42no—
131#380—42no—
132C4×C22⋊C4—42no—
133C4×C22⋊C4C2×C4116yes—
134C4×C22⋊C4C2×C4116yescomplemented
135C2×#95—42no—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C25C2411yescharacteristic
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C2×#41—232no—
168C2×#41—416no—
169C2×#41—416no—
170C2×#41—432no—
171C2×#41—432no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8C22×C414yes—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—42no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—42no—
188C22×D4C22×C414yes—
189C22×D4C22×C414yes—
190C22×D4C22×C414yes—
191C22×D4#4118yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—41no—
209C23×C4—41no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4C22×C414yes—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233#117—416no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22×C418yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C414yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C22×C418yes—
273C2×C22⋊C4C22×C418yes—
274C2×C22⋊C4C22×C418yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—416no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C22×C418yes—
288C2×C42C22×C418yes—
289C2×C42C22×C418yes—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C22×C418yes—
315#95C22×C418yes—
316#95C22×C418yes—
317#95C22×C418yes—
318C24—22no—
319C24—21no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24C23×C412yes—
338C24C23×C412yes—
339C24C23×C412yes—
340C24C2×#4114yes—
341C24C2×#4114yes—
342C24C2511yescharacteristic
343#41—4128no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—44no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4#117116yes—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
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—24no—
421C22×C4—24no—
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—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—84no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—88no—
451C22×C4C23×C412yes—
452C22×C4C23×C414yes—
453C22×C4C23×C414yes—
454C22×C4C23×C414yes—
455C22×C4C23×C414yes—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)14yes—
463C4⋊C4—216no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C42—216no—
480C42—48no—
481C42—48no—
482C42—84no—
483C42—88no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—44no—
529C23—44no—
530C23C2×#11714yes—
531C23C2×#11714yes—
532C23C24×C411yescharacteristic
533C23C22×#4112yes—
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23#58514yes—
538C23#58512yes—
539C23#58512yes—
540Q8—416no—
541D4—416no—
542D4—416no—
543D4—416no—
544C2×C4—232no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—416no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
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×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—816no—
617C2×C4C2×#11714yes—
618C2×C4#51814yes—
619C2×C4#51814yes—
620C22—24no—
621C22—24no—
622C22—28no—
623C22—28no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—48no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—44no—
641C22—84no—
642C22—82no—
643C22—82no—
644C22C22×#11711yescharacteristic
645C22C2×#51811yescharacteristic
646C22C2×#51811yescharacteristic
647C22#342812yes—
648C22#342812yes—
649C22#349012yes—
650C22#349012yes—
651C22#352114yes—
652C22#352114yes—
653C22#352912yes—
654C22#352912yes—
655C22#353912yes—
656C22#354011yescharacteristic
657C22#354011yescharacteristic
658C22#356214yes—
659C22#356214yes—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—84no—
669C4—84no—
670C4—84no—
671C4—84no—
672C2—44no—
673C2—44no—
674C2—82no—
675C2#3417611yescharacteristic
676C2#3419512yes—
677C2#3434211yescharacteristic
678C2#3436111yescharacteristic
679C2#3440212yes—
680C2#4572914yes—
681C2#4573012yes—
682C2#4573112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14683C2semidirect
3. #14666C2semidirect
4. #14516C2semidirect
5. #14513C2semidirect
6. #14432C2semidirect
7. #14431C2semidirect
9. #13944C2semidirect
11. #13919C2semidirect
12. #13913C2semidirect
13. #13901C2semidirect
14. #13892C2semidirect
15. #13832C2semidirect
19. #2297C4semidirect
92. C2×#117C2×C4semidirect
111. C22×C42C2×C4semidirect
118. #391C2×C4semidirect
126. #386C2×C4semidirect
127. #386C2×C4semidirect
134. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
#146832
#146662
#145162
#145132
#144322
#144312
#139562
#139442
#139372
#139192
#139132
#139012
#138922
#138322