Finite Group Databasebeta

← #559845

Subgroups — #559845

Order 512.

3862 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#2377C212yesmaximal
3#14685C211yesmaximal, characteristic, complemented
4#14684C211yesmaximal, characteristic, complemented
5#14660C212yesmaximal, complemented
6#14505C214yesmaximal, complemented
7#14475C214yesmaximal, complemented
8#14474C212yesmaximal, complemented
9#14473C212yesmaximal, complemented
10#13962C212yesmaximal, complemented
11#13943C212yesmaximal, complemented
12#13927C212yesmaximal, complemented
13#13916C212yesmaximal, complemented
14#13894C212yesmaximal, complemented
15#13894C212yesmaximal, complemented
16C2×#529—216no—
17C2×#529C4116yescomplemented
18C22×#126C2212yes—
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×#117C2212yes—
28C23×C42C2211yescharacteristic
29#2377—216no—
30C2×#390C2214yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#381C2212yes—
37C2×#381C2211yescharacteristic
38C2×#381C2212yes—
39C2×#381C2211yescharacteristic
40C2×#380C2212yes—
41C22×#95C2212yes—
42#1446C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2212yes—
57C4⋊#95C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60#1434C2212yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2211yescharacteristic
65#1434C2212yes—
66#1434C2211yescharacteristic
67#1434C2212yes—
68#1434C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C24×C4C2311yescharacteristic
86#529—2128no—
87C2×#126—216no—
88C2×#126C2×C4116yescomplemented
89C2×#125—28no—
90C2×#125C2×C418yescomplemented
91C2×#124—28no—
92C2×#124C2×C418yes—
93C2×#123—216no—
94C2×#123C2×C4116yescomplemented
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8C2×C418yescomplemented
97C2×#121C2×C418yes—
98C2×#121D4116yes—
99C2×C4×Q8—28no—
100C2×C4×Q8C2×C418yescomplemented
101C2×C4×D4—216no—
102C2×C4×D4—28no—
103C2×C4×D4C2×C418yes—
104C2×C4×D4C2×C4116yescomplemented
105C2×#117—216no—
106C2×#117C2×C4116yescomplemented
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42—28no—
127C22×C42C2×C418yescomplemented
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2314yes—
134#390—216no—
135#388—216no—
136#387—216no—
137#386—216no—
138#386—216no—
139#385—216no—
140#381—28no—
141#381—42no—
142#381—44no—
143#381—42no—
144#380—28no—
145C2×#95—44no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C25C2412yes—
174C22×Q8—24no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8C22×C414yes—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4C22×C414yes—
184C22×D4C22×C418yes—
185C22×D4C2×D4116yes—
186C22×D4C2×D418yes—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C22×C418yes—
208C23×C4C22×C414yes—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C2×D418yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#126—264no—
226#125—232no—
227#124—232no—
228#123—264no—
229C22⋊Q8—232no—
230#121C22⋊C4164yes—
231C4×Q8—232no—
232C4×D4—232no—
233C4×D4—264no—
234#117—264no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4C2×D418yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C2×D4116yes—
281C2×C42—232no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42C22×C414yes—
291C2×C42C22×C418yes—
292C2×C42C22×C414yes—
293C2×C42C22×C414yes—
294C2×C42C22×C414yes—
295#95—28no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—42no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24C23×C414yes—
338C24C23×C412yes—
339C24C22×D418yes—
340C24C22×D414yes—
341C24C2511yescharacteristic
342C2×Q8—28no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—416no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4C2×C22⋊C4132yes—
354C2×D4C2×C22⋊C4116yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—216no—
386C22×C4—28no—
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—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
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—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4C2×C22⋊C4116yes—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C23×C414yes—
443C22×C4C23×C412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D418yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C4⋊C4C2×C22⋊C4116yes—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—84no—
465C22⋊C4—88no—
466C22⋊C4—84no—
467C22⋊C4C2×C22⋊C4132yes—
468C42—28no—
469C42—28no—
470C42—28no—
471C42—28no—
472C42—216no—
473C42—88no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—28no—
494C23—24no—
495C23—28no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23C22×C22⋊C418yes—
513C23C22×C22⋊C414yes—
514C23C24×C411yescharacteristic
515C23C23×D412yes—
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23#58512yes—
519C23#58512yes—
520C23#58512yes—
521C23#58512yes—
522C23#58512yes—
523Q8—416no—
524D4—416no—
525D4—432no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—416no—
556C2×C4—416no—
557C2×C4—432no—
558C2×C4—416no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
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—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
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—832no—
596C2×C4C22×C22⋊C414yes—
597C2×C4C22×C22⋊C414yes—
598C2×C4C22×C22⋊C418yes—
599C2×C4#52014yes—
600C22—24no—
601C22—24no—
602C22—28no—
603C22—28no—
604C22—24no—
605C22—28no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22C23×C22⋊C411yescharacteristic
621C22C2×#52011yescharacteristic
622C22#342811yescharacteristic
623C22#342811yescharacteristic
624C22#342811yescharacteristic
625C22#342811yescharacteristic
626C22#342811yescharacteristic
627C22#348514yes—
628C22#348514yes—
629C22#348812yes—
630C22#348912yes—
631C22#352912yes—
632C22#353112yes—
633C22#356212yes—
634C22#356512yes—
635C22#356614yes—
636C22#356614yes—
637C4—24no—
638C4—24no—
639C4—28no—
640C4—24no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—44no—
646C4—84no—
647C4—84no—
648C4—84no—
649C4—84no—
650C2—44no—
651C2—48no—
652C2#3417211yescharacteristic
653C2#3417211yescharacteristic
654C2#3417511yescharacteristic
655C2#3435011yescharacteristic
656C2#3438511yescharacteristic
657C2#3438711yescharacteristic
658C2#3439511yescharacteristic
659C2#4576414yes—
660C2#4576514yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14685C2semidirect
4. #14684C2semidirect
5. #14660C2semidirect
6. #14505C2semidirect
7. #14475C2semidirect
8. #14474C2semidirect
9. #14473C2semidirect
10. #13962C2semidirect
11. #13943C2semidirect
12. #13927C2semidirect
13. #13916C2semidirect
14. #13894C2semidirect
15. #13894C2semidirect
17. C2×#529C4semidirect
88. C2×#126C2×C4semidirect
90. C2×#125C2×C4semidirect
94. C2×#123C2×C4semidirect
96. C2×C22⋊Q8C2×C4semidirect
100. C2×C4×Q8C2×C4semidirect
104. C2×C4×D4C2×C4semidirect
106. C2×#117C2×C4semidirect
127. C22×C42C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#23772
#146852
#146842
#146602
#145052
#144752
#144742
#144732
#139622
#139432
#139272
#139162
#138942