Finite Group Databasebeta

← #626069

Subgroups — #626069

Order 512.

6094 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#537C211yesmaximal, characteristic, complemented
2C2×#2404C212yesmaximal, complemented
3C2×#2617C212yesmaximal, complemented
4C22×#388C211yesmaximal, characteristic, complemented
5#15613C218yesmaximal, complemented
6#15320C218yesmaximal, complemented
7#14913C212yesmaximal, complemented
8#14716C212yesmaximal, complemented
9#14698C212yesmaximal, complemented
10#14339C211yesmaximal, characteristic, complemented
11#13956C212yesmaximal, complemented
12C23×#41C2212yes—
13C2×#537—216no—
14C2×#537—28no—
15C22×C22⋊Q8C2214yescomplemented
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×#121C2214yescomplemented
19C22×#120C2214yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25#2617—216no—
26#2404—216no—
27C2×#397C2214yes—
28C2×#394C2218yes—
29C2×#393C2214yes—
30C2×#392C2214yes—
31C2×#392C2214yes—
32C2×#392C2214yes—
33C2×#390C2214yescomplemented
34C2×#388—28no—
35C2×#388C2218yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#386C2218yes—
39C2×#386C2214yes—
40C2×#381C2212yes—
41C2×#380C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1446C2214yescomplemented
45#1442C2214yescomplemented
46#1441C2218yescomplemented
47#1440C2218yescomplemented
48#1439C2214yes—
49#1438C2218yes—
50#1438C2214yescomplemented
51#1438C2214yescomplemented
52#1437C2218yescomplemented
53C4⋊#95C2214yescomplemented
54C22⋊#95C2212yes—
55C22⋊#95C2218yescomplemented
56C26C2311yescharacteristic
57C22×#41—216no—
58C22×#41—28no—
59C22×#41—28no—
60C22×#41—28no—
61C22×#41—48no—
62C23×Q8C2312yes—
63C23×D4C2314yes—
64C23×D4C2312yes—
65C23×D4C2312yes—
66C23×D4C2318yes—
67C23×D4C2312yes—
68C24×C4C2314yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75#537—2128no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×#121—216no—
80C2×#121—216no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—216no—
85C2×#121—216no—
86C2×#121D4116yescomplemented
87C2×#120—216no—
88C2×#120—216no—
89C2×#120—48no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4D418yescomplemented
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4D418yescomplemented
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2318yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C42—24no—
114C22×C42—44no—
115C22×C42C2314yes—
116#397—216no—
117#394—232no—
118#393—216no—
119#392—216no—
120#392—216no—
121#392—48no—
122#390—48no—
123#388—232no—
124#386—216no—
125#386—216no—
126#386—232no—
127#386—48no—
128#381—28no—
129#380—28no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2318yes—
142C25—22no—
143C25—22no—
144C25—24no—
145C25—24no—
146C25—42no—
147C25—42no—
148C25—42no—
149C25—42no—
150C25C2×D414yes—
151C25C2412yes—
152C25C2414yes—
153C25C2411yescharacteristic
154C2×#41—264no—
155C2×#41—464no—
156C2×#41—464no—
157C2×#41—432no—
158C2×#41—432no—
159C22×Q8—28no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—44no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—216no—
173C22×D4—216no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—44no—
181C22×D4—48no—
182C22×D4—44no—
183C22×D4—48no—
184C22×D4—48no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—48no—
188C22×D4—44no—
189C22×D4—48no—
190C22×D4C2×D4116yes—
191C22×D4C2×D418yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—28no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4C2×D414yes—
224C23×C4#4114yes—
225C23×C4C2412yes—
226C23×C4C2418yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C22⋊Q8—2128no—
234#121—264no—
235#121—2128no—
236#121—264no—
237#120—464no—
238C2×C4⋊C4—232no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D4116yes—
249C2×C4⋊C4C2×D418yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—232no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—416no—
274C2×C22⋊C4—416no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4C2×D4116yes—
282C2×C22⋊C4C2×D4116yes—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—44no—
286C2×C42—416no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—216no—
293#95—216no—
294#95—44no—
295#95—48no—
296#95—44no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—28no—
302C24—22no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—24no—
309C24—28no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—28no—
315C24—28no—
316C24—28no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—216no—
321C24—48no—
322C24—48no—
323C24—44no—
324C24—48no—
325C24—44no—
326C24—44no—
327C24—44no—
328C24—48no—
329C24—44no—
330C24—44no—
331C24—48no—
332C24—88no—
333C24—88no—
334C24—88no—
335C24—88no—
336C24C22×D414yes—
337C24C22×D418yes—
338C24C22×D412yes—
339C24C22×D414yes—
340C24C22×D412yes—
341C24C2×#4112yes—
342C24H(5,𝔽2)14yes—
343C24C2511yescharacteristic
344#41—4256no—
345C2×Q8—216no—
346C2×Q8—416no—
347C2×Q8—416no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×D4—216no—
351C2×D4—232no—
352C2×D4—232no—
353C2×D4—232no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—432no—
365C2×D4—432no—
366C2×D4—48no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—88no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—816no—
373C2×D4—816no—
374C2×D4—816no—
375C2×D4—816no—
376C2×D4—816no—
377C2×D4—816no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—232no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—416no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4#120116yes—
442C22×C4#121116yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D418yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)14yes—
453C4⋊C4—216no—
454C4⋊C4—232no—
455C4⋊C4—216no—
456C4⋊C4—88no—
457C22⋊C4—232no—
458C22⋊C4—232no—
459C22⋊C4—232no—
460C22⋊C4—416no—
461C22⋊C4—432no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—816no—
465C22⋊C4—816no—
466C22⋊C4—816no—
467C42—416no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—28no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—216no—
487C23—28no—
488C23—216no—
489C23—416no—
490C23—416no—
491C23—42no—
492C23—416no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—48no—
497C23—48no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—832no—
524C23—88no—
525C23—88no—
526C23—88no—
527C23—88no—
528C23—832no—
529C23C2×#12014yes—
530C23C2×#12114yes—
531C23#53514yes—
532C23D4218yes—
533C23#54618yes—
534C23C23×D411yescharacteristic
535C23C23×D412yes—
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)12yes—
540C23#58512yes—
541C23#58512yes—
542C23#58512yes—
543Q8—432no—
544D4—432no—
545D4—464no—
546D4—816no—
547D4—816no—
548D4—864no—
549C2×C4—24no—
550C2×C4—232no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—416no—
569C2×C4—464no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4C2×#12018yes—
597C2×C4C2×#12118yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—216no—
605C22—216no—
606C22—48no—
607C22—44no—
608C22—48no—
609C22—44no—
610C22—48no—
611C22—44no—
612C22—416no—
613C22—416no—
614C22—416no—
615C22—832no—
616C22—82no—
617C22—84no—
618C22—82no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—88no—
625C22—88no—
626C22—88no—
627C22#240818yes—
628C22C22×#12011yescharacteristic
629C22C22×#12111yescharacteristic
630C22C2×#53511yescharacteristic
631C22#344912yes—
632C22#346012yes—
633C22C2×D4212yes—
634C22C2×#54612yes—
635C22#348411yescharacteristic
636C22#348511yescharacteristic
637C22#348711yescharacteristic
638C22#348711yescharacteristic
639C22#349312yes—
640C22#349614yes—
641C22#349612yes—
642C22#350014yes—
643C22#353414yes—
644C22#356014yes—
645C4—28no—
646C4—28no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—48no—
651C4—82no—
652C4—88no—
653C2—28no—
654C2—88no—
655C2—84no—
656C2C2×#240811yescharacteristic
657C2#3515011yescharacteristic
658C2#3516811yescharacteristic
659C2#3534711yescharacteristic
660C2#3567611yescharacteristic
661C2#3671011yescharacteristic
662C2#3696111yescharacteristic
663C2#3935414yes—
664C2#3941514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#537C2semidirect
2. C2×#2404C2semidirect
3. C2×#2617C2semidirect
4. C22×#388C2semidirect
5. #15613C2semidirect
6. #15320C2semidirect
7. #14913C2semidirect
8. #14716C2semidirect
9. #14698C2semidirect
10. #14339C2semidirect
11. #13956C2semidirect
15. C22×C22⋊Q8C22semidirect
18. C22×#121C22semidirect
33. C2×#390C22semidirect
44. #1446C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1440C22semidirect
50. #1438C22semidirect
51. #1438C22semidirect
52. #1437C22semidirect
53. C4⋊#95C22semidirect
55. C22⋊#95C22semidirect
86. C2×#121D4semidirect
91. C22×C4⋊C4D4semidirect
102. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5372
C2×#24042
C2×#26172
C22×#3882
#156132
#153202
#149132
#147162
#146982
#143392
#139562