Finite Group Databasebeta

← #626902

Subgroups — #626902

Order 512.

3282 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic
3#15555C214yesmaximal, complemented
4#15545C214yesmaximal, complemented
5#15392C214yesmaximal, complemented
6#15378C214yesmaximal, complemented
7#14766C214yesmaximal, complemented
8#14702C211yesmaximal, characteristic, complemented
9#14686C211yesmaximal, characteristic, complemented
10#14353C211yesmaximal, characteristic, complemented
11#14281C211yesmaximal, characteristic, complemented
12#13877C214yesmaximal
13C2×#1437C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2212yes—
18C23×C42C2211yescharacteristic
19#2626—28no—
20C2×#399C2212yes—
21C2×#398C2214yescomplemented
22C2×#396C2214yes—
23C2×#393C2212yes—
24#2315—28no—
25C2×#391C2212yescomplemented
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#385C2212yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#384C2212yescomplemented
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2214yes—
37C2×#380C2214yes—
38C2×#379C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41#1446C2212yescomplemented
42#1445C2214yescomplemented
43#1445C2212yescomplemented
44#1444C2214yescomplemented
45#1444C2214yescomplemented
46#1444C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1437C2214yes—
58#1437C2212yes—
59#1437C2212yes—
60#1436C2212yescomplemented
61#1436C2212yescomplemented
62C4⋊#95C2214yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—48no—
96C22×C22⋊C4D418yes—
97C22×C22⋊C4Q8116yescomplemented
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42—24no—
110C22×C42—44no—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#399—28no—
115#398—48no—
116#396—216no—
117#393—28no—
118#391—44no—
119#385—28no—
120#385—216no—
121#385—216no—
122#385—48no—
123#385—48no—
124#384—44no—
125#380—28no—
126#380—24no—
127#380—24no—
128#380—24no—
129#380—48no—
130#380—48no—
131#379—28no—
132C2×#95—24no—
133C2×#95—44no—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25—22no—
156C25—22no—
157C25—24no—
158C25—42no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25C2×D414yes—
163C25C2×Q818yes—
164C25C2412yes—
165C25C2411yescharacteristic
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—42no—
170C22×Q8—42no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—28no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2×Q818yes—
199C23×C4C2×Q818yes—
200C23×C4#4114yes—
201C23×C4#4114yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C22⋊Q8—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C22⋊C4—232no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—432no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—432no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4C2×D418yes—
277C2×C22⋊C4C2×D418yes—
278C2×C22⋊C4C2×Q8116yes—
279C2×C22⋊C4C2×Q8116yes—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—416no—
286C2×C42—42no—
287C2×C42—42no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—28no—
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—44no—
307#95—44no—
308#95—44no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—28no—
321C24—28no—
322C24—216no—
323C24—216no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—82no—
331C24—82no—
332C24—84no—
333C24—84no—
334C24—82no—
335C24—82no—
336C24—84no—
337C24—84no—
338C24—84no—
339C24—84no—
340C24C4⋊Q8116yes—
341C24C22×D414yes—
342C24C22×D412yes—
343C24C22×Q818yes—
344C24C22×Q814yes—
345C24C2×#4112yes—
346C24C2×#4112yes—
347C24H(5,𝔽2)14yes—
348C24C2511yescharacteristic
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—82no—
352C2×Q8—82no—
353C2×Q8—84no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—216no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
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—44no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—416no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—82no—
429C22×C4—84no—
430C22×C4—82no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4#123116yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×Q818yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q814yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C4H(5,𝔽2)14yes—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—232no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—28no—
475C42—48no—
476C42—44no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—232no—
504C23—432no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—88no—
516C23—816no—
517C23—816no—
518C23—816no—
519C23—88no—
520C23—816no—
521C23—816no—
522C23C2×#12314yes—
523C23C2×C4⋊Q818yes—
524C23C2×C4⋊Q814yes—
525C23#53514yes—
526C23#55418yes—
527C23#55418yes—
528C23C23×D411yescharacteristic
529C23C23×Q812yes—
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)12yes—
535C23#58512yes—
536C23#58512yes—
537Q8—88no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—416no—
561C2×C4—432no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
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—42no—
591C2×C4—42no—
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—816no—
605C2×C4—88no—
606C2×C4—832no—
607C2×C4C2×#12314yes—
608C2×C4C2×#12314yes—
609C2×C4#54318yes—
610C22—24no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—28no—
615C22—24no—
616C22—28no—
617C22—28no—
618C22—216no—
619C22—28no—
620C22—44no—
621C22—48no—
622C22—48no—
623C22—48no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—832no—
628C22#250218yes—
629C22C22×#12311yescharacteristic
630C22C22×C4⋊Q811yescharacteristic
631C22C2×#53511yescharacteristic
632C22C2×#54312yes—
633C22#345812yes—
634C22C2×#55412yes—
635C22C2×#55412yes—
636C22#349011yescharacteristic
637C22#349111yescharacteristic
638C22#349412yes—
639C22#349612yes—
640C22#349612yes—
641C22#350112yes—
642C22#350112yes—
643C22#350512yes—
644C22#350811yescharacteristic
645C22#351011yescharacteristic
646C22#353412yes—
647C22#353712yes—
648C22#355812yes—
649C22#356012yes—
650C4—24no—
651C4—24no—
652C4—28no—
653C4—48no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—84no—
658C4—84no—
659C4—84no—
660C4—82no—
661C2—28no—
662C2—48no—
663C2C2×#250211yescharacteristic
664C2#3500212yes—
665C2#3536011yescharacteristic
666C2#3554111yescharacteristic
667C2#3673912yes—
668C2#3787212yes—
669C2#3787612yes—
670C2#3949212yes—
671C2#3949512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15555C2semidirect
4. #15545C2semidirect
5. #15392C2semidirect
6. #15378C2semidirect
7. #14766C2semidirect
8. #14702C2semidirect
9. #14686C2semidirect
10. #14353C2semidirect
11. #14281C2semidirect
21. C2×#398C22semidirect
25. C2×#391C22semidirect
31. C2×#384C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1444C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
97. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C2×#23152
#155552
#155452
#153922
#153782
#147662
#147022
#146862
#143532
#142812
#138772
C2×#14372