Finite Group Databasebeta

← #732766

Subgroups — #732766

Order 512.

3598 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2436C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic, complemented
3C2×#2639C211yesmaximal, characteristic
4C2×#2364C211yesmaximal, characteristic
5#16805C212yesmaximal, complemented
6#16506C212yesmaximal, complemented
7#15858C214yesmaximal, complemented
8#15311C214yesmaximal, complemented
9#15125C212yesmaximal, complemented
10#15050C212yesmaximal, complemented
11#14737C214yesmaximal, complemented
12#14159C211yesmaximal, characteristic, complemented
13#13899C214yesmaximal, complemented
14C2×#1446C211yesmaximal, characteristic, complemented
15#13782C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×#124C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×D4C2212yes—
22#2639—28no—
23#2436—28no—
24#2434—28no—
25C2×#400C2212yes—
26C2×#397C2212yes—
27C2×#397C2214yes—
28C2×#395C2214yes—
29C2×#394C2212yes—
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33#2364—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#380C2214yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2214yes—
52#1446C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1446C2214yes—
56#1446C2211yescharacteristic
57#1445C2211yescharacteristic
58#1445C2211yescharacteristic
59#1445C2214yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2212yes—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—48no—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—48no—
95C2×#124—28no—
96C2×#124—48no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×#121—24no—
101C2×#121—24no—
102C2×#121D418yes—
103C2×C4×D4—28no—
104C2×C4×D4—216no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#400—28no—
128#397—28no—
129#397—48no—
130#395—216no—
131#394—28no—
132#393—28no—
133#393—28no—
134#393—28no—
135#391—28no—
136#391—28no—
137#388—28no—
138#387—28no—
139#386—216no—
140#386—216no—
141#386—28no—
142#386—28no—
143#385—28no—
144#385—28no—
145#385—28no—
146#380—48no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4D418yes—
150C2×#95—24no—
151C2×#95—24no—
152C2×#95—48no—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8—432no—
197C22×Q8—432no—
198C22×Q8—432no—
199C22×Q8—416no—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—24no—
203C22×D4—28no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4C2×D418yes—
208C22×D4C2×D414yes—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C4⋊Q8—432no—
247#124—432no—
248C22⋊Q8—416no—
249#121—232no—
250C4×D4—432no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×D418yes—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4#4118yes—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42C2×D418yes—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C2×D418yes—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24C22×D414yes—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C2×#4112yes—
353C24C2×#4114yes—
354C242−1+414yes—
355C24C2511yescharacteristic
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—416no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—416no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—4128no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—44no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×D4—216no—
372C2×D4—232no—
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—84no—
379C2×D4—84no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
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—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—84no—
452C22×C4—88no—
453C22×C4—84no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—88no—
458C22×C4#121116yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4118yes—
469C22×C4H(5,𝔽2)18yes—
470C22×C42−1+414yes—
471C4⋊C4—28no—
472C4⋊C4—416no—
473C4⋊C4—416no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—88no—
478C22⋊C4—28no—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4#124116yescomplemented
490C42—48no—
491C42—48no—
492C42—48no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23C2×#12114yes—
528C23C2×#12414yes—
529C23#53614yes—
530C23#54718yes—
531C23#55618yes—
532C23C23×D411yescharacteristic
533C23C23×D411yescharacteristic
534C23C22×#4111yescharacteristic
535C23C22×#4112yes—
536C23C2×H(5,𝔽2)14yes—
537C23C2×2−1+412yes—
538C23C2×2−1+412yes—
539C23#58512yes—
540Q8—432no—
541Q8—432no—
542Q8—88no—
543D4—432no—
544D4—88no—
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—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—416no—
566C2×C4—432no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
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—816no—
612C2×C4—816no—
613C2×C4—816no—
614C2×C4C2×#12114yes—
615C2×C4C2×#12114yes—
616C2×C4C2×#12418yes—
617C22—24no—
618C22—216no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—24no—
623C22—28no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—44no—
628C22—48no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#254718yes—
636C22C22×#12111yescharacteristic
637C22C22×#12411yescharacteristic
638C22C2×#53611yescharacteristic
639C22#346112yes—
640C22C2×#54712yes—
641C22C2×#55612yes—
642C22#348412yes—
643C22#348412yes—
644C22#348912yes—
645C22#349012yes—
646C22#349911yescharacteristic
647C22#350011yescharacteristic
648C22#351414yes—
649C22#352014yes—
650C22#354114yes—
651C22#355214yes—
652C4—24no—
653C4—24no—
654C4—28no—
655C4—48no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—84no—
663C4—82no—
664C4—82no—
665C2—28no—
666C2—84no—
667C2C2×#254711yescharacteristic
668C2#3530811yescharacteristic
669C2#3534811yescharacteristic
670C2#3593912yes—
671C2#3823314yes—
672C2#4059112yes—
673C2#4300412yes—
674C2#4389212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2434C2semidirect
5. #16805C2semidirect
6. #16506C2semidirect
7. #15858C2semidirect
8. #15311C2semidirect
9. #15125C2semidirect
10. #15050C2semidirect
11. #14737C2semidirect
12. #14159C2semidirect
13. #13899C2semidirect
14. C2×#1446C2semidirect
15. #13782C2semidirect
489. C22⋊C4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#24362
C2×#24342
C2×#26392
C2×#23642
#168052
#165062
#158582
#153112
#151252
#150502
#147372
#141592
#138992
C2×#14462
#137822