Finite Group Databasebeta

← #5025099

Subgroups — #5025099

Order 512.

3414 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×#2641C211yesmaximal, characteristic, complemented
3C2×#2633C211yesmaximal, characteristic, complemented
4C2×#2380C211yesmaximal, characteristic, complemented
5#15496C214yesmaximal, complemented
6#15493C214yesmaximal, complemented
7#15479C214yesmaximal, complemented
8#15313C214yesmaximal, complemented
9#15277C214yesmaximal, complemented
10#15274C214yesmaximal, complemented
11#14289C211yesmaximal, characteristic, complemented
12#14164C211yesmaximal, characteristic, complemented
13#13959C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#127C2212yes—
16C22×#125C2211yescharacteristic
17C22×#121C2214yes—
18C22×C4×D4C2211yescharacteristic
19#2641—28no—
20#2633—28no—
21#2541—28no—
22C2×#399C2212yes—
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#397C2214yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#392C2214yes—
33#2380—28no—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#385C2214yes—
43C2×#383C2211yescharacteristic
44C2×#383C2211yescharacteristic
45C2×#380C2214yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×#376C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1442C2212yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2214yes—
63#1438C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2314yes—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2312yes—
77C24×C4C2312yes—
78C24×C4C2314yes—
79C2×#127—28no—
80C2×#127—24no—
81C2×#127—24no—
82C2×#127—44no—
83C2×#127—44no—
84C2×#127D418yes—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125—24no—
88C2×#121—216no—
89C2×#121—48no—
90C2×#121—48no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#399—28no—
120#398—28no—
121#397—28no—
122#397—48no—
123#394—28no—
124#394—28no—
125#394—216no—
126#394—28no—
127#394—48no—
128#394—48no—
129#392—216no—
130#390—28no—
131#390—28no—
132#390—28no—
133#386—216no—
134#386—216no—
135#386—216no—
136#386—48no—
137#386—48no—
138#385—216no—
139#383—24no—
140#383D418yescomplemented
141#380—48no—
142C4×C4⋊C4—24no—
143#376—24no—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C25C2414yes—
165C25C2412yes—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—42no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4C2×D418yes—
202C22×D4C2×D418yes—
203C22×D4C2×D418yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#127—232no—
228#127—432no—
229#125—416no—
230#121—464no—
231C4×D4—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×D418yes—
324#95C2×D418yes—
325#95C2×D418yes—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24C22×D418yes—
351C24H(5,𝔽2)14yes—
352C24C2511yescharacteristic
353C2×D4—28no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—416no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—44no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—88no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
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—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
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—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—88no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4C22×D418yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)18yes—
463C4⋊C4—216no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—48no—
486C42—48no—
487C42#121116yes—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)14yes—
551C23#58512yes—
552C23#58512yes—
553C23#58512yes—
554D4—416no—
555D4—816no—
556D4—816no—
557D4—816no—
558D4—832no—
559D4—832no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
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—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—816no—
618C2×C4—832no—
619C2×C4C2×#12114yes—
620C2×C4C2×#12114yes—
621C2×C4C2×#12114yes—
622C22—44no—
623C22—48no—
624C22—48no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—82no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—84no—
639C22—82no—
640C22—84no—
641C22—84no—
642C22—82no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22C22×#12111yescharacteristic
649C22#348411yescharacteristic
650C22#348412yes—
651C22#348512yes—
652C22#348511yescharacteristic
653C22#348812yes—
654C22#349412yes—
655C22#349812yes—
656C22#350012yes—
657C22#350812yes—
658C22#351011yescharacteristic
659C22#351011yescharacteristic
660C22#353414yes—
661C22#353414yes—
662C22#353914yes—
663C22#356214yes—
664C4—24no—
665C4—24no—
666C4—24no—
667C4—48no—
668C4—48no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—84no—
675C2—84no—
676C2—88no—
677C2#3551411yescharacteristic
678C2#3552811yescharacteristic
679C2#3561711yescharacteristic
680C2#5065014yes—
681C2#5071312yes—
682C2#5108812yes—
683C2#5200812yes—
684C2#5200912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×#2641C2semidirect
3. C2×#2633C2semidirect
4. C2×#2380C2semidirect
5. #15496C2semidirect
6. #15493C2semidirect
7. #15479C2semidirect
8. #15313C2semidirect
9. #15277C2semidirect
10. #15274C2semidirect
11. #14289C2semidirect
12. #14164C2semidirect
13. #13959C2semidirect
140. #383D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×#26412
C2×#26332
C2×#23802
#154962
#154932
#154792
#153132
#152772
#152742
#142892
#141642
#139592