Finite Group Databasebeta

← #4420412

Subgroups — #4420412

Order 512.

3392 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2615C211yesmaximal, characteristic
2C2×#2339C211yesmaximal, characteristic, complemented
3#16698C212yesmaximal, complemented
4#16635C214yesmaximal, complemented
5#16361C212yesmaximal, complemented
6#15375C214yesmaximal, complemented
7#15369C214yesmaximal, complemented
8#15019C212yesmaximal, complemented
9#14913C212yesmaximal, complemented
10#14730C214yesmaximal, complemented
11#14659C211yesmaximal, characteristic, complemented
12#14347C211yesmaximal, characteristic, complemented
13#14295C211yesmaximal, characteristic, complemented
14#14152C211yesmaximal, characteristic, complemented
15#13956C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#121C2212yes—
19C22×C4×D4C2212yes—
20#2615—28no—
21C2×#394C2214yes—
22C2×#393C2212yes—
23C2×#392C2214yes—
24C2×#392C2212yes—
25#2339—28no—
26C2×#391C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#381C2212yes—
34C2×#381C2211yescharacteristic
35C2×#381C2214yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42#1446C2212yes—
43#1445C2214yes—
44#1445C2211yescharacteristic
45#1445C2212yes—
46#1445C2211yescharacteristic
47#1444C2212yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2212yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2214yes—
65#1438C2212yes—
66#1438C2211yescharacteristic
67#1438C2212yes—
68#1438C2211yescharacteristic
69#1438C2212yes—
70#1437C2214yes—
71#1436C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—48no—
80C22×#41D418yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2314yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×#121—28no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—48no—
97C2×C4×D4—28no—
98C2×C4×D4—48no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120#394—48no—
121#393—28no—
122#392—28no—
123#392—216no—
124#391—28no—
125#386—28no—
126#386—28no—
127#386—48no—
128#386—48no—
129#386—48no—
130#386—48no—
131#381—28no—
132#381—24no—
133#381—48no—
134#380—24no—
135#380—28no—
136#380—24no—
137C4×C22⋊C4—24no—
138C2×#95—28no—
139C2×#95—24no—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25C2412yes—
168C25C2411yescharacteristic
169C25C2412yes—
170C2×#41—232no—
171C2×#41—432no—
172C2×#41—432no—
173C2×#41—432no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8C2×D414yes—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4C2×D414yes—
202C22×D4C2×D418yes—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D418yes—
224C23×C4C2×D414yes—
225C23×C4#4114yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#121—432no—
240#121—416no—
241C4×D4—432no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
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×C42—24no—
296C2×C42—44no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24C22×D412yes—
348C24C22×D414yes—
349C24C2×#4112yes—
350C24C2×#4112yes—
351C24H(5,𝔽2)18yes—
352C24C2511yescharacteristic
353#41—4128no—
354C2×Q8—28no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—28no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—48no—
372C2×D4—416no—
373C2×D4—44no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—88no—
382C2×D4—88no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—216no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—88no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4#123116yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)14yes—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—88no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—84no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—84no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—48no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23C2×#12314yes—
549C23#53418yes—
550C23C23×D411yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C2×H(5,𝔽2)14yes—
554C23C2×H(5,𝔽2)12yes—
555C23#58512yes—
556C23#58512yes—
557C23#58512yes—
558Q8—416no—
559D4—232no—
560D4—432no—
561D4—88no—
562D4—816no—
563D4—832no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—416no—
576C2×C4—432no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
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—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—816no—
627C2×C4—816no—
628C2×C4—816no—
629C2×C4C2×#12314yes—
630C2×C4C2×#12314yes—
631C22—24no—
632C22—28no—
633C22—24no—
634C22—24no—
635C22—28no—
636C22—28no—
637C22—44no—
638C22—48no—
639C22—44no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—84no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22C22×#12311yescharacteristic
660C22C2×#53412yes—
661C22#346012yes—
662C22#348812yes—
663C22#349012yes—
664C22#349012yes—
665C22#349312yes—
666C22#350011yescharacteristic
667C22#350011yescharacteristic
668C22#350811yescharacteristic
669C22#351011yescharacteristic
670C22#351212yes—
671C22#353314yes—
672C22#353414yes—
673C22#354214yes—
674C22#356014yes—
675C4—24no—
676C4—24no—
677C4—44no—
678C4—44no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4—82no—
687C2—44no—
688C2—48no—
689C2—162no—
690C2#3499211yescharacteristic
691C2#3525411yescharacteristic
692C2#3561811yescharacteristic
693C2#4105214yes—
694C2#4193312yes—
695C2#4193612yes—
696C2#4788712yes—
697C2#4959512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2615C2
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×#2339C2semidirect
3. #16698C2semidirect
4. #16635C2semidirect
5. #16361C2semidirect
6. #15375C2semidirect
7. #15369C2semidirect
8. #15019C2semidirect
9. #14913C2semidirect
10. #14730C2semidirect
11. #14659C2semidirect
12. #14347C2semidirect
13. #14295C2semidirect
14. #14152C2semidirect
15. #13956C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26152
C2×#23392
#166982
#166352
#163612
#153752
#153692
#150192
#149132
#147302
#146592
#143472
#142952
#141522
#139562