Finite Group Databasebeta

← #625037

Subgroups — #625037

Order 512.

2950 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2325C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic, complemented
3#16038C214yesmaximal, complemented
4#16030C214yesmaximal, complemented
5#15979C214yesmaximal, complemented
6#15973C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic, complemented
8#14936C211yesmaximal, characteristic, complemented
9#14926C211yesmaximal, characteristic
10#14913C211yesmaximal, characteristic
11#14696C212yesmaximal, complemented
12#14349C211yesmaximal, characteristic, complemented
13#14256C212yesmaximal, complemented
14#14015C212yesmaximal, complemented
15#13968C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2325—28no—
23C2×#388C2212yes—
24C2×#388C2214yes—
25C2×#387—28no—
26C2×#387C2212yes—
27C2×#387C2214yes—
28C2×#387C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#381C2212yes—
38C2×#380C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1445C2214yes—
44#1445C2212yes—
45#1445C2212yes—
46#1444C2212yes—
47#1444C2212yes—
48#1444C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2212yes—
53#1442C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2212yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1434C2214yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—44no—
90C2×#121—28no—
91C2×#121—24no—
92C2×C4×D4—28no—
93C2×C4×D4—44no—
94C2×C4×D4—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42—24no—
120C22×C42—44no—
121C22×C42C2314yes—
122C22×C42C2312yes—
123#388—28no—
124#388—48no—
125#387—28no—
126#387—216no—
127#387—216no—
128#386—28no—
129#386—28no—
130#386—28no—
131#386—28no—
132#386—44no—
133#385—28no—
134#385—44no—
135#385—48no—
136#381—44no—
137#380—24no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95—24no—
141C2×#95D418yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4#4118yes—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4#4114yes—
218C23×C4#4114yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232#123—432no—
233#121—232no—
234C4×D4—432no—
235C2×C4⋊C4—232no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
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×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42—416no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—42no—
313#95—42no—
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#95C2×D418yes—
323#95C2×D418yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24C22×D412yes—
339C24C2×#4114yes—
340C24C2×#4112yes—
341C24C2×#4114yes—
342C24C2×#4112yes—
343C24H(5,𝔽2)14yes—
344C24C2511yescharacteristic
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—216no—
348C2×D4—28no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—82no—
354C2×D4—82no—
355C2×D4—88no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C42−1+414yes—
461C4⋊C4—232no—
462C4⋊C4—28no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—88no—
480C42—416no—
481C42—48no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—28no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23#52514yes—
518C23#52514yes—
519C23#53514yes—
520C23#55118yes—
521C23#55318yes—
522C23C23×D411yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4112yes—
526C23C2×H(5,𝔽2)12yes—
527C23C2×H(5,𝔽2)12yes—
528C23C2×2−1+412yes—
529C23#58512yes—
530C23#58512yes—
531D4—432no—
532D4—88no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
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—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—816no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—216no—
614C22—216no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22#259818yes—
630C22C2×#52511yescharacteristic
631C22C2×#52511yescharacteristic
632C22C2×#53511yescharacteristic
633C22#345912yes—
634C22#346112yes—
635C22C2×#55112yes—
636C22C2×#55312yes—
637C22#348411yescharacteristic
638C22#348511yescharacteristic
639C22#348511yescharacteristic
640C22#348611yescharacteristic
641C22#350612yes—
642C22#350712yes—
643C22#352014yes—
644C22#352114yes—
645C22#353414yes—
646C22#355914yes—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—84no—
658C4—84no—
659C4—82no—
660C2—28no—
661C2—84no—
662C2C2×#259811yescharacteristic
663C2#3498611yescharacteristic
664C2#3509211yescharacteristic
665C2#3535711yescharacteristic
666C2#3545211yescharacteristic
667C2#3672111yescharacteristic
668C2#3695811yescharacteristic
669C2#3824514yes—
670C2#3927914yes—
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. C22×#387C2semidirect
3. #16038C2semidirect
4. #16030C2semidirect
5. #15979C2semidirect
6. #15973C2semidirect
7. #15015C2semidirect
8. #14936C2semidirect
11. #14696C2semidirect
12. #14349C2semidirect
13. #14256C2semidirect
14. #14015C2semidirect
15. #13968C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23252
C22×#3872
#160382
#160302
#159792
#159732
#150152
#149362
#149262
#149132
#146962
#143492
#142562
#140152
#139682