Finite Group Databasebeta

← #725969

Subgroups — #725969

Order 512.

2846 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#397C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic, complemented
3#16575C214yesmaximal, complemented
4#15708C214yesmaximal, complemented
5#15704C214yesmaximal, complemented
6#14919C212yesmaximal, complemented
7#14918C212yesmaximal
8#14859C214yesmaximal, complemented
9#14338C211yesmaximal, characteristic
10#14149C211yesmaximal, characteristic, complemented
11#13966C211yesmaximal, characteristic, complemented
12#13958C212yesmaximal, complemented
13#13904C211yesmaximal, characteristic, complemented
14#13868C212yesmaximal
15#13775C211yesmaximal, characteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C2×#397—28no—
21C2×#397C2212yes—
22C2×#397C2214yes—
23C2×#397C2212yes—
24#2344—28no—
25C2×#389C2212yescomplemented
26C2×#387C2212yescomplemented
27C2×#387C2212yescomplemented
28C2×#385C2212yes—
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#381C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#379C2211yescharacteristic
35C2×#379C2211yescharacteristic
36C2×#379C2212yes—
37C2×C4×C4⋊C4C2212yescomplemented
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42#1445C2212yescomplemented
43#1445C2214yescomplemented
44#1445C2212yescomplemented
45#1445C2214yescomplemented
46#1445C2214yescomplemented
47#1443C2214yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yescomplemented
66C4⋊#95C2214yescomplemented
67#1434C2214yescomplemented
68#1434C2214yescomplemented
69#1434C2212yescomplemented
70#1434C2212yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C2×#117—28no—
83C2×#117—28no—
84C2×#117—44no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4D418yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120#397—28no—
121#397—216no—
122#397—28no—
123#389—28no—
124#387—28no—
125#387—28no—
126#385—44no—
127#381—28no—
128#381—44no—
129#381—44no—
130#380—24no—
131#380—24no—
132#379—24no—
133#379—24no—
134#379—44no—
135C4×C4⋊C4—28no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138C2×#95—28no—
139C2×#95—24no—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25—22no—
169C25—22no—
170C25—22no—
171C25—22no—
172C25—22no—
173C25—22no—
174C25—44no—
175C25—44no—
176C25#4114yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4C2×D418yes—
196C23×C4C2×D414yes—
197C23×C4#4114yes—
198C23×C4#4114yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#117—432no—
213C2×C4⋊C4—232no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D414yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
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—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—432no—
268C2×C22⋊C4—416no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
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#4114yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—24no—
319C24—24no—
320C24—28no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—28no—
333C24—416no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—44no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—84no—
342C24—84no—
343C24—84no—
344C24—84no—
345C24—88no—
346C24—88no—
347C24C22×D412yes—
348C24C2×#4114yes—
349C24C2×#4112yes—
350C24C2×#4112yes—
351C24C2×#4112yes—
352C24C2×#4112yes—
353C24H(5,𝔽2)12yes—
354C24H(5,𝔽2)12yes—
355C24C2511yescharacteristic
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
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—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—88no—
437C22×C4#123116yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C4⋊C4—28no—
448C4⋊C4—216no—
449C4⋊C4—28no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C22⋊C4—28no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—48no—
470C42—48no—
471C42—48no—
472C42—84no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—432no—
489C23—42no—
490C23—416no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—816no—
509C23—816no—
510C23—88no—
511C23—88no—
512C23—816no—
513C23—816no—
514C23—816no—
515C23C2×#12314yes—
516C23#52914yes—
517C23#54014yes—
518C23#54614yes—
519C23#54614yes—
520C23#55314yes—
521C23#55514yes—
522C23C23×D411yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23#58514yes—
530C23#58514yes—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—416no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—432no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
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—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
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—88no—
603C2×C4—816no—
604C2×C4C2×#12314yes—
605C2×C4C2×#12314yes—
606C22—24no—
607C22—24no—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—832no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22#242018yes—
627C22C22×#12311yescharacteristic
628C22C2×#52911yescharacteristic
629C22#344912yes—
630C22C2×#54011yescharacteristic
631C22C2×#54611yescharacteristic
632C22C2×#54611yescharacteristic
633C22C2×#55311yescharacteristic
634C22C2×#55511yescharacteristic
635C22#348712yes—
636C22#349112yes—
637C22#349112yes—
638C22#349114yes—
639C22#351012yes—
640C22#351012yes—
641C22#352114yes—
642C22#353314yes—
643C22#356014yes—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—84no—
656C4—82no—
657C2—28no—
658C2—84no—
659C2C2×#242011yescharacteristic
660C2#3509711yescharacteristic
661C2#3567911yescharacteristic
662C2#3601312yes—
663C2#3784312yes—
664C2#3803112yes—
665C2#4138912yes—
666C2#4351112yes—
667C2#4446412yes—
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
2. C2×#2344C2semidirect
3. #16575C2semidirect
4. #15708C2semidirect
5. #15704C2semidirect
6. #14919C2semidirect
8. #14859C2semidirect
10. #14149C2semidirect
11. #13966C2semidirect
12. #13958C2semidirect
13. #13904C2semidirect
25. C2×#389C22semidirect
26. C2×#387C22semidirect
27. C2×#387C22semidirect
37. C2×C4×C4⋊C4C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1443C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3972
C2×#23442
#165752
#157082
#157042
#149192
#149182
#148592
#143382
#141492
#139662
#139582
#139042
#138682
#137752