Finite Group Databasebeta

← #637919

Subgroups — #637919

Order 512.

4984 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic, complemented
2C2×#2623C212yesmaximal, complemented
3C2×#2613C212yesmaximal, complemented
4C22×#390C211yesmaximal, characteristic, complemented
5#16811C212yesmaximal, complemented
6#16810C212yesmaximal, complemented
7#16809C214yesmaximal, complemented
8#15172C212yesmaximal, complemented
9#15170C212yesmaximal, complemented
10#15007C214yesmaximal, complemented
11#14734C212yesmaximal, complemented
12#14716C212yesmaximal, complemented
13#13995C214yesmaximal, complemented
14C4×#381C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#120C2214yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2854—28no—
24#2623—216no—
25#2613—216no—
26C2×#400C2214yescomplemented
27C2×#399C2214yescomplemented
28C2×#399C2214yescomplemented
29C2×#398C2214yescomplemented
30C2×#397C2214yescomplemented
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#392C2214yes—
34C2×#392C2214yes—
35C2×#392C2214yescomplemented
36C2×#390—28no—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#390C2211yescharacteristic, complemented
42C2×#390C2211yescharacteristic, complemented
43C2×#390C2218yescomplemented
44C2×#389C2211yescharacteristic, complemented
45C2×#389C2211yescharacteristic, complemented
46C2×#388C2212yescomplemented
47C2×#388C2212yescomplemented
48C2×#386C2212yes—
49C2×#386C2214yes—
50C2×#385C2214yes—
51C2×#381C2212yes—
52C2×#380C2211yescharacteristic
53C2×#379C2214yes—
54C2×C4×C22⋊C4C2212yes—
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2218yescomplemented
58#1439C2218yescomplemented
59#1438C2218yescomplemented
60#1438C2218yescomplemented
61#1437C2218yescomplemented
62#1434C2212yescomplemented
63#1434C2212yescomplemented
64C22⋊#95C2212yes—
65C4×#95C2212yescomplemented
66C4×#95C2212yescomplemented
67C26C2312yes—
68C23×D4—28no—
69C23×D4—24no—
70C23×D4—28no—
71C23×D4—44no—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C23×D4C2318yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123—28no—
84C2×#121—24no—
85C2×#121—44no—
86C2×#120—216no—
87C2×#120—216no—
88C2×#120—48no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2318yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4—48no—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—44no—
119C22×C42D418yescomplemented
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#400—216no—
126#399—216no—
127#399—48no—
128#398—216no—
129#397—216no—
130#394—216no—
131#394—216no—
132#392—216no—
133#392—216no—
134#392—48no—
135#390—24no—
136#390—24no—
137#390—28no—
138#390—28no—
139#390—28no—
140#390—28no—
141#390—416no—
142#389—24no—
143#389—24no—
144#388—28no—
145#388—28no—
146#386—28no—
147#386—216no—
148#385—216no—
149#381D4116yescomplemented
150#380—24no—
151#379—216no—
152C4×C22⋊C4—44no—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25—24no—
164C25—24no—
165C25—28no—
166C25—28no—
167C25—44no—
168C25—44no—
169C25—44no—
170C25—44no—
171C25#4118yes—
172C25C2412yes—
173C25C2414yes—
174C25C2411yescharacteristic
175C22×D4—232no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—216no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—216no—
190C22×D4—24no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—464no—
194C22×D4—416no—
195C22×D4—48no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—48no—
199C22×D4—48no—
200C22×D4—48no—
201C22×D4—48no—
202C22×D4—416no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2418yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#123—432no—
231#121—416no—
232#120—464no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
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—216no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—432no—
287C2×C22⋊C4—432no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4C2×D4116yes—
295C2×C42—28no—
296C2×C42—416no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307#95—24no—
308#95—216no—
309#95—24no—
310#95—24no—
311#95—216no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95C2×D4116yes—
316#95C2×D4116yes—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—216no—
322C24—24no—
323C24—28no—
324C24—24no—
325C24—24no—
326C24—28no—
327C24—24no—
328C24—28no—
329C24—24no—
330C24—28no—
331C24—24no—
332C24—232no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—416no—
343C24—416no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—416no—
349C24—416no—
350C24—48no—
351C24—48no—
352C24—416no—
353C24—816no—
354C24—816no—
355C24—816no—
356C24—816no—
357C24C22×D412yes—
358C24C22×D414yes—
359C24C2×#4118yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)14yes—
362C24C2511yescharacteristic
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—432no—
366C2×D4—432no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—4128no—
370C2×D4—432no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—432no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—88no—
383C2×D4—816no—
384C2×D4—816no—
385C2×D4—816no—
386C2×D4—816no—
387C2×D4—816no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—232no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—816no—
443C22×C4#127116yes—
444C22×C4C22×D4116yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4H(5,𝔽2)14yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—816no—
463C22⋊C4—232no—
464C22⋊C4—432no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—816no—
476C22⋊C4—816no—
477C22⋊C4—816no—
478C42—44no—
479C42—44no—
480C42—44no—
481C42—44no—
482C42—88no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—216no—
495C23—216no—
496C23—28no—
497C23—28no—
498C23—432no—
499C23—432no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—44no—
513C23—44no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—432no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—48no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—864no—
537C23—816no—
538C23—864no—
539C23—816no—
540C23—816no—
541C23—816no—
542C23#52514yes—
543C23C2×#12714yes—
544C23#53414yes—
545C23#546116yes—
546C23C23×D411yescharacteristic
547C23C23×D412yes—
548C23C22×#4112yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)12yes—
552C23#58512yes—
553C23#58512yes—
554D4—432no—
555D4—432no—
556D4—88no—
557D4—864no—
558D4—816no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—232no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—216no—
579C2×C4—216no—
580C2×C4—216no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4C2×#12714yes—
611C2×C4C2×#12714yes—
612C2×C4#53514yes—
613C2×C4#53514yes—
614C22—24no—
615C22—232no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—48no—
622C22—416no—
623C22—416no—
624C22—48no—
625C22—48no—
626C22—864no—
627C22—84no—
628C22—88no—
629C22—88no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—82no—
634C22—82no—
635C22—88no—
636C22—88no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22#253918yes—
643C22C2×#52511yescharacteristic
644C22C22×#12711yescharacteristic
645C22C2×#53411yescharacteristic
646C22C2×#53511yescharacteristic
647C22C2×#53511yescharacteristic
648C22#344911yescharacteristic
649C22#344911yescharacteristic
650C22#345914yes—
651C22C2×#54614yes—
652C22#349814yes—
653C22#350014yes—
654C22#350514yes—
655C22#353414yes—
656C22#353414yes—
657C4—24no—
658C4—24no—
659C4—24no—
660C4—24no—
661C4—416no—
662C4—42no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—84no—
668C2—28no—
669C2—88no—
670C2—162no—
671C2#3459311yescharacteristic
672C2C2×#253911yescharacteristic
673C2#3510211yescharacteristic
674C2#3527411yescharacteristic
675C2#3532911yescharacteristic
676C2#3744011yescharacteristic
677C2#3744611yescharacteristic
678C2#3943018yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2854C2semidirect
2. C2×#2623C2semidirect
3. C2×#2613C2semidirect
4. C22×#390C2semidirect
5. #16811C2semidirect
6. #16810C2semidirect
7. #16809C2semidirect
8. #15172C2semidirect
9. #15170C2semidirect
10. #15007C2semidirect
11. #14734C2semidirect
12. #14716C2semidirect
13. #13995C2semidirect
14. C4×#381C2semidirect
26. C2×#400C22semidirect
27. C2×#399C22semidirect
28. C2×#399C22semidirect
29. C2×#398C22semidirect
30. C2×#397C22semidirect
35. C2×#392C22semidirect
41. C2×#390C22semidirect
42. C2×#390C22semidirect
43. C2×#390C22semidirect
44. C2×#389C22semidirect
45. C2×#389C22semidirect
46. C2×#388C22semidirect
47. C2×#388C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1438C22semidirect
60. #1438C22semidirect
61. #1437C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
65. C4×#95C22semidirect
66. C4×#95C22semidirect
119. C22×C42D4semidirect
149. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#26232
C2×#26132
C22×#3902
#168112
#168102
#168092
#151722
#151702
#150072
#147342
#147162
#139952
C4×#3812