Finite Group Databasebeta

← #948169

Subgroups — #948169

Order 512.

4336 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C212yesmaximal, complemented
2#17224C212yesmaximal, complemented
3#17216C212yesmaximal, complemented
4#16272C214yesmaximal, complemented
5#16249C214yesmaximal, complemented
6#15966C211yesmaximal, characteristic, complemented
7#15947C211yesmaximal, characteristic, complemented
8#15266C212yesmaximal, complemented
9#15160C212yesmaximal, complemented
10#14838C214yesmaximal, complemented
11#14311C212yesmaximal, complemented
12#13995C214yesmaximal, complemented
13#13851C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#120C2214yes—
16C22×#120C2212yes—
17C22×#120C2212yescomplemented
18C23×C22⋊C4C2212yescomplemented
19#2404—216no—
20C2×#397C2212yescomplemented
21C2×#394C2214yes—
22C2×#394C2214yescomplemented
23C2×#393C2214yescomplemented
24C2×#392C2212yes—
25C2×#392C2214yes—
26C2×#392C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yescomplemented
33C2×#386C2212yescomplemented
34C2×#385C2214yescomplemented
35C2×#381C2212yescomplemented
36C2×#379C2214yescomplemented
37C2×C4×C22⋊C4C2211yescharacteristic, complemented
38C2×C4×C22⋊C4C2214yescomplemented
39C2×C4×C22⋊C4C2211yescharacteristic, complemented
40#1446C2211yescharacteristic, complemented
41#1446C2211yescharacteristic, complemented
42#1445C2212yescomplemented
43#1443C2214yescomplemented
44#1442C2212yescomplemented
45#1442C2214yescomplemented
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2212yescomplemented
54#1439C2214yescomplemented
55#1438C2211yescharacteristic
56#1438C2211yescharacteristic
57#1438C2212yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1438C2214yescomplemented
63#1438C2214yescomplemented
64#1437C2214yescomplemented
65#1437C2214yescomplemented
66C4⋊#95C2212yescomplemented
67#1434C2212yescomplemented
68C26C2312yes—
69C22×#41—42no—
70C22×#41—42no—
71C22×#41—48no—
72C22×#41D418yescomplemented
73C23×Q8C2311yescharacteristic
74C23×D4C2314yes—
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C23×D4C2314yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×#120—28no—
84C2×#120—28no—
85C2×#120—216no—
86C2×#120—48no—
87C2×#120—48no—
88C2×#120—48no—
89C2×#120—48no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116#397—44no—
117#394—48no—
118#394—48no—
119#393—216no—
120#392—216no—
121#392—28no—
122#392—44no—
123#390—44no—
124#390—44no—
125#386—28no—
126#386—28no—
127#386—216no—
128#386—216no—
129#386—48no—
130#385—48no—
131#381—28no—
132#379—48no—
133C4×C22⋊C4—42no—
134C4×C22⋊C4—42no—
135C4×C22⋊C4—48no—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C25—24no—
156C25—28no—
157C25—28no—
158C25—42no—
159C25—42no—
160C25—44no—
161C25—44no—
162C25—44no—
163C25—44no—
164C25C2411yescharacteristic
165C25C2414yes—
166C25C2412yes—
167C2×#41—416no—
168C2×#41—432no—
169C2×#41—432no—
170C2×#41—432no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—44no—
174C22×Q8C2×D414yes—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—44no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4C2×D414yes—
206C22×D4C2×D418yes—
207C22×D4#4118yescomplemented
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D418yes—
221C23×C4C2×D414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232#120—464no—
233#120—432no—
234#120—432no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
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—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—432no—
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×C42—24no—
297C2×C42—24no—
298C2×C42—44no—
299C2×C42—44no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—42no—
308#95—41no—
309#95—41no—
310#95—41no—
311#95—44no—
312#95—44no—
313#95—41no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—42no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—21no—
338C24—21no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—416no—
348C24—48no—
349C24—416no—
350C24—416no—
351C24—48no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—48no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24—88no—
361C24—88no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24—84no—
366C24—84no—
367C24—88no—
368C24C22×D412yes—
369C24C22×D414yes—
370C24C2×#4118yes—
371C24H(5,𝔽2)18yes—
372C24C2511yescharacteristic
373#41—4128no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—416no—
384C2×D4—48no—
385C2×D4—416no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—84no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C22×C4—24no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—216no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—88no—
482C22⋊C4—232no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—84no—
501C42—84no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—22no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—432no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—41no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—41no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—41no—
568C23—41no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—832no—
575C23—816no—
576C23—832no—
577C23—832no—
578C23—832no—
579C23—832no—
580C23—816no—
581C23—832no—
582C23#53418yes—
583C23C23×D411yescharacteristic
584C23C22×#4112yes—
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)14yes—
588C23C2×H(5,𝔽2)12yes—
589C23C2×H(5,𝔽2)12yes—
590C23#58512yes—
591Q8—416no—
592D4—416no—
593D4—432no—
594D4—816no—
595D4—816no—
596D4—832no—
597D4—832no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—432no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—816no—
654C2×C4#53414yes—
655C22—216no—
656C22—28no—
657C22—48no—
658C22—48no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—42no—
668C22—48no—
669C22—42no—
670C22—42no—
671C22—48no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—864no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—82no—
680C22—84no—
681C22—84no—
682C22—82no—
683C22—84no—
684C22—84no—
685C22—82no—
686C22—84no—
687C22—82no—
688C22C2×#53412yes—
689C22C2×#53411yescharacteristic
690C22#345912yes—
691C22D4×#4112yes—
692C22#348111yescharacteristic
693C22#349014yes—
694C22#349812yes—
695C22#351914yes—
696C22#352911yescharacteristic
697C22#353212yes—
698C22#353212yes—
699C22#353414yes—
700C22#353712yes—
701C22#354911yescharacteristic
702C22#354911yescharacteristic
703C22#355814yes—
704C4—24no—
705C4—44no—
706C4—42no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—84no—
711C4—84no—
712C4—82no—
713C4—84no—
714C2—48no—
715C2—82no—
716C2—88no—
717C2#3729711yescharacteristic
718C2#3743611yescharacteristic
719C2#3842414yes—
720C2#4091212yes—
721C2#4209112yes—
722C2#4424112yes—
723C2#5216311yescharacteristic
724C2#5242212yes—
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×#2404C2semidirect
2. #17224C2semidirect
3. #17216C2semidirect
4. #16272C2semidirect
5. #16249C2semidirect
6. #15966C2semidirect
7. #15947C2semidirect
8. #15266C2semidirect
9. #15160C2semidirect
10. #14838C2semidirect
11. #14311C2semidirect
12. #13995C2semidirect
13. #13851C2semidirect
17. C22×#120C22semidirect
18. C23×C22⋊C4C22semidirect
20. C2×#397C22semidirect
22. C2×#394C22semidirect
23. C2×#393C22semidirect
32. C2×#386C22semidirect
33. C2×#386C22semidirect
34. C2×#385C22semidirect
35. C2×#381C22semidirect
36. C2×#379C22semidirect
37. C2×C4×C22⋊C4C22semidirect
38. C2×C4×C22⋊C4C22semidirect
39. C2×C4×C22⋊C4C22semidirect
40. #1446C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
62. #1438C22semidirect
63. #1438C22semidirect
64. #1437C22semidirect
65. #1437C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
72. C22×#41D4semidirect
207. C22×D4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
#172242
#172162
#162722
#162492
#159662
#159472
#152662
#151602
#148382
#143112
#139952
#138512