Finite Group Databasebeta

← #619651

Subgroups — #619651

Order 512.

3881 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2849C211yesmaximal, characteristic
2C2×#2622C211yesmaximal, characteristic
3C2×#2312C211yesmaximal, characteristic, complemented
4C22×#381C211yesmaximal, characteristic, complemented
5#15929C214yesmaximal, complemented
6#15921C214yesmaximal, complemented
7#15915C214yesmaximal, complemented
8#15914C214yesmaximal, complemented
9#14935C212yesmaximal, complemented
10#14927C212yesmaximal, complemented
11#14681C211yesmaximal, characteristic, complemented
12#14262C211yesmaximal, characteristic, complemented
13#14255C211yesmaximal, characteristic
14#13996C212yesmaximal, complemented
15#13972C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#120C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2849—28no—
22#2622—28no—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27#2312—28no—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2211yescharacteristic
33C2×#386C2211yescharacteristic
34C2×#386C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2211yescharacteristic
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#381—28no—
41C2×#381C2212yes—
42C2×#381C2214yes—
43C2×#381C2214yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2212yes—
47C2×#379C2211yescharacteristic
48C2×#379C2211yescharacteristic
49C2×#379C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C22×#95C2212yes—
52C22×#95C2212yes—
53#1445C2214yescomplemented
54#1443C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1440C2212yes—
63#1440C2212yes—
64#1440C2214yescomplemented
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1438C2212yes—
70#1438C2212yes—
71#1437C2214yescomplemented
72#1436C2214yescomplemented
73C4⋊#95C2214yescomplemented
74#1434C2214yescomplemented
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C26C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—24no—
84C24×C4—28no—
85C24×C4—24no—
86C24×C4—24no—
87C24×C4—44no—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—44no—
97C2×#120—24no—
98C2×#120—24no—
99C2×#120—24no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4D418yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2314yes—
132#397—28no—
133#397—28no—
134#394—28no—
135#394—28no—
136#388—216no—
137#388—216no—
138#386—24no—
139#386—24no—
140#386—28no—
141#386—28no—
142#386—44no—
143#385—24no—
144#385—24no—
145#385—28no—
146#385—28no—
147#385—44no—
148#381—28no—
149#381—28no—
150#381—216no—
151#381—216no—
152#381—44no—
153#380—28no—
154#379—24no—
155#379—24no—
156#379D418yes—
157C4×C22⋊C4—28no—
158C2×#95—28no—
159C2×#95—28no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25—24no—
183C25—24no—
184C25—44no—
185C25—44no—
186C25C2×D414yes—
187C25C2×D414yes—
188C25#4114yes—
189C25C2411yescharacteristic
190C25C2411yescharacteristic
191C25C2411yescharacteristic
192C25C2411yescharacteristic
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—44no—
206C23×C4—232no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—432no—
235C23×C4—432no—
236C23×C4—416no—
237C23×C4—416no—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4#4118yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2414yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2414yes—
252#123—432no—
253#120—416no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C22⋊C4—232no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4C2×D418yes—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×D418yes—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—44no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345C24—24no—
346C24—24no—
347C24—28no—
348C24—28no—
349C24—216no—
350C24—216no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—48no—
372C24—416no—
373C24—416no—
374C24—416no—
375C24—416no—
376C24#121116yes—
377C24C22×D412yes—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C2×#4112yes—
384C24C2×#4112yes—
385C24C2×#4112yes—
386C24C2×#4114yes—
387C24H(5,𝔽2)14yes—
388C24C2511yescharacteristic
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—84no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
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—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
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—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—4128no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—88no—
489C22×C4—88no—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D418yes—
493C22×C4C2×#4118yes—
494C4⋊C4—416no—
495C4⋊C4—88no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—88no—
520C42—88no—
521C23—28no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—232no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—416no—
553C23—416no—
554C23—432no—
555C23—464no—
556C23—432no—
557C23—464no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23C2×#12114yes—
582C23C2×#12114yes—
583C23C2×#12114yes—
584C23#52514yes—
585C23#53514yes—
586C23#54618yes—
587C23#55118yes—
588C23C23×D411yescharacteristic
589C23C23×D411yescharacteristic
590C23C22×#4111yescharacteristic
591C23C22×#4112yes—
592C23C2×H(5,𝔽2)12yes—
593C23#58514yes—
594C23#58514yes—
595D4—88no—
596D4—88no—
597D4—88no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—432no—
611C2×C4—432no—
612C2×C4—44no—
613C2×C4—432no—
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—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4—816no—
665C22—28no—
666C22—216no—
667C22—28no—
668C22—24no—
669C22—24no—
670C22—216no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—464no—
678C22—44no—
679C22—44no—
680C22—48no—
681C22—48no—
682C22—44no—
683C22—44no—
684C22—48no—
685C22—48no—
686C22—82no—
687C22—84no—
688C22—82no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—84no—
693C22#241118yes—
694C22C22×#12111yescharacteristic
695C22C2×#52511yescharacteristic
696C22C2×#53511yescharacteristic
697C22#345514yes—
698C22C2×#54612yes—
699C22C2×#55112yes—
700C22#348512yes—
701C22#348512yes—
702C22#348512yes—
703C22#348512yes—
704C22#349112yes—
705C22#349112yes—
706C22#349314yes—
707C22#353314yes—
708C22#353314yes—
709C4—48no—
710C4—44no—
711C4—44no—
712C4—48no—
713C4—82no—
714C4—84no—
715C4—82no—
716C4—84no—
717C4—81no—
718C4—81no—
719C4—81no—
720C2—28no—
721C2—44no—
722C2—44no—
723C2—161no—
724C2C2×#241111yescharacteristic
725C2#3509312yes—
726C2#3544711yescharacteristic
727C2#3544711yescharacteristic
728C2#3695612yes—
729C2#3766714yes—
730C2#3785514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2312C2semidirect
4. C22×#381C2semidirect
5. #15929C2semidirect
6. #15921C2semidirect
7. #15915C2semidirect
8. #15914C2semidirect
9. #14935C2semidirect
10. #14927C2semidirect
11. #14681C2semidirect
12. #14262C2semidirect
14. #13996C2semidirect
15. #13972C2semidirect
53. #1445C22semidirect
54. #1443C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
64. #1440C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
71. #1437C22semidirect
72. #1436C22semidirect
73. C4⋊#95C22semidirect
74. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
C2×#26222
C2×#23122
C22×#3812
#159292
#159212
#159152
#159142
#149352
#149272
#146812
#142622
#142552
#139962
#139722