Finite Group Databasebeta

← #641079

Subgroups — #641079

Order 512.

4916 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C2×#2412C214yesmaximal, complemented
3C22×#380C211yesmaximal, characteristic, complemented
4#17059C212yesmaximal, complemented
5#16807C214yesmaximal, complemented
6#16363C212yesmaximal, complemented
7#15069C212yesmaximal, complemented
8#14936C212yesmaximal, complemented
9#14926C212yesmaximal, complemented
10#14836C214yesmaximal, complemented
11#14777C212yesmaximal, complemented
12#14353C212yesmaximal, complemented
13#14271C212yesmaximal, complemented
14C2×C22⋊#95C211yesmaximal, characteristic, complemented
15C25×C4C2211yescharacteristic
16C22×#123C2214yes—
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#120C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24C23×C22⋊C4C2212yes—
25#2847—28no—
26#2412—232no—
27C2×#399C2214yescomplemented
28C2×#397C2218yes—
29C2×#394C2218yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#386C2218yes—
35C2×#385C2214yes—
36C2×#381C2214yes—
37C2×#380—28no—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2218yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C22×#95C2212yes—
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1445C2214yescomplemented
50#1445C2218yescomplemented
51#1445C2214yescomplemented
52#1444C2214yescomplemented
53#1444C2214yescomplemented
54#1444C2214yescomplemented
55#1444C2214yescomplemented
56#1444C2214yescomplemented
57#1439C2218yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2214yescomplemented
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C26C2311yescharacteristic
67C23×D4C2312yes—
68C23×D4C2312yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4—28no—
72C24×C4—44no—
73C24×C4—44no—
74C24×C4D418yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C2×#123—216no—
85C2×#123—48no—
86C2×#123—48no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#121—28no—
91C2×#121—44no—
92C2×#121—44no—
93C2×#120—28no—
94C2×#120—216no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2318yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—216no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4D4116yescomplemented
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125#399—48no—
126#397—232no—
127#394—232no—
128#386—28no—
129#386—28no—
130#386—28no—
131#386—232no—
132#386—216no—
133#385—216no—
134#381—216no—
135#380—24no—
136#380—24no—
137#380—24no—
138#380—232no—
139#380—28no—
140#380—28no—
141#380—28no—
142#380—28no—
143C2×#95—28no—
144C2×#95—28no—
145C2×#95—44no—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C25—24no—
160C25—22no—
161C25—24no—
162C25—24no—
163C25—22no—
164C25—42no—
165C25—42no—
166C25C2×D418yes—
167C25C2×D414yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—232no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—416no—
210C23×C4—416no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—464no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D4116yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4#4118yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2418yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243#123—464no—
244#121—416no—
245#121—416no—
246#121—416no—
247#120—432no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—48no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—432no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—432no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4C2×D4116yes—
299C2×C22⋊C4C2×D4116yes—
300#95—28no—
301#95—28no—
302#95—216no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—216no—
307#95—216no—
308#95—216no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—44no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—22no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—28no—
323C24—216no—
324C24—28no—
325C24—28no—
326C24—28no—
327C24—216no—
328C24—232no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—44no—
346C24—44no—
347C24—416no—
348C24—416no—
349C24—48no—
350C24—48no—
351C24—416no—
352C24—48no—
353C24—48no—
354C24#127116yes—
355C24C22×D414yes—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C22×D414yes—
361C24C2×#4114yes—
362C24H(5,𝔽2)12yes—
363C24H(5,𝔽2)14yes—
364C242−1+414yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—432no—
381C2×D4—44no—
382C2×D4—88no—
383C2×D4—816no—
384C22×C4—232no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—232no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—4128no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—416no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—816no—
462C22×C4#123116yescomplemented
463C22×C4C22×D412yes—
464C22×C4C22×D4116yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)14yes—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—432no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—816no—
480C22⋊C4—232no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—432no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—816no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—232no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—28no—
527C23—28no—
528C23—232no—
529C23—416no—
530C23—416no—
531C23—416no—
532C23—416no—
533C23—416no—
534C23—464no—
535C23—464no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—48no—
552C23—48no—
553C23—44no—
554C23—44no—
555C23—48no—
556C23—48no—
557C23—44no—
558C23—44no—
559C23C2×#12314yes—
560C23C2×#12714yes—
561C23C2×#12714yes—
562C23C2×#12714yes—
563C23#53418yes—
564C23#53414yes—
565C23#53614yes—
566C23#53714yes—
567C23#53718yes—
568C23#547116yes—
569C23C23×D411yescharacteristic
570C23C23×D412yes—
571C23C22×#4112yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)12yes—
575C23C2×2−1+412yes—
576C23C2×2−1+411yescharacteristic
577C23#58512yes—
578D4—88no—
579D4—816no—
580D4—816no—
581D4—816no—
582C2×C4—232no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—216no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—216no—
597C2×C4—216no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—832no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4C2×#12314yes—
635C2×C4C2×#12314yes—
636C2×C4#53414yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—216no—
650C22—216no—
651C22—216no—
652C22—216no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—464no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—48no—
662C22—48no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—88no—
671C22—88no—
672C22#250118yes—
673C22#265618yes—
674C22C22×#12311yescharacteristic
675C22C22×#12711yescharacteristic
676C22C2×#53411yescharacteristic
677C22C2×#53412yes—
678C22C2×#53411yescharacteristic
679C22C2×#53611yescharacteristic
680C22C2×#53711yescharacteristic
681C22C2×#53712yes—
682C22#344911yescharacteristic
683C22C2×#54714yes—
684C22#349014yes—
685C22#349412yes—
686C22#349512yes—
687C22#349614yes—
688C22#353514yes—
689C22#353612yes—
690C22#353812yes—
691C4—24no—
692C4—24no—
693C4—24no—
694C4—42no—
695C4—416no—
696C4—84no—
697C4—82no—
698C4—84no—
699C4—82no—
700C4—82no—
701C2—28no—
702C2—28no—
703C2—44no—
704C2—162no—
705C2C2×#265611yescharacteristic
706C2C2×#250111yescharacteristic
707C2#3499211yescharacteristic
708C2#3500111yescharacteristic
709C2#3526311yescharacteristic
710C2#3744211yescharacteristic
711C2#3744711yescharacteristic
712C2#3786014yes—
713C2#3786114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
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×#2412C2semidirect
3. C22×#380C2semidirect
4. #17059C2semidirect
5. #16807C2semidirect
6. #16363C2semidirect
7. #15069C2semidirect
8. #14936C2semidirect
9. #14926C2semidirect
10. #14836C2semidirect
11. #14777C2semidirect
12. #14353C2semidirect
13. #14271C2semidirect
14. C2×C22⋊#95C2semidirect
27. C2×#399C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1438C22semidirect
112. C22×C22⋊C4D4semidirect
462. C22×C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C2×#24122
C22×#3802
#170592
#168072
#163632
#150692
#149362
#149262
#148362
#147772
#143532
#142712
C2×C22⋊#952