Finite Group Databasebeta

← #761054

Subgroups — #761054

Order 512.

4902 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#386C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#16884C214yesmaximal, complemented
4#15091C212yesmaximal, complemented
5#15019C212yesmaximal, complemented
6#14842C214yesmaximal, complemented
7#14820C212yesmaximal, complemented
8#14776C212yesmaximal, complemented
9#14020C214yesmaximal, complemented
10#13966C212yesmaximal, complemented
11#13903C212yesmaximal, complemented
12#13869C212yesmaximal, complemented
13#13867C212yesmaximal
14C4×#379C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic, complemented
18C23×C4⋊C4C2211yescharacteristic, complemented
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C2×#397C2214yescomplemented
24C2×#386—28no—
25C2×#386C2214yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#381C2214yes—
32C2×#380—28no—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yescomplemented
36C2×#380C2212yescomplemented
37C2×#379C2212yes—
38C2×#379C2214yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yescomplemented
41C2×C4×C22⋊C4C2212yescomplemented
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2212yescomplemented
45#1445C2214yescomplemented
46#1443C2218yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1440C2218yescomplemented
51#1440C2218yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1438C2218yes—
56#1437C2218yescomplemented
57#1434C2218yescomplemented
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2212yes—
61C22⋊#95C2214yescomplemented
62C22⋊#95C2214yescomplemented
63C22⋊#95C2212yescomplemented
64C22⋊#95C2212yescomplemented
65C26C2311yescharacteristic
66C26C2311yescharacteristic
67C23×D4—24no—
68C23×D4—28no—
69C23×D4—28no—
70C23×D4—44no—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4—28no—
74C24×C4—44no—
75C24×C4—44no—
76C24×C4D418yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—48no—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2318yes—
117C22×C42C2314yes—
118#397—48no—
119#386—216no—
120#386—28no—
121#386—28no—
122#386—28no—
123#386—28no—
124#386—48no—
125#381—48no—
126#380—28no—
127#380—28no—
128#380—28no—
129#380—28no—
130#379—28no—
131#379—48no—
132C4×C22⋊C4—28no—
133C4×C22⋊C4—44no—
134C4×C22⋊C4—44no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95—28no—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2314yes—
150C2×#95C2318yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C25—24no—
154C25—24no—
155C25—24no—
156C25—24no—
157C25—22no—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—24no—
162C25—42no—
163C25—42no—
164C25—42no—
165C25—42no—
166C25—42no—
167C25C2×D414yes—
168C25#4118yes—
169C25C2412yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×D4—232no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—44no—
183C22×D4—464no—
184C22×D4—416no—
185C22×D4—416no—
186C22×D4#4118yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—232no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—416no—
212C23×C4—416no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—464no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4#4118yes—
235C23×C4C2418yes—
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—416no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—416no—
284C2×C22⋊C4—432no—
285C2×C22⋊C4—416no—
286C2×C22⋊C4—416no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C42—28no—
293C2×C42—44no—
294C2×C42—44no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—216no—
303#95—28no—
304#95—48no—
305#95—44no—
306#95—48no—
307#95—48no—
308#95—48no—
309#95—48no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—28no—
319C24—28no—
320C24—28no—
321C24—28no—
322C24—232no—
323C24—232no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—28no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—44no—
340C24—44no—
341C24—416no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—416no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—44no—
354C24—44no—
355C24—416no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24#124116yes—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C22×D412yes—
363C24C2×#4114yes—
364C24C2×#4114yes—
365C24C2×#4114yes—
366C24C2×#4114yes—
367C24C2×#4114yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)14yes—
370C24H(5,𝔽2)14yes—
371C24C2511yescharacteristic
372C2×D4—216no—
373C2×D4—216no—
374C2×D4—232no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—4128no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—416no—
383C22×C4—232no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—4128no—
446C22×C4—416no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—416no—
462C22×C4—88no—
463C22×C4—816no—
464C22×C4—816no—
465C22×C4#123116yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)14yes—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—816no—
490C22⋊C4—816no—
491C42—88no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—232no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—232no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—416no—
528C23—416no—
529C23—416no—
530C23—416no—
531C23—4128no—
532C23—416no—
533C23—416no—
534C23—416no—
535C23—416no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—48no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23—44no—
552C23—48no—
553C23—864no—
554C23—88no—
555C23—88no—
556C23—88no—
557C23—88no—
558C23C2×#12314yes—
559C23#52514yes—
560C23C2×#12414yes—
561C23C2×#12414yes—
562C23C2×#12414yes—
563C23#53414yes—
564C23#53514yes—
565C23#551116yes—
566C23#553116yes—
567C23C23×D411yescharacteristic
568C23C22×#4112yes—
569C23C22×#4112yes—
570C23C22×#4112yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)12yes—
574C23#58512yes—
575D4—432no—
576D4—432no—
577C2×C4—232no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—816no—
628C2×C4—816no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4C2×#12314yes—
632C2×C4C2×#12314yes—
633C2×C4#53414yes—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—232no—
640C22—232no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—464no—
652C22—42no—
653C22—42no—
654C22—416no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—416no—
660C22—44no—
661C22—832no—
662C22—82no—
663C22—88no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22#250418yes—
668C22#256118yes—
669C22C22×#12311yescharacteristic
670C22C2×#52511yescharacteristic
671C22C22×#12411yescharacteristic
672C22C2×#53411yescharacteristic
673C22C2×#53411yescharacteristic
674C22C2×#53511yescharacteristic
675C22#344911yescharacteristic
676C22C2×#55114yes—
677C22C2×#55314yes—
678C22#347514yes—
679C22#349014yes—
680C22#350514yes—
681C22#351914yes—
682C22#353414yes—
683C4—24no—
684C4—24no—
685C4—24no—
686C4—44no—
687C4—42no—
688C4—84no—
689C4—84no—
690C4—88no—
691C4—82no—
692C4—82no—
693C2—28no—
694C2—28no—
695C2—44no—
696C2—84no—
697C2C2×#250411yescharacteristic
698C2#3480911yescharacteristic
699C2C2×#256111yescharacteristic
700C2#3499211yescharacteristic
701C2#3501911yescharacteristic
702C2#3507611yescharacteristic
703C2#3744011yescharacteristic
704C2#5930118yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#380C2semidirect
3. #16884C2semidirect
4. #15091C2semidirect
5. #15019C2semidirect
6. #14842C2semidirect
7. #14820C2semidirect
8. #14776C2semidirect
9. #14020C2semidirect
10. #13966C2semidirect
11. #13903C2semidirect
12. #13869C2semidirect
14. C4×#379C2semidirect
17. C23×C4⋊C4C22semidirect
18. C23×C4⋊C4C22semidirect
23. C2×#397C22semidirect
35. C2×#380C22semidirect
36. C2×#380C22semidirect
40. C2×C4×C22⋊C4C22semidirect
41. C2×C4×C22⋊C4C22semidirect
44. C22×#95C22semidirect
45. #1445C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
56. #1437C22semidirect
57. #1434C22semidirect
61. C22⋊#95C22semidirect
62. C22⋊#95C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3862
C22×#3802
#168842
#150912
#150192
#148422
#148202
#147762
#140202
#139662
#139032
#138692
#138672
C4×#3792