Finite Group Databasebeta

← #613445

Subgroups — #613445

Order 512.

5832 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic, complemented
3C2×#2591C211yesmaximal, characteristic, complemented
4C2×#2539C211yesmaximal, characteristic
5#15069C212yesmaximal, complemented
6#14911C212yesmaximal, complemented
7#14893C214yesmaximal, complemented
8#14886C214yesmaximal, complemented
9#14706C212yesmaximal, complemented
10#14694C212yesmaximal, complemented
11#14071C218yesmaximal, complemented
12#13923C211yesmaximal, characteristic, complemented
13#13903C211yesmaximal, characteristic, complemented
14C4×#381C211yesmaximal, characteristic, complemented
15C25×C4C2211yescharacteristic
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C22×#117—28no—
21C22×#117—28no—
22C22×#117—28no—
23C22×#117C2214yes—
24C22×#117C2214yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C42C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2854—28no—
32#2591—28no—
33#2539—28no—
34C2×#400C2212yescomplemented
35C2×#399C2212yes—
36C2×#394C2212yescomplemented
37C2×#392C2212yes—
38C2×#390C2214yes—
39C2×#390C2212yescomplemented
40C2×#389C2212yescomplemented
41C2×#388C2214yescomplemented
42C2×#388C2214yescomplemented
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2218yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#379C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1444C2218yescomplemented
54#1441C2214yescomplemented
55#1439C2214yes—
56#1439C2218yes—
57#1439C2218yescomplemented
58#1438C2218yes—
59#1437C2214yes—
60#1437C2218yescomplemented
61#1436C2218yescomplemented
62#1436C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2218yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C4×#95C2214yescomplemented
69C26C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2311yescharacteristic
73C24×C4—24no—
74C24×C4—24no—
75C24×C4—24no—
76C24×C4—216no—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—44no—
88C2×#123—44no—
89C2×#123—44no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—24no—
93C2×#120—28no—
94C2×#120—28no—
95C2×#120—44no—
96C2×#117—2128no—
97C2×#117—216no—
98C2×#117—216no—
99C2×#117—216no—
100C2×#117—216no—
101C2×#117—216no—
102C2×#117—216no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4D418yescomplemented
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4D418yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2318yes—
136C22×C42—24no—
137C22×C42—24no—
138C22×C42—24no—
139C22×C42—24no—
140C22×C42—24no—
141C22×C42D418yescomplemented
142C22×C42C2314yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2312yes—
146C22×C42C2312yes—
147#400—28no—
148#399—28no—
149#394—28no—
150#392—28no—
151#390—216no—
152#390—28no—
153#389—28no—
154#388—216no—
155#388—216no—
156#386—28no—
157#386—28no—
158#386—28no—
159#386—232no—
160#381—28no—
161#381—44no—
162#380—24no—
163#379—44no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—24no—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2314yes—
173C2×#95C2318yes—
174C2×#95C2312yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C25—22no—
178C25—22no—
179C25—28no—
180C25—24no—
181C25—24no—
182C25—44no—
183C25C2×D414yes—
184C25C2412yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—216no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—28no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—48no—
203C23×C4—232no—
204C23×C4—264no—
205C23×C4—264no—
206C23×C4—264no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—28no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4#41116yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251#123—432no—
252#123—432no—
253#121—416no—
254#120—432no—
255#117—2128no—
256#117—2128no—
257C2×C4⋊C4—232no—
258C2×C4⋊C4—232no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C22⋊C4—232no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—232no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
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—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—216no—
307C2×C22⋊C4—216no—
308C2×C22⋊C4—216no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4C2×D418yes—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4#4118yes—
320C2×C42—232no—
321C2×C42—232no—
322C2×C42—28no—
323C2×C42—28no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D418yes—
334C2×C42C2×D418yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—216no—
342#95—216no—
343#95—216no—
344#95—48no—
345#95—48no—
346C24—24no—
347C24—24no—
348C24—216no—
349C24—216no—
350C24—28no—
351C24—216no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—28no—
358C24—216no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—28no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—28no—
368C24—48no—
369C24—432no—
370C24—432no—
371C24—48no—
372C24—48no—
373C24—44no—
374C24—44no—
375C24—48no—
376C24C22×D414yes—
377C24C22×D412yes—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C2×#4118yes—
381C24C2×#4114yes—
382C24H(5,𝔽2)14yes—
383C24C2511yescharacteristic
384C2×D4—416no—
385C2×D4—432no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—416no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—88no—
399C2×D4—816no—
400C22×C4—232no—
401C22×C4—2256no—
402C22×C4—232no—
403C22×C4—232no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—232no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—232no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—232no—
423C22×C4—232no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—216no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4#120116yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4118yes—
487C22×C4C2×#4118yes—
488C4⋊C4—232no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—416no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—44no—
495C22⋊C4—232no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—88no—
508C22⋊C4—816no—
509C42—232no—
510C42—216no—
511C42—216no—
512C42—84no—
513C42—84no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—216no—
519C23—24no—
520C23—24no—
521C23—216no—
522C23—232no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—216no—
529C23—28no—
530C23—216no—
531C23—464no—
532C23—416no—
533C23—432no—
534C23—464no—
535C23—432no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—416no—
540C23—44no—
541C23—48no—
542C23—44no—
543C23—48no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—48no—
556C23—48no—
557C23—44no—
558C23—44no—
559C23—48no—
560C23—48no—
561C23C2×#12014yes—
562C23#52518yes—
563C23#53214yes—
564C23#53514yes—
565C23#54618yes—
566C23#55018yes—
567C23C23×D411yescharacteristic
568C23C23×D411yescharacteristic
569C23C23×D411yescharacteristic
570C23C22×#4114yes—
571C23C22×#4112yes—
572C23C2×H(5,𝔽2)12yes—
573C23#58514yes—
574D4—88no—
575D4—832no—
576D4—816no—
577C2×C4—232no—
578C2×C4—232no—
579C2×C4—2128no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—216no—
602C2×C4—28no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4C2×#12014yes—
635C2×C4C2×#12014yes—
636C2×C4#52518yes—
637C2×C4#52518yes—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—216no—
642C22—28no—
643C22—216no—
644C22—28no—
645C22—216no—
646C22—464no—
647C22—44no—
648C22—48no—
649C22—44no—
650C22—416no—
651C22—48no—
652C22—48no—
653C22—82no—
654C22—82no—
655C22—84no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22#243118yes—
663C22C22×#12011yescharacteristic
664C22C2×#52512yes—
665C22C2×#52512yes—
666C22C2×#52512yes—
667C22C2×#53211yescharacteristic
668C22C2×#53511yescharacteristic
669C22#344912yes—
670C22C2×#54612yes—
671C22C2×#55012yes—
672C22#347518yes—
673C22#348512yes—
674C22#348512yes—
675C22#349418yes—
676C4—28no—
677C4—28no—
678C4—24no—
679C4—24no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—48no—
684C4—82no—
685C4—84no—
686C4—84no—
687C2—28no—
688C2—48no—
689C2—162no—
690C2C2×#243111yescharacteristic
691C2#3487712yes—
692C2#3508811yescharacteristic
693C2#3530112yes—
694C2#3634014yes—
695C2#3651714yes—
696C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2539C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#117C2semidirect
2. C2×#2854C2semidirect
3. C2×#2591C2semidirect
5. #15069C2semidirect
6. #14911C2semidirect
7. #14893C2semidirect
8. #14886C2semidirect
9. #14706C2semidirect
10. #14694C2semidirect
11. #14071C2semidirect
12. #13923C2semidirect
13. #13903C2semidirect
14. C4×#381C2semidirect
34. C2×#400C22semidirect
36. C2×#394C22semidirect
39. C2×#390C22semidirect
40. C2×#389C22semidirect
41. C2×#388C22semidirect
42. C2×#388C22semidirect
53. #1444C22semidirect
54. #1441C22semidirect
57. #1439C22semidirect
60. #1437C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1434C22semidirect
68. C4×#95C22semidirect
108. C22×C4⋊C4D4semidirect
141. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1172
C2×#28542
C2×#25912
C2×#25392
#150692
#149112
#148932
#148862
#147062
#146942
#140712
#139232
#139032
C4×#3812