Finite Group Databasebeta

← #640629

Subgroups — #640629

Order 512.

4208 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#16895C214yesmaximal, complemented
4#16816C214yesmaximal, complemented
5#15233C212yesmaximal, complemented
6#15219C212yesmaximal, complemented
7#15195C212yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#14718C214yesmaximal, complemented
10#14259C214yesmaximal, complemented
11#14137C211yesmaximal, characteristic, complemented
12#14137C211yesmaximal, characteristic, complemented
13#13881C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#120C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24#2847—28no—
25C2×#394C2218yes—
26C2×#390C2212yescomplemented
27C2×#390C2212yescomplemented
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#385C2214yes—
34C2×#384C2212yescomplemented
35C2×#382C2212yescomplemented
36C2×#380—28no—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2212yes—
50#1444C2214yescomplemented
51#1443C2218yescomplemented
52#1442C2214yescomplemented
53#1442C2218yescomplemented
54#1440C2218yes—
55#1440C2218yescomplemented
56#1439C2218yes—
57#1439C2218yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1438C2218yes—
61C4⋊#95C2212yescomplemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4—24no—
75C24×C4—24no—
76C24×C4—44no—
77C24×C4—44no—
78C24×C4D418yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#121—24no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—44no—
93C2×#121—44no—
94C2×#120—28no—
95C2×#120—48no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2318yes—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—48no—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2318yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#394—416no—
127#390—44no—
128#390—44no—
129#386—28no—
130#386—28no—
131#386—28no—
132#386—216no—
133#386—48no—
134#385—48no—
135#384—44no—
136#382—44no—
137#380—24no—
138#380—24no—
139#380—24no—
140#380—28no—
141#380—28no—
142#380—28no—
143#380—28no—
144#379—28no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—44no—
148C4×C22⋊C4—44no—
149C2×#95—28no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2318yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C25—22no—
165C25—22no—
166C25—24no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—42no—
171C25—42no—
172C25C2×D414yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—232no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—416no—
217C23×C4—416no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—432no—
229C23×C4—432no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4#4114yes—
237C23×C4#4114yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249#121—416no—
250#121—416no—
251#121—416no—
252#120—432no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—48no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—432no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4#4118yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—216no—
325#95—44no—
326#95—44no—
327#95—48no—
328#95—48no—
329#95—48no—
330#95—48no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—216no—
336C24—216no—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—28no—
355C24—28no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—44no—
366C24—44no—
367C24—416no—
368C24—416no—
369C24—416no—
370C24—416no—
371C24—48no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2×#4112yes—
379C24C2×#4112yes—
380C24C2×#4114yes—
381C24H(5,𝔽2)14yes—
382C24H(5,𝔽2)14yes—
383C242−1+414yes—
384C24C2511yescharacteristic
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—816no—
402C2×D4—816no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
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—48no—
485C22×C4—816no—
486C22×C4—816no—
487C22×C4#123116yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C2×#4118yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C4H(5,𝔽2)14yes—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—816no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—816no—
520C22⋊C4—816no—
521C42—48no—
522C42—48no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—232no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—416no—
557C23—416no—
558C23—416no—
559C23—416no—
560C23—416no—
561C23—464no—
562C23—464no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—416no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—48no—
590C23—48no—
591C23—48no—
592C23—48no—
593C23C2×#12314yes—
594C23#53214yes—
595C23#53514yes—
596C23#53514yes—
597C23#53614yes—
598C23#55318yes—
599C23#55518yes—
600C23C23×D411yescharacteristic
601C23C22×#4111yescharacteristic
602C23C22×#4111yescharacteristic
603C23C22×#4112yes—
604C23C2×H(5,𝔽2)12yes—
605C23C2×H(5,𝔽2)12yes—
606C23C2×H(5,𝔽2)12yes—
607C23C2×H(5,𝔽2)12yes—
608C23C2×2−1+412yes—
609D4—816no—
610D4—816no—
611D4—816no—
612D4—88no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—232no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—416no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—416no—
639C2×C4—416no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—48no—
662C2×C4—48no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—48no—
666C2×C4—44no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4—816no—
671C2×C4—816no—
672C2×C4C2×#12314yes—
673C2×C4C2×#12314yes—
674C2×C4#53414yes—
675C2×C4#53414yes—
676C2×C4#53818yes—
677C2×C4#53818yes—
678C22—24no—
679C22—24no—
680C22—216no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—216no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—464no—
693C22—42no—
694C22—42no—
695C22—48no—
696C22—48no—
697C22—42no—
698C22—42no—
699C22—48no—
700C22—48no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—82no—
706C22—82no—
707C22—88no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—88no—
712C22—82no—
713C22—82no—
714C22#250018yes—
715C22C22×#12311yescharacteristic
716C22C2×#53211yescharacteristic
717C22C2×#53411yescharacteristic
718C22C2×#53411yescharacteristic
719C22C2×#53511yescharacteristic
720C22C2×#53511yescharacteristic
721C22C2×#53611yescharacteristic
722C22C2×#53812yes—
723C22C2×#53812yes—
724C22C2×#55312yes—
725C22C2×#55512yes—
726C22#349212yes—
727C22#349212yes—
728C22#351914yes—
729C22#352014yes—
730C22#353614yes—
731C22#353714yes—
732C4—24no—
733C4—48no—
734C4—42no—
735C4—42no—
736C4—84no—
737C4—84no—
738C4—84no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4#262218yes—
743C2—28no—
744C2—44no—
745C2—44no—
746C2—162no—
747C2C2×#262211yescharacteristic
748C2C2×#250011yescharacteristic
749C2#3507811yescharacteristic
750C2#3508511yescharacteristic
751C2#3522711yescharacteristic
752C2#3745011yescharacteristic
753C2#3745111yescharacteristic
754C2#3893814yes—
755C2#3914414yes—
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. C22×#380C2semidirect
3. #16895C2semidirect
4. #16816C2semidirect
5. #15233C2semidirect
6. #15219C2semidirect
7. #15195C2semidirect
8. #15130C2semidirect
9. #14718C2semidirect
10. #14259C2semidirect
11. #14137C2semidirect
12. #14137C2semidirect
13. #13881C2semidirect
14. #13880C2semidirect
15. C4×#379C2semidirect
26. C2×#390C22semidirect
27. C2×#390C22semidirect
34. C2×#384C22semidirect
35. C2×#382C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
55. #1440C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
61. C4⋊#95C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C22×#3802
#168952
#168162
#152332
#152192
#151952
#151302
#147182
#142592
#141372
#138812
#138802
C4×#3792