Finite Group Databasebeta

← #848398

Subgroups — #848398

Order 512.

4224 conjugacy classes of subgroups.

The subgroup lattice has 799 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2539C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic, complemented
3#16473C212yesmaximal, complemented
4#16282C212yesmaximal, complemented
5#15962C214yesmaximal, complemented
6#15730C212yesmaximal, complemented
7#15635C212yesmaximal, complemented
8#14912C212yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14718C212yesmaximal, complemented
11#14694C212yesmaximal, complemented
12#14536C214yesmaximal, complemented
13#14135C212yesmaximal, complemented
14#14039C212yesmaximal, complemented
15#13868C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2539—28no—
25C2×#399C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#394C2214yes—
28C2×#394C2214yescomplemented
29C2×#392C2211yescharacteristic
30C2×#392C2211yescharacteristic
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#387C2214yescomplemented
34C2×#386C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#385—28no—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#381C2211yescharacteristic
47C2×#379C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2212yes—
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2214yes—
57#1440C2214yescomplemented
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1438C2214yes—
66#1437C2212yes—
67#1437C2212yes—
68#1437C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4—24no—
83C24×C4—48no—
84C24×C4—42no—
85C24×C4—42no—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—44no—
95C2×#121—24no—
96C2×#121—42no—
97C2×#121—42no—
98C2×#120—28no—
99C2×#120—28no—
100C2×#120—44no—
101C2×#117—44no—
102C2×#117—42no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—48no—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—42no—
115C22×C22⋊C4—42no—
116C22×C22⋊C4D418yes—
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⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139#399—24no—
140#399—24no—
141#394—216no—
142#394—48no—
143#392—24no—
144#392—24no—
145#390—28no—
146#390—28no—
147#387—48no—
148#386—48no—
149#386—48no—
150#386—44no—
151#386—44no—
152#386—44no—
153#386—44no—
154#385—28no—
155#385—28no—
156#385—28no—
157#385—28no—
158#385—28no—
159#381—42no—
160#379—44no—
161C4×C22⋊C4—24no—
162C2×#95—28no—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
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×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C25—24no—
183C25—22no—
184C25—22no—
185C25—24no—
186C25—24no—
187C25—24no—
188C25—44no—
189C25C2×D414yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C25C2411yescharacteristic
193C25C2411yescharacteristic
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—28no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—41no—
211C22×D4—41no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—28no—
232C23×C4—28no—
233C23×C4—44no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—44no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—432no—
243C23×C4—416no—
244C23×C4—432no—
245C23×C4—432no—
246C23×C4C2×D414yes—
247C23×C4C2×D414yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2414yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2414yes—
259#123—432no—
260#121—416no—
261#120—432no—
262#117—416no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—432no—
284C2×C22⋊C4—232no—
285C2×C22⋊C4—24no—
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—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—416no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—41no—
315C2×C22⋊C4—41no—
316C2×C22⋊C4—41no—
317C2×C22⋊C4—41no—
318C2×C22⋊C4—41no—
319C2×C22⋊C4—41no—
320C2×C22⋊C4—41no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4#4118yes—
344C2×C22⋊C4#4118yes—
345C2×C42—24no—
346C2×C42—42no—
347C2×C42—44no—
348C2×C42—44no—
349#95—28no—
350#95—28no—
351#95—22no—
352#95—22no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—42no—
358#95—42no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—216no—
372C24—216no—
373C24—28no—
374C24—28no—
375C24—28no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—21no—
385C24—24no—
386C24—24no—
387C24—22no—
388C24—216no—
389C24—22no—
390C24—24no—
391C24—22no—
392C24—22no—
393C24—24no—
394C24—24no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—24no—
400C24—48no—
401C24—416no—
402C24—416no—
403C24—416no—
404C24—416no—
405C24—48no—
406C24—48no—
407C24—48no—
408C24—44no—
409C24—44no—
410C24C22×D412yes—
411C24C22×D412yes—
412C24C22×D412yes—
413C24C2×#4114yes—
414C24C2×#4114yes—
415C24H(5,𝔽2)12yes—
416C24H(5,𝔽2)12yes—
417C24H(5,𝔽2)14yes—
418C242−1+414yes—
419C24C2511yescharacteristic
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—416no—
423C2×D4—416no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—88no—
438C2×D4—88no—
439C2×D4—84no—
440C2×D4—84no—
441C22×C4—216no—
442C22×C4—216no—
443C22×C4—28no—
444C22×C4—216no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
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—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—28no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—416no—
505C22×C4—416no—
506C22×C4—416no—
507C22×C4—416no—
508C22×C4—416no—
509C22×C4—416no—
510C22×C4—416no—
511C22×C4—416no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—4128no—
517C22×C4—48no—
518C22×C4—48no—
519C22×C4—48no—
520C22×C4—48no—
521C22×C4—48no—
522C22×C4—44no—
523C22×C4—44no—
524C22×C4—84no—
525C22×C4—84no—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C2×#4114yes—
531C22×C4C2×#4118yes—
532C22×C4C2×#4114yes—
533C22×C4H(5,𝔽2)18yes—
534C4⋊C4—416no—
535C4⋊C4—416no—
536C4⋊C4—416no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C22⋊C4—216no—
542C22⋊C4—216no—
543C22⋊C4—216no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—88no—
556C22⋊C4—88no—
557C22⋊C4—88no—
558C22⋊C4—88no—
559C22⋊C4—88no—
560C22⋊C4—88no—
561C42—48no—
562C42—84no—
563C23—22no—
564C23—22no—
565C23—28no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—28no—
570C23—28no—
571C23—24no—
572C23—24no—
573C23—28no—
574C23—232no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—24no—
581C23—28no—
582C23—24no—
583C23—24no—
584C23—24no—
585C23—24no—
586C23—24no—
587C23—24no—
588C23—28no—
589C23—28no—
590C23—28no—
591C23—432no—
592C23—432no—
593C23—416no—
594C23—464no—
595C23—432no—
596C23—432no—
597C23—42no—
598C23—42no—
599C23—416no—
600C23—41no—
601C23—41no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—41no—
606C23—41no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—44no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—44no—
621C23—44no—
622C23—42no—
623C23—42no—
624C23—44no—
625C23—44no—
626C23—44no—
627C23—44no—
628C23—44no—
629C23—44no—
630C23—42no—
631C23—44no—
632C23—44no—
633C23—44no—
634C23—44no—
635C23#52514yes—
636C23#53414yes—
637C23#53614yes—
638C23#54018yes—
639C23#54618yes—
640C23#55914yes—
641C23#56014yes—
642C23C23×D411yescharacteristic
643C23C22×#4112yes—
644C23C22×#4112yes—
645C23C2×H(5,𝔽2)11yescharacteristic
646C23C2×H(5,𝔽2)11yescharacteristic
647C23C2×H(5,𝔽2)12yes—
648C23C2×H(5,𝔽2)14yes—
649C23C2×2−1+412yes—
650D4—816no—
651D4—816no—
652D4—88no—
653D4—816no—
654C2×C4—28no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—28no—
664C2×C4—28no—
665C2×C4—24no—
666C2×C4—28no—
667C2×C4—24no—
668C2×C4—28no—
669C2×C4—28no—
670C2×C4—28no—
671C2×C4—28no—
672C2×C4—432no—
673C2×C4—432no—
674C2×C4—432no—
675C2×C4—432no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—416no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—42no—
702C2×C4—44no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—42no—
707C2×C4—42no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—44no—
711C2×C4—44no—
712C2×C4—44no—
713C2×C4—44no—
714C2×C4—44no—
715C2×C4—44no—
716C2×C4—42no—
717C2×C4—44no—
718C2×C4—44no—
719C2×C4—44no—
720C2×C4—44no—
721C2×C4—44no—
722C2×C4—88no—
723C22—24no—
724C22—28no—
725C22—216no—
726C22—28no—
727C22—28no—
728C22—28no—
729C22—24no—
730C22—28no—
731C22—28no—
732C22—24no—
733C22—24no—
734C22—464no—
735C22—42no—
736C22—42no—
737C22—44no—
738C22—48no—
739C22—48no—
740C22—42no—
741C22—42no—
742C22—44no—
743C22—44no—
744C22—44no—
745C22—44no—
746C22—44no—
747C22—82no—
748C22—82no—
749C22—82no—
750C22—84no—
751C22—84no—
752C22—82no—
753C22—84no—
754C22—82no—
755C22—82no—
756C22—84no—
757C22—82no—
758C22#270918yes—
759C22C2×#52511yescharacteristic
760C22C2×#53411yescharacteristic
761C22C2×#53611yescharacteristic
762C22C2×#54012yes—
763C22C2×#54612yes—
764C22C2×#55911yescharacteristic
765C22#348114yes—
766C22#348412yes—
767C22#349212yes—
768C22#349212yes—
769C22#351914yes—
770C22#352014yes—
771C22C2×#56011yescharacteristic
772C22#353212yes—
773C22#353212yes—
774C22#353614yes—
775C4—48no—
776C4—42no—
777C4—42no—
778C4—44no—
779C4—42no—
780C4—42no—
781C4—48no—
782C4—44no—
783C4—84no—
784C4—82no—
785C4—84no—
786C2—28no—
787C2—44no—
788C2—44no—
789C2—162no—
790C2#3505711yescharacteristic
791C2#3631612yes—
792C2#3671411yescharacteristic
793C2#3802112yes—
794C2#3854314yes—
795C2#4574312yes—
796C2C2×#270911yescharacteristic
797C2#5962412yes—
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
2. C22×#385C2semidirect
3. #16473C2semidirect
4. #16282C2semidirect
5. #15962C2semidirect
6. #15730C2semidirect
7. #15635C2semidirect
8. #14912C2semidirect
9. #14719C2semidirect
10. #14718C2semidirect
11. #14694C2semidirect
12. #14536C2semidirect
13. #14135C2semidirect
14. #14039C2semidirect
15. #13868C2semidirect
28. C2×#394C22semidirect
33. C2×#387C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
57. #1440C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
68. #1437C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25392
C22×#3852
#164732
#162822
#159622
#157302
#156352
#149122
#147192
#147182
#146942
#145362
#141352
#140392
#138682