Finite Group Databasebeta

← #894653

Subgroups — #894653

Order 512.

4294 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2679C211yesmaximal, characteristic
3C2×#2472C214yesmaximal, complemented
4#17069C212yesmaximal, complemented
5#16878C212yesmaximal, complemented
6#15613C214yesmaximal, complemented
7#15268C214yesmaximal, complemented
8#15070C212yesmaximal, complemented
9#14968C214yesmaximal, complemented
10#14700C212yesmaximal, complemented
11#14357C212yesmaximal, complemented
12#13967C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×C22⋊Q8C2214yescomplemented
16C22×#120C2212yes—
17C22×#120C2214yes—
18C22×C4×D4C2211yescharacteristic
19C23×C22⋊C4C2212yescomplemented
20#2683—28no—
21#2679—28no—
22#2472—232no—
23C2×#397C2214yescomplemented
24C2×#396C2214yescomplemented
25C2×#395C2212yescomplemented
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2212yes—
30C2×#393C2214yescomplemented
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2212yescomplemented
37C2×#388C2214yescomplemented
38C2×#388C2214yescomplemented
39C2×#388C2214yescomplemented
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#386C2214yes—
43C2×#379C2214yescomplemented
44C2×#379C2212yescomplemented
45C2×C4×C4⋊C4C2211yescharacteristic, complemented
46C2×C4×C22⋊C4C2214yescomplemented
47C2×C43C2211yescharacteristic, complemented
48#1446C2212yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1438C2214yes—
58#1438C2214yes—
59#1438C2212yes—
60#1437C2214yescomplemented
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C4⋊#95C2211yescharacteristic, complemented
66C4⋊#95C2211yescharacteristic, complemented
67#1434C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C22⋊#95C2212yescomplemented
70C26C2312yes—
71C23×Q8C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#124—216no—
79C2×#124—44no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—48no—
83C2×#120—28no—
84C2×#120—216no—
85C2×#120—28no—
86C2×#120—28no—
87C2×#120—216no—
88C2×#120—48no—
89C2×C4×D4—24no—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—216no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#397—216no—
126#396—216no—
127#395—28no—
128#394—216no—
129#394—28no—
130#394—216no—
131#394—216no—
132#393—216no—
133#390—24no—
134#390—28no—
135#390—44no—
136#390D418yes—
137#390D4116yescomplemented
138#389—28no—
139#388—216no—
140#388—216no—
141#388—48no—
142#386—28no—
143#386—216no—
144#386—216no—
145#379—216no—
146#379—44no—
147C4×C4⋊C4—42no—
148C4×C22⋊C4—216no—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C43—42no—
164C25—28no—
165C25—24no—
166C25—24no—
167C25—24no—
168C25—42no—
169C25—44no—
170C25—42no—
171C25—44no—
172C25—44no—
173C25—42no—
174C25—42no—
175C25C2414yes—
176C25C2412yes—
177C22×Q8—24no—
178C22×Q8—28no—
179C22×Q8—44no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4C2×D418yes—
205C22×D4C2×D414yes—
206C22×D4#4114yes—
207C22×D4#4114yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#124—432no—
230C22⋊Q8—464no—
231#120—464no—
232#120—432no—
233C4×D4—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D414yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D4116yes—
308C2×C22⋊C4C2×D4116yes—
309C2×C22⋊C4C2×D4116yes—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—28no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—41no—
319C2×C42—42no—
320C2×C42—41no—
321C2×C42—41no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—41no—
325C2×C42C2×D418yes—
326C2×C42C2×D414yes—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—44no—
337#95—44no—
338#95—42no—
339#95—44no—
340#95—44no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—48no—
364C24—416no—
365C24—416no—
366C24—416no—
367C24—48no—
368C24—416no—
369C24—48no—
370C24—48no—
371C24—48no—
372C24—84no—
373C24—84no—
374C24—84no—
375C24—84no—
376C24—88no—
377C24—88no—
378C24—88no—
379C24—88no—
380C24—88no—
381C24—88no—
382C24—84no—
383C24—84no—
384C24—88no—
385C24—88no—
386C24C22×D418yes—
387C24C22×D414yes—
388C24C2×#4114yes—
389C24C2×#4114yes—
390C24C2511yescharacteristic
391C2×Q8—416no—
392C2×Q8—48no—
393C2×Q8—88no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×D4—216no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—48no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—84no—
421C2×D4#124116yescomplemented
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—48no—
482C22×C4—88no—
483C22×C4—84no—
484C22×C4—88no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4C22×D414yes—
488C22×C4C22×D418yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)14yes—
499C4⋊C4—216no—
500C4⋊C4—44no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—416no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C22⋊C4—232no—
508C22⋊C4—232no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—84no—
524C42—28no—
525C42—44no—
526C42—48no—
527C42—48no—
528C42—44no—
529C42—48no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—432no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—832no—
589C23—816no—
590C23—816no—
591C23—816no—
592C23—832no—
593C23—832no—
594C23—832no—
595C23—816no—
596C23—816no—
597C23—816no—
598C23C2×#12418yes—
599C23C23×D411yescharacteristic
600C23C23×D412yes—
601C23C22×#4111yescharacteristic
602C23C22×#4111yescharacteristic
603C23C2×H(5,𝔽2)11yescharacteristic
604C23C2×H(5,𝔽2)11yescharacteristic
605C23C2×H(5,𝔽2)12yes—
606C23#58511yescharacteristic
607C23#58512yes—
608C23#58511yescharacteristic
609C23#58512yes—
610Q8—816no—
611D4—416no—
612D4—816no—
613D4—816no—
614D4—832no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—816no—
680C2×C4—816no—
681C2×C4C2×#12414yes—
682C2×C4D4214yes—
683C2×C4D4218yes—
684C22—216no—
685C22—28no—
686C22—48no—
687C22—44no—
688C22—48no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—48no—
694C22—44no—
695C22—48no—
696C22—48no—
697C22—48no—
698C22—48no—
699C22—48no—
700C22—48no—
701C22—48no—
702C22—864no—
703C22—84no—
704C22—84no—
705C22—84no—
706C22—84no—
707C22—84no—
708C22—84no—
709C22—84no—
710C22—84no—
711C22C22×#12411yescharacteristic
712C22#344912yes—
713C22#345911yescharacteristic
714C22#345911yescharacteristic
715C22C2×D4212yes—
716C22C2×D4211yescharacteristic
717C22D4×#4112yes—
718C22D4×#4112yes—
719C22#349011yescharacteristic
720C22#349111yescharacteristic
721C22#349612yes—
722C22#349612yes—
723C22#349611yescharacteristic
724C22#349611yescharacteristic
725C22#350012yes—
726C22#351412yes—
727C22#351412yes—
728C22#352911yescharacteristic
729C22#353412yes—
730C22#354511yescharacteristic
731C22#354911yescharacteristic
732C22#356012yes—
733C22#356812yes—
734C4—24no—
735C4—24no—
736C4—28no—
737C4—44no—
738C4—48no—
739C4—44no—
740C4—42no—
741C4—84no—
742C4—82no—
743C4—82no—
744C4—84no—
745C2—48no—
746C2—88no—
747C2#3499311yescharacteristic
748C2D4×#12412yes—
749C2#3715011yescharacteristic
750C2#3843212yes—
751C2#3935412yes—
752C2#3945212yes—
753C2#4623711yescharacteristic
754C2#4628711yescharacteristic
755C2#5210311yescharacteristic
756C2#5981411yescharacteristic
757C2#5981611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2683C2semidirect
3. C2×#2472C2semidirect
4. #17069C2semidirect
5. #16878C2semidirect
6. #15613C2semidirect
7. #15268C2semidirect
8. #15070C2semidirect
9. #14968C2semidirect
10. #14700C2semidirect
11. #14357C2semidirect
12. #13967C2semidirect
13. C4×#390C2semidirect
15. C22×C22⋊Q8C22semidirect
19. C23×C22⋊C4C22semidirect
23. C2×#397C22semidirect
24. C2×#396C22semidirect
25. C2×#395C22semidirect
30. C2×#393C22semidirect
36. C2×#389C22semidirect
37. C2×#388C22semidirect
38. C2×#388C22semidirect
39. C2×#388C22semidirect
43. C2×#379C22semidirect
44. C2×#379C22semidirect
45. C2×C4×C4⋊C4C22semidirect
46. C2×C4×C22⋊C4C22semidirect
47. C2×C43C22semidirect
48. #1446C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
68. C22⋊#95C22semidirect
69. C22⋊#95C22semidirect
137. #390D4semidirect
421. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#26792
C2×#24722
#170692
#168782
#156132
#152682
#150702
#149682
#147002
#143572
#139672
C4×#3902