Finite Group Databasebeta

← #636396

Subgroups — #636396

Order 512.

4018 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2549C211yesmaximal, characteristic, complemented
2C2×#2626C212yesmaximal
3C22×#380C211yesmaximal, characteristic
4#16305C214yesmaximal, complemented
5#16303C214yesmaximal, complemented
6#14771C214yesmaximal, complemented
7#14770C214yesmaximal, complemented
8#14728C212yesmaximal, complemented
9#14720C212yesmaximal, complemented
10#14680C211yesmaximal, characteristic
11#14262C212yesmaximal
12#14149C211yesmaximal, characteristic, complemented
13#13965C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C4⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20#2626—216no—
21#2549—28no—
22C2×#399C2214yes—
23C2×#398C2212yescomplemented
24C2×#398C2214yescomplemented
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#395C2212yescomplemented
28C2×#395C2214yescomplemented
29C2×#393C2214yes—
30C2×#391C2212yescomplemented
31C2×#389C2212yescomplemented
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#380—28no—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×#380C2214yes—
44C2×#379C2212yes—
45C2×#379C2211yescharacteristic
46C2×#379C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2214yes—
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1444C2212yescomplemented
53#1444C2214yescomplemented
54#1444C2214yescomplemented
55#1444C2214yescomplemented
56#1444C2214yescomplemented
57#1442C2212yescomplemented
58#1442C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2212yescomplemented
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C22⋊#95C2214yes—
78C26C2311yescharacteristic
79C23×Q8C2312yes—
80C24×C4—24no—
81C24×C4—24no—
82C24×C4—48no—
83C24×C4Q818yescomplemented
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—48no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#399—216no—
126#398—28no—
127#398—48no—
128#396—216no—
129#396—216no—
130#395—28no—
131#395—48no—
132#393—216no—
133#391—28no—
134#389—28no—
135#385—28no—
136#385—28no—
137#385—28no—
138#385—44no—
139#380—24no—
140#380—24no—
141#380—28no—
142#380—216no—
143#380—28no—
144#380—28no—
145#380—48no—
146#379—28no—
147#379D418yes—
148#379Q818yescomplemented
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—48no—
152C2×#95—24no—
153C2×#95D418yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C25—22no—
179C25—22no—
180C25—22no—
181C25—22no—
182C25—44no—
183C25—44no—
184C25C2×D414yes—
185C25C2×Q814yes—
186C25C2×Q814yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—44no—
193C22×Q8—44no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—232no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—432no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—432no—
218C23×C4—432no—
219C23×C4C2×D418yes—
220C23×C4C2×D414yes—
221C23×C4C2×Q814yes—
222C23×C4C2×Q818yes—
223C23×C4C2×Q818yes—
224C23×C4C2×Q814yes—
225C23×C4#4114yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C22⋊Q8—432no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
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—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
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—432no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D414yes—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×D4116yes—
310C2×C22⋊C4C2×Q814yes—
311C2×C22⋊C4C2×Q814yes—
312C2×C22⋊C4C2×Q8116yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—44no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C2×D418yes—
339#95C2×D414yes—
340#95C2×D414yes—
341C24—24no—
342C24—24no—
343C24—28no—
344C24—28no—
345C24—28no—
346C24—28no—
347C24—28no—
348C24—28no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—48no—
358C24—416no—
359C24—48no—
360C24—416no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—416no—
366C24—416no—
367C24C4⋊Q8116yes—
368C24C22×D414yes—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C22×Q814yes—
372C24C22×Q814yes—
373C24C22×Q812yes—
374C24C22×Q812yes—
375C24C2×#4112yes—
376C24H(5,𝔽2)12yes—
377C242−1+412yes—
378C24C2511yescharacteristic
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—84no—
382C2×Q8—88no—
383C2×Q8—88no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—216no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—4128no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4C22⋊Q8116yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q814yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q814yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)18yes—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—84no—
506C42—84no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—232no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—432no—
533C23—416no—
534C23—416no—
535C23—432no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23C2×C22⋊Q814yes—
549C23C2×C4⋊Q818yes—
550C23C2×C4⋊Q814yes—
551C23#54414yes—
552C23D4214yes—
553C23#54714yes—
554C23D4×Q814yes—
555C23#55414yes—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C23×Q811yescharacteristic
559C23C23×Q811yescharacteristic
560C23C22×#4111yescharacteristic
561C23C2×H(5,𝔽2)14yes—
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×2−1+411yescharacteristic
564C23#58514yes—
565Q8—816no—
566C2×C4—232no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—416no—
592C2×C4—432no—
593C2×C4—432no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—816no—
635C2×C4—816no—
636C2×C4C2×C22⋊Q814yes—
637C2×C4C2×C22⋊Q814yes—
638C2×C4#54318yes—
639C22—24no—
640C22—28no—
641C22—28no—
642C22—28no—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—464no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—48no—
655C22—48no—
656C22—44no—
657C22—48no—
658C22—48no—
659C22#244018yes—
660C22C22×C22⋊Q811yescharacteristic
661C22C22×C4⋊Q811yescharacteristic
662C22#344912yes—
663C22#345512yes—
664C22C2×#54312yes—
665C22C2×#54411yescharacteristic
666C22#345812yes—
667C22C2×D4211yescharacteristic
668C22C2×#54711yescharacteristic
669C22C2×D4×Q811yescharacteristic
670C22C2×#55411yescharacteristic
671C22#348412yes—
672C22#349012yes—
673C22#349614yes—
674C22#350114yes—
675C22#354114yes—
676C22#354214yes—
677C4—28no—
678C4—24no—
679C4—24no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4—84no—
690C2—28no—
691C2—48no—
692C2#3463411yescharacteristic
693C2C2×#244011yescharacteristic
694C2#3498211yescharacteristic
695C2#3500212yes—
696C2#3747612yes—
697C2#3933212yes—
698C2#3934412yes—
699C2#3958812yes—
700C2#3960012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2549C2semidirect
4. #16305C2semidirect
5. #16303C2semidirect
6. #14771C2semidirect
7. #14770C2semidirect
8. #14728C2semidirect
9. #14720C2semidirect
12. #14149C2semidirect
13. #13965C2semidirect
14. #13908C2semidirect
23. C2×#398C22semidirect
24. C2×#398C22semidirect
27. C2×#395C22semidirect
28. C2×#395C22semidirect
30. C2×#391C22semidirect
31. C2×#389C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
83. C24×C4Q8semidirect
148. #379Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25492
C2×#26262
C22×#3802
#163052
#163032
#147712
#147702
#147282
#147202
#146802
#142622
#141492
#139652
#139082
C4×#3792