Finite Group Databasebeta

← #613466

Subgroups — #613466

Order 512.

4114 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2549C211yesmaximal, characteristic, complemented
2C22×C4×C22⋊C4C211yesmaximal, characteristic
3#14775C214yesmaximal, complemented
4#14770C214yesmaximal, complemented
5#14726C212yesmaximal
6#14717C212yesmaximal
7#14702C212yesmaximal, complemented
8#14695C212yesmaximal, complemented
9#14680C211yesmaximal, characteristic
10#14213C214yesmaximal, complemented
11#14210C214yesmaximal, complemented
12#14149C211yesmaximal, characteristic, complemented
13#13966C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C22⋊C4C2212yes—
19C23×C42C2212yes—
20#2549—28no—
21C2×#398C2212yescomplemented
22C2×#396C2214yes—
23C2×#395C2212yescomplemented
24C2×#393C2214yes—
25C2×#391C2212yescomplemented
26C2×#391C2214yescomplemented
27C2×#389C2212yescomplemented
28C2×#387C2214yescomplemented
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#379C2211yescharacteristic
36C2×#379C2211yescharacteristic
37C2×#379C2212yes—
38C2×C4×C22⋊C4—28no—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2214yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1444C2212yescomplemented
48#1442C2212yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63C4⋊#95C2214yescomplemented
64#1434C2214yescomplemented
65#1434C2212yescomplemented
66#1434C2212yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—48no—
82C24×C4Q818yescomplemented
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—216no—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42—28no—
116C22×C42—48no—
117C22×C42C2314yes—
118C22×C42C2314yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121#398—28no—
122#396—216no—
123#395—28no—
124#393—216no—
125#391—28no—
126#391—48no—
127#389—28no—
128#387—48no—
129#385—28no—
130#385—28no—
131#380—24no—
132#380—24no—
133#380—48no—
134#380—44no—
135#379—24no—
136#379—24no—
137#379—44no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—216no—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—48no—
144C2×#95—24no—
145C2×#95D418yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C25—22no—
173C25—28no—
174C25—22no—
175C25—22no—
176C25—22no—
177C25—22no—
178C25—22no—
179C25—44no—
180C25C2×Q814yes—
181C25C2412yes—
182C25C2411yescharacteristic
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—28no—
186C22×Q8—42no—
187C22×Q8—44no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—232no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—432no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—432no—
212C23×C4—432no—
213C23×C4C2×D418yes—
214C23×C4C2×D414yes—
215C23×C4C2×Q814yes—
216C23×C4C2×Q818yes—
217C23×C4C2×Q818yes—
218C23×C4C2×Q814yes—
219C23×C4#4114yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C22⋊Q8—432no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C22⋊C4—264no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4#41116yes—
292C2×C22⋊C4#4114yes—
293C2×C22⋊C4#4114yes—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—432no—
303C2×C42—44no—
304C2×C42—44no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95C2×D418yes—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—24no—
332C24—24no—
333C24—216no—
334C24—216no—
335C24—28no—
336C24—216no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—28no—
347C24—28no—
348C24—48no—
349C24—416no—
350C24—416no—
351C24—416no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—44no—
356C24—44no—
357C24—44no—
358C24—44no—
359C24C22×D412yes—
360C24C22×Q814yes—
361C24C22×Q812yes—
362C24C2×#4118yes—
363C24C2×#4112yes—
364C24C2×#4112yes—
365C24C2×#4112yes—
366C24H(5,𝔽2)12yes—
367C242−1+412yes—
368C24C2511yescharacteristic
369C2×Q8—48no—
370C2×Q8—416no—
371C2×Q8—48no—
372C2×Q8—84no—
373C2×Q8—88no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—4128no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4C22⋊Q8116yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q814yes—
463C22×C4C2×#4118yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4118yes—
469C22×C4H(5,𝔽2)18yes—
470C4⋊C4—416no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—88no—
478C4⋊C4—84no—
479C22⋊C4—232no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—232no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C42—416no—
492C42—416no—
493C42—84no—
494C42—84no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—232no—
506C23—24no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—432no—
519C23—432no—
520C23—416no—
521C23—432no—
522C23—432no—
523C23—432no—
524C23—432no—
525C23—416no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23C2×C22⋊Q814yes—
537C23#53318yes—
538C23#53314yes—
539C23#54414yes—
540C23#54614yes—
541C23#54814yes—
542C23#55514yes—
543C23#55614yes—
544C23C23×D411yescharacteristic
545C23C23×Q811yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4114yes—
548C23C22×#4111yescharacteristic
549C23C22×#4111yescharacteristic
550C23C2×H(5,𝔽2)14yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×2−1+411yescharacteristic
553Q8—816no—
554C2×C4—232no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—416no—
580C2×C4—432no—
581C2×C4—432no—
582C2×C4—432no—
583C2×C4—432no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4C2×C22⋊Q814yes—
624C2×C4C2×C22⋊Q814yes—
625C2×C4#52518yes—
626C2×C4#54318yes—
627C22—28no—
628C22—24no—
629C22—28no—
630C22—24no—
631C22—28no—
632C22—28no—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—464no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—48no—
647C22#243918yes—
648C22C22×C22⋊Q811yescharacteristic
649C22C2×#52512yes—
650C22C2×#53312yes—
651C22C2×#53311yescharacteristic
652C22#344412yes—
653C22C2×#54312yes—
654C22C2×#54411yescharacteristic
655C22C2×#54611yescharacteristic
656C22C2×#54811yescharacteristic
657C22C2×#55511yescharacteristic
658C22C2×#55611yescharacteristic
659C22#348114yes—
660C22#348114yes—
661C22#348412yes—
662C22#349012yes—
663C22#351914yes—
664C22#352014yes—
665C4—28no—
666C4—24no—
667C4—24no—
668C4—28no—
669C4—44no—
670C4—48no—
671C4—44no—
672C4—44no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—84no—
678C2—28no—
679C2—48no—
680C2C2×#243911yescharacteristic
681C2#3495812yes—
682C2#3498211yescharacteristic
683C2#3507912yes—
684C2#3631612yes—
685C2#3633912yes—
686C2#3652812yes—
687C2#3653112yes—
688C2#3662511yescharacteristic
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
3. #14775C2semidirect
4. #14770C2semidirect
7. #14702C2semidirect
8. #14695C2semidirect
10. #14213C2semidirect
11. #14210C2semidirect
12. #14149C2semidirect
13. #13966C2semidirect
14. #13908C2semidirect
21. C2×#398C22semidirect
23. C2×#395C22semidirect
25. C2×#391C22semidirect
26. C2×#391C22semidirect
27. C2×#389C22semidirect
28. C2×#387C22semidirect
47. #1444C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. C4⋊#95C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
82. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25492
C22×C4×C22⋊C42
#147752
#147702
#147262
#147172
#147022
#146952
#146802
#142132
#142102
#141492
#139662
#139082
C4×#3792