Finite Group Databasebeta

← #613530

Subgroups — #613530

Order 512.

4034 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2569C211yesmaximal, characteristic, complemented
2C22×C4×C22⋊C4C211yesmaximal, characteristic
3#14774C214yesmaximal, complemented
4#14771C214yesmaximal, complemented
5#14734C212yesmaximal
6#14726C212yesmaximal
7#14697C212yesmaximal, complemented
8#14695C212yesmaximal, complemented
9#14680C211yesmaximal, characteristic
10#14215C214yesmaximal, complemented
11#14210C214yesmaximal, complemented
12#14149C211yesmaximal, characteristic, complemented
13#13965C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C42C2212yes—
19#2569—28no—
20C2×#400C2212yescomplemented
21C2×#399C2214yes—
22C2×#398C2212yescomplemented
23C2×#396C2214yes—
24C2×#389C2214yescomplemented
25C2×#387C2212yescomplemented
26C2×#387C2212yescomplemented
27C2×#387C2214yescomplemented
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2214yes—
37C2×#379C2211yescharacteristic
38C2×#379C2211yescharacteristic
39C2×#379C2212yes—
40C2×C4×C22⋊C4—28no—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2214yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2214yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1444C2212yescomplemented
50#1444C2214yescomplemented
51#1442C2212yescomplemented
52#1441C2214yescomplemented
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65C4⋊#95C2214yescomplemented
66C4⋊#95C2212yescomplemented
67C4⋊#95C2212yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—48no—
81C24×C4Q818yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—216no—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42—28no—
113C22×C42—48no—
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118#400—28no—
119#399—216no—
120#398—28no—
121#396—216no—
122#389—48no—
123#387—28no—
124#387—28no—
125#387—48no—
126#385—28no—
127#385—28no—
128#385—44no—
129#385—44no—
130#380—24no—
131#380—28no—
132#380—24no—
133#380—28no—
134#380—48no—
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×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
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
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—232no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—432no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—432no—
207C23×C4—432no—
208C23×C4C2×D418yes—
209C23×C4C2×D414yes—
210C23×C4C2×Q814yes—
211C23×C4C2×Q818yes—
212C23×C4C2×Q818yes—
213C23×C4C2×Q814yes—
214C23×C4#4114yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C22⋊C4—264no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4#41116yes—
287C2×C22⋊C4#4114yes—
288C2×C22⋊C4#4114yes—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—432no—
298C2×C42—44no—
299C2×C42—44no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×D418yes—
324#95C2×D414yes—
325#95C2×D414yes—
326C24—24no—
327C24—24no—
328C24—216no—
329C24—216no—
330C24—28no—
331C24—216no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—28no—
342C24—28no—
343C24—48no—
344C24—416no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—48no—
349C24—48no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—44no—
354C24C22×D412yes—
355C24C22×Q814yes—
356C24C22×Q812yes—
357C24C2×#4118yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C24C2×#4112yes—
361C24H(5,𝔽2)12yes—
362C242−1+412yes—
363C24C2511yescharacteristic
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—4128no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4C22⋊Q8116yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q814yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4118yes—
459C22×C4H(5,𝔽2)18yes—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C22⋊C4—232no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—232no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—416no—
483C42—416no—
484C42—84no—
485C42—84no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—232no—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—432no—
510C23—432no—
511C23—416no—
512C23—432no—
513C23—432no—
514C23—432no—
515C23—432no—
516C23—42no—
517C23—416no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23C2×C22⋊Q814yes—
528C23#53318yes—
529C23#53314yes—
530C23#54414yes—
531C23#54714yes—
532C23#55014yes—
533C23#55214yes—
534C23#55314yes—
535C23C23×D411yescharacteristic
536C23C23×Q811yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4114yes—
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)14yes—
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×2−1+411yescharacteristic
544C2×C4—232no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—416no—
570C2×C4—432no—
571C2×C4—432no—
572C2×C4—432no—
573C2×C4—432no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4C2×C22⋊Q814yes—
614C2×C4C2×C22⋊Q814yes—
615C2×C4#52518yes—
616C2×C4#54318yes—
617C22—28no—
618C22—24no—
619C22—28no—
620C22—24no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—464no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22#244118yes—
638C22C22×C22⋊Q811yescharacteristic
639C22C2×#52512yes—
640C22C2×#53312yes—
641C22C2×#53311yescharacteristic
642C22#344412yes—
643C22C2×#54312yes—
644C22C2×#54411yescharacteristic
645C22C2×#54711yescharacteristic
646C22C2×#55011yescharacteristic
647C22C2×#55211yescharacteristic
648C22C2×#55311yescharacteristic
649C22#348114yes—
650C22#348114yes—
651C22#348412yes—
652C22#349012yes—
653C22#351914yes—
654C22#352014yes—
655C4—28no—
656C4—24no—
657C4—24no—
658C4—28no—
659C4—48no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—82no—
668C2—28no—
669C2—48no—
670C2C2×#244111yescharacteristic
671C2#3495812yes—
672C2#3498211yescharacteristic
673C2#3507912yes—
674C2#3631712yes—
675C2#3633812yes—
676C2#3652912yes—
677C2#3653012yes—
678C2#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×#2569C2semidirect
3. #14774C2semidirect
4. #14771C2semidirect
7. #14697C2semidirect
8. #14695C2semidirect
10. #14215C2semidirect
11. #14210C2semidirect
12. #14149C2semidirect
13. #13965C2semidirect
14. #13908C2semidirect
20. C2×#400C22semidirect
22. C2×#398C22semidirect
24. C2×#389C22semidirect
25. C2×#387C22semidirect
26. C2×#387C22semidirect
27. C2×#387C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25692
C22×C4×C22⋊C42
#147742
#147712
#147342
#147262
#146972
#146952
#146802
#142152
#142102
#141492
#139652
#139082
C4×#3792