Finite Group Databasebeta

← #536049

Subgroups — #536049

Order 512.

4506 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#394C211yesmaximal, characteristic
2C2×#2342C212yesmaximal, complemented
3#14566C214yesmaximal, complemented
4#14540C214yesmaximal, complemented
5#14199C214yesmaximal, complemented
6#14182C214yesmaximal, complemented
7#13883C212yesmaximal, complemented
8#13873C212yesmaximal, complemented
9#13868C212yesmaximal, complemented
10#13867C212yesmaximal
11C22⋊C42C212yesmaximal, complemented
12#13795C211yesmaximal, characteristic, complemented
13#13776C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#117C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20C2×#394C4116yescomplemented
21C2×#394C2212yes—
22C2×#394C2212yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25#2342—216no—
26C2×#388C2214yescomplemented
27C2×#388C2214yescomplemented
28C2×#388C2214yescomplemented
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yescomplemented
32C2×#381C2211yescharacteristic
33C2×#381C2211yescharacteristic
34C2×#380C2212yes—
35C2×#379C2212yes—
36C2×#379C2212yes—
37C2×#379C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2214yescomplemented
41C2×C4×C22⋊C4C2214yescomplemented
42C22×#95C2211yescharacteristic
43#1443C2214yescomplemented
44#1442C2214yescomplemented
45#1440C2214yescomplemented
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1438C2214yes—
50#1438C2214yes—
51#1437C2214yescomplemented
52#1437C2214yescomplemented
53#1437C2214yescomplemented
54#1437C2214yescomplemented
55#1437C2214yescomplemented
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59C4⋊#95C2214yescomplemented
60C4⋊#95C2214yescomplemented
61#1434C2214yescomplemented
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2214yescomplemented
69C26C2312yes—
70C23×D4—44no—
71C23×D4—48no—
72C23×D4C2×C418yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#117—28no—
82C2×#117—28no—
83C2×#117—44no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4C2×C418yescomplemented
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—48no—
96C22×C22⋊C4C2×C4116yescomplemented
97C22×C22⋊C4C2×C4116yescomplemented
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2314yes—
118C22×C42C2314yes—
119C22×C42C2314yes—
120#394C2×C4116yes—
121#394C2×C4116yescomplemented
122#394C2×C4116yescomplemented
123#394C2×C4116yescomplemented
124#388—216no—
125#388—216no—
126#388—48no—
127#386—48no—
128#386—48no—
129#385—216no—
130#381—24no—
131#381D418yescomplemented
132#380—28no—
133#379—28no—
134#379—28no—
135#379—44no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—216no—
138C4×C22⋊C4—44no—
139C4×C22⋊C4—48no—
140C2×#95C2×C418yescomplemented
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C25—28no—
160C25—24no—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—24no—
165C25—24no—
166C25—44no—
167C25—44no—
168C25—44no—
169C25C22×C418yes—
170C25C2412yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—416no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—432no—
183C22×D4—464no—
184C22×D4C22×C414yes—
185C22×D4C22×C418yes—
186C22×D4C22×C414yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C22×C414yes—
205C23×C4C22×C414yes—
206C23×C4C22×C418yes—
207C23×C4C22×C414yes—
208C23×C4C2×D414yes—
209C23×C4#4114yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220#117—432no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—416no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C418yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—264no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
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—432no—
265C2×C22⋊C4—432no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C22×C418yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4#4118yes—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C22×C418yes—
314#95C22×C414yes—
315#95C22×C414yes—
316#95C2×D4116yescomplemented
317C24—24no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—216no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—216no—
340C24—216no—
341C24—216no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24—48no—
352C24—48no—
353C24—48no—
354C24—48no—
355C24—88no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24—88no—
361C24C23×C414yes—
362C24C23×C414yes—
363C24C23×C412yes—
364C24C22×D412yes—
365C24C2×#4112yes—
366C24C2×#4114yes—
367C24H(5,𝔽2)18yes—
368C24H(5,𝔽2)14yes—
369C24C2511yescharacteristic
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—4128no—
381C2×D4—416no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—216no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4C4×D4116yes—
451C22×C4C23×C414yes—
452C22×C4C23×C414yes—
453C22×C4C23×C412yes—
454C22×C4C23×C412yes—
455C22×C4C23×C414yes—
456C22×C4C23×C414yes—
457C22×C4C22×D418yes—
458C22×C4C22×D414yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4114yes—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—88no—
465C22⋊C4—216no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C42—416no—
485C42—88no—
486C42—88no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—432no—
515C23—432no—
516C23—432no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—432no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—816no—
545C23—816no—
546C23—816no—
547C23—832no—
548C23—832no—
549C23—832no—
550C23—832no—
551C23C2×C4×D414yes—
552C23C4×#4114yes—
553C23#51818yes—
554C23#51814yes—
555C23#54618yes—
556C23#55318yes—
557C23C24×C411yescharacteristic
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C22×#4112yes—
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)14yes—
563C23#58514yes—
564D4—416no—
565D4—432no—
566D4—416no—
567C2×C4—232no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
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—28no—
592C2×C4—28no—
593C2×C4—416no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
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—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—88no—
625C2×C4—816no—
626C2×C4—832no—
627C2×C4C2×C4×D418yes—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—28no—
632C22—216no—
633C22—216no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—864no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22#236418yes—
658C22C22×C4×D411yescharacteristic
659C22C2×C4×#4111yescharacteristic
660C22C2×#51811yescharacteristic
661C22C2×#51812yes—
662C22#342812yes—
663C22C2×#54612yes—
664C22C2×#55312yes—
665C22#348814yes—
666C22#349414yes—
667C22#351314yes—
668C22#351314yes—
669C22#353214yes—
670C22#354514yes—
671C4—28no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—48no—
678C4—48no—
679C4—84no—
680C4—84no—
681C2—28no—
682C2—48no—
683C2—84no—
684C2C2×#236411yescharacteristic
685C2#3420412yes—
686C2#3423812yes—
687C2#3435611yescharacteristic
688C2#3436611yescharacteristic
689C2#4400114yes—
690C2#4400214yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2342C2semidirect
3. #14566C2semidirect
4. #14540C2semidirect
5. #14199C2semidirect
6. #14182C2semidirect
7. #13883C2semidirect
8. #13873C2semidirect
9. #13868C2semidirect
11. C22⋊C42C2semidirect
12. #13795C2semidirect
13. #13776C2semidirect
20. C2×#394C4semidirect
26. C2×#388C22semidirect
27. C2×#388C22semidirect
28. C2×#388C22semidirect
31. C2×#385C22semidirect
40. C2×C4×C22⋊C4C22semidirect
41. C2×C4×C22⋊C4C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1440C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
51. #1437C22semidirect
52. #1437C22semidirect
53. #1437C22semidirect
54. #1437C22semidirect
55. #1437C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. #1434C22semidirect
68. C22⋊#95C22semidirect
85. C22×C4⋊C4C2×C4semidirect
96. C22×C22⋊C4C2×C4semidirect
97. C22×C22⋊C4C2×C4semidirect
121. #394C2×C4semidirect
122. #394C2×C4semidirect
123. #394C2×C4semidirect
131. #381D4semidirect
140. C2×#95C2×C4semidirect
316. #95C2×D4semidirect

Maximal subgroups

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

subgroupindex
C22×#3942
C2×#23422
#145662
#145402
#141992
#141822
#138832
#138732
#138682
#138672
C22⋊C422
#137952
#137762