Finite Group Databasebeta

← #719121

Subgroups — #719121

Order 512.

3086 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C212yesmaximal
2C22×#397C211yesmaximal, characteristic
3C2×#2344C211yesmaximal, characteristic, complemented
4#16528C214yesmaximal, complemented
5#15645C214yesmaximal, complemented
6#15641C214yesmaximal, complemented
7#14919C212yesmaximal, complemented
8#14918C212yesmaximal
9#14784C214yesmaximal, complemented
10#14338C211yesmaximal, characteristic
11#14149C211yesmaximal, characteristic, complemented
12#14128C212yesmaximal, complemented
13#13965C211yesmaximal, characteristic, complemented
14#13904C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20#2482—216no—
21C2×#399C2214yes—
22C2×#397—28no—
23C2×#397C2214yes—
24C2×#397C2214yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27#2344—28no—
28C2×#389C2212yescomplemented
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2212yescomplemented
32C2×#387C2212yescomplemented
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#379C2212yes—
42C2×#379C2211yescharacteristic
43C2×#379C2211yescharacteristic
44C2×C4×C4⋊C4C2212yescomplemented
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C22×#95C2212yes—
48C22×#95C2211yescharacteristic
49#1445C2214yescomplemented
50#1445C2212yescomplemented
51#1445C2212yescomplemented
52#1444C2214yescomplemented
53#1444C2214yescomplemented
54#1443C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1440C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C4⋊#95C2212yescomplemented
72C4⋊#95C2214yescomplemented
73C4⋊#95C2212yescomplemented
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C22⋊#95C2211yescharacteristic
78C26C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C2×#117—28no—
84C2×#117—28no—
85C2×#117—44no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123#399—216no—
124#397—28no—
125#397—216no—
126#397—28no—
127#397—216no—
128#389—28no—
129#388—216no—
130#388—216no—
131#387—28no—
132#387—28no—
133#385—28no—
134#385—28no—
135#385—44no—
136#381—28no—
137#381—28no—
138#381—28no—
139#380—24no—
140#380—24no—
141#379—28no—
142#379—24no—
143#379D418yes—
144C4×C4⋊C4—28no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C2×#95—28no—
148C2×#95—24no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25—22no—
176C25—22no—
177C25—22no—
178C25—22no—
179C25—22no—
180C25—22no—
181C25—44no—
182C25—44no—
183C25C2×D414yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2×D418yes—
203C23×C4C2×D414yes—
204C23×C4#4114yes—
205C23×C4#4114yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#117—432no—
220C2×C4⋊C4—232no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—432no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4C2×D4116yes—
292C2×C22⋊C4#4114yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—44no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—28no—
343C24—28no—
344C24—416no—
345C24—48no—
346C24—48no—
347C24—416no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24—84no—
353C24—84no—
354C24—84no—
355C24—84no—
356C24—88no—
357C24—88no—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C2×#4112yes—
362C24C2×#4112yes—
363C24C2×#4112yes—
364C24H(5,𝔽2)12yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
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—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—88no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—88no—
446C22×C4#123116yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C4⋊C4—28no—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—48no—
483C42—48no—
484C42—48no—
485C42—84no—
486C23—24no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—432no—
502C23—42no—
503C23—416no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—816no—
522C23—816no—
523C23—88no—
524C23—88no—
525C23—816no—
526C23—816no—
527C23—816no—
528C23C2×#12314yes—
529C23#53214yes—
530C23#54214yes—
531C23D4214yes—
532C23#54714yes—
533C23#54714yes—
534C23#55314yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4111yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23#58514yes—
543C23#58514yes—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—432no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
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×C4—44no—
614C2×C4—44no—
615C2×C4—816no—
616C2×C4—88no—
617C2×C4C2×#12314yes—
618C2×C4C2×#12314yes—
619C22—28no—
620C22—24no—
621C22—24no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—48no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—832no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22#242218yes—
640C22C22×#12311yescharacteristic
641C22C2×#53211yescharacteristic
642C22#344912yes—
643C22C2×#54211yescharacteristic
644C22C2×D4211yescharacteristic
645C22C2×#54711yescharacteristic
646C22C2×#54711yescharacteristic
647C22C2×#55311yescharacteristic
648C22#348514yes—
649C22#348712yes—
650C22#349112yes—
651C22#349112yes—
652C22#351012yes—
653C22#351012yes—
654C22#352114yes—
655C22#353314yes—
656C22#356414yes—
657C4—24no—
658C4—24no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—82no—
668C4—82no—
669C4—84no—
670C2—28no—
671C2—84no—
672C2C2×#242211yescharacteristic
673C2#3509711yescharacteristic
674C2#3567911yescharacteristic
675C2#3601212yes—
676C2#3765512yes—
677C2#3804712yes—
678C2#4139112yes—
679C2#4351012yes—
680C2#4446512yes—
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
3. C2×#2344C2semidirect
4. #16528C2semidirect
5. #15645C2semidirect
6. #15641C2semidirect
7. #14919C2semidirect
9. #14784C2semidirect
11. #14149C2semidirect
12. #14128C2semidirect
13. #13965C2semidirect
14. #13904C2semidirect
28. C2×#389C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
44. C2×C4×C4⋊C4C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1443C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C22×#3972
C2×#23442
#165282
#156452
#156412
#149192
#149182
#147842
#143382
#141492
#141282
#139652
#139042
#137752