Finite Group Databasebeta

← #1031590

Subgroups — #1031590

Order 512.

2834 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2784C212yesmaximal, complemented
2#17196C212yesmaximal, complemented
3#16556C212yesmaximal, complemented
4#16320C214yesmaximal, complemented
5#15962C214yesmaximal, complemented
6#15429C212yesmaximal, complemented
7#15144C212yesmaximal, complemented
8#15134C212yesmaximal, complemented
9#15128C211yesmaximal, characteristic, complemented
10#14154C211yesmaximal, characteristic
11#14029C212yesmaximal, complemented
12#14025C212yesmaximal
13#13977C212yesmaximal, complemented
14#13920C212yesmaximal, complemented
15C4×#400C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×C4×D4C2212yes—
19#2784—216no—
20C2×#400C2211yescharacteristic
21C2×#396C2214yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#393C2214yes—
25C2×#393C2214yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#1446C2212yes—
46#1445C2211yescharacteristic
47#1445C2214yes—
48#1444C2212yes—
49#1443C2212yes—
50#1442C2211yescharacteristic
51#1442C2212yes—
52#1441C2212yes—
53#1441C2211yescharacteristic
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2214yes—
65#1438C2212yes—
66#1438C2214yes—
67#1436C2214yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2214yes—
73C4×#95C2212yes—
74C4×#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2314yes—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—44no—
83C2×#127—42no—
84C2×#127—42no—
85C2×#127Q818yescomplemented
86C2×C4×D4—28no—
87C2×C4×D4—44no—
88C2×C4×D4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118#400D418yescomplemented
119#396—216no—
120#396—48no—
121#395—28no—
122#393—216no—
123#393—48no—
124#390—28no—
125#390—28no—
126#390—44no—
127#390—44no—
128#390—44no—
129#390—44no—
130#390—44no—
131#389—44no—
132#389—44no—
133#387—216no—
134#386—216no—
135#386—48no—
136#385—216no—
137#385—48no—
138#384—28no—
139#380—216no—
140C4×C4⋊C4—42no—
141C4×C4⋊C4—42no—
142C4×C4⋊C4—42no—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C25C2414yes—
169C22×Q8—24no—
170C22×Q8—42no—
171C22×Q8—44no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—42no—
190C22×D4C2×Q818yes—
191C22×D4C2×Q818yes—
192C22×D4C2×Q818yes—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8—416no—
215#127—416no—
216C4×D4—432no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
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⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4#4118yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
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—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
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⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C2×Q814yes—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—42no—
311#95—42no—
312#95—41no—
313#95—42no—
314#95—41no—
315#95—41no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C2×D414yes—
323#95C2×D418yes—
324#95C2×D418yes—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24C22×Q818yes—
340C24C2511yescharacteristic
341C2×Q8—48no—
342C2×Q8—84no—
343C2×Q8—88no—
344C2×Q8—84no—
345C2×Q8—88no—
346C2×D4—216no—
347C2×D4—216no—
348C2×D4—216no—
349C2×D4—216no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—416no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—84no—
359C2×D4—84no—
360C2×D4—84no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—88no—
364C2×D4—84no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C4⋊C4—216no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—416no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C42—28no—
482C42—48no—
483C42—48no—
484C42—44no—
485C42—44no—
486C42—84no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C23×D411yescharacteristic
523C23C23×Q811yescharacteristic
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+411yescharacteristic
531C23#58512yes—
532C23#58511yescharacteristic
533Q8—816no—
534D4—432no—
535D4—816no—
536D4—816no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—832no—
618C2×C4#53818yes—
619C2×C4#54314yes—
620C2×C4D4×Q814yes—
621C2×C4#55418yes—
622C2×C4#55514yes—
623C2×C4#56014yes—
624C22—48no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22C2×#53812yes—
640C22C2×#54311yescharacteristic
641C22C2×D4×Q811yescharacteristic
642C22D4×#4112yes—
643C22C2×#55412yes—
644C22C2×#55511yescharacteristic
645C22#349611yescharacteristic
646C22#349812yes—
647C22#349811yescharacteristic
648C22#350112yes—
649C22#350111yescharacteristic
650C22#351912yes—
651C22#352212yes—
652C22#352412yes—
653C22C2×#56011yescharacteristic
654C22#353611yescharacteristic
655C22#353711yescharacteristic
656C22#354612yes—
657C22#354612yes—
658C22#355111yescharacteristic
659C22#355412yes—
660C22#355812yes—
661C22#356111yescharacteristic
662C4—24no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—48no—
669C4—44no—
670C4—42no—
671C4—82no—
672C4—84no—
673C4—81no—
674C4—84no—
675C4#278418yescomplemented
676C2—88no—
677C2#3936711yescharacteristic
678C2#3940512yes—
679C2#4249011yescharacteristic
680C2#4284212yes—
681C2#4495212yes—
682C2#4537811yescharacteristic
683C2#4548711yescharacteristic
684C2C2×#278411yescharacteristic
685C2#5296011yescharacteristic
686C2#5945812yes—
687C2#5984111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2784C2semidirect
2. #17196C2semidirect
3. #16556C2semidirect
4. #16320C2semidirect
5. #15962C2semidirect
6. #15429C2semidirect
7. #15144C2semidirect
8. #15134C2semidirect
9. #15128C2semidirect
11. #14029C2semidirect
13. #13977C2semidirect
14. #13920C2semidirect
15. C4×#400C2semidirect
85. C2×#127Q8semidirect
118. #400D4semidirect
675. C4#2784semidirect

Maximal subgroups

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

subgroupindex
C2×#27842
#171962
#165562
#163202
#159622
#154292
#151442
#151342
#151282
#141542
#140292
#140252
#139772
#139202
C4×#4002