Finite Group Databasebeta

← #676661

Subgroups — #676661

Order 512.

3742 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C4×#127C211yesmaximal, characteristic
5#15331C218yesmaximal, complemented
6#15231C212yesmaximal, complemented
7#15222C212yesmaximal, complemented
8#15181C214yesmaximal, complemented
9#14714C211yesmaximal, characteristic, complemented
10#14025C212yesmaximal
11#13941C212yesmaximal, complemented
12#13906C211yesmaximal, characteristic, complemented
13#13872C214yesmaximal, complemented
14C4×#389C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2214yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2563—28no—
23#2563—28no—
24C2×#393C2214yes—
25C2×#393C2214yes—
26D4×C4⋊C4—28no—
27C4×#127—28no—
28C2×#391C2214yes—
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2214yes—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#386C2218yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×#380C2212yes—
44C2×#380C2214yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C43C2211yescharacteristic
52#1442C2218yes—
53#1441C2218yes—
54#1439C2218yes—
55#1439C2218yes—
56#1439C2218yes—
57#1439C2214yes—
58#1438C2218yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C22⋊#95C2218yes—
66C4×#95C2212yes—
67C23×Q8C2312yes—
68C23×D4C2311yescharacteristic
69C23×D4C2314yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2314yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×#127—24no—
78C2×#127—24no—
79C2×#127Q818yescomplemented
80C2×C4×D4—28no—
81C2×C4×D4—216no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—216no—
88C2×C4×D4—24no—
89C2×C4×D4—44no—
90C2×C4×D4—48no—
91C2×C4×D4D418yes—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—48no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2318yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#393—216no—
124#393—216no—
125#391—48no—
126#390—24no—
127#390—28no—
128#390—24no—
129#390—216no—
130#390—28no—
131#390—44no—
132#389—28no—
133#389—28no—
134#386—416no—
135#385—216no—
136#384—24no—
137#384—24no—
138#382—28no—
139#382—28no—
140#380—216no—
141#380—28no—
142#380—28no—
143#380—28no—
144#380—48no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—44no—
147C4×C4⋊C4D418yescomplemented
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2318yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C43—24no—
161C25C2412yes—
162C25C2412yes—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—28no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—28no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4C2×D414yes—
185C22×D4C2×Q814yes—
186C22×D4C2×Q8116yes—
187C22×D4C2×Q814yes—
188C22×D4#4114yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—48no—
201C23×C4—44no—
202C23×C4—48no—
203C23×C4C2×D418yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—416no—
217#127—232no—
218C4×D4—232no—
219C4×D4—464no—
220C4×D4—432no—
221C4×D4—416no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—432no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4C2×D418yes—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—42no—
297C2×C42C2×D418yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×Q814yes—
301C2×C42#4118yes—
302C2×C42#4114yes—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—216no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—216no—
311#95—44no—
312#95—48no—
313#95—48no—
314#95C2×D418yes—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24C22×D414yes—
334C24C22×Q814yes—
335C24C22×Q814yes—
336C24C2×#4114yes—
337C24C2511yescharacteristic
338C2×Q8—416no—
339C2×Q8—48no—
340C2×Q8—88no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×D4—232no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—232no—
353C2×D4—28no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4C22⋊Q8116yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—216no—
389C22×C4—232no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—88no—
423C22×C4—816no—
424C22×C4—816no—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+412yes—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—84no—
460C22⋊C4—216no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—816no—
468C22⋊C4—816no—
469C42—28no—
470C42—216no—
471C42—216no—
472C42—48no—
473C42—44no—
474C42—48no—
475C42#121116yes—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23C2×C22⋊Q818yes—
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C23×Q811yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23C2×2−1+411yescharacteristic
532C23#58512yes—
533Q8—816no—
534D4—232no—
535D4—464no—
536D4—416no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
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—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—432no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—832no—
593C2×C4—816no—
594C2×C4C2×#12114yes—
595C2×C4C2×#12114yes—
596C2×C4C2×#12114yes—
597C2×C4C2×C22⋊Q814yes—
598C2×C4#53214yes—
599C2×C4#53514yes—
600C2×C4#53714yes—
601C2×C4#54614yes—
602C2×C4#54618yes—
603C2×C4D4×Q814yes—
604C2×C4#55214yes—
605C2×C4#55418yes—
606C2×C4#55414yes—
607C2×C4#55918yes—
608C22—216no—
609C22—28no—
610C22—44no—
611C22—48no—
612C22—48no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—48no—
619C22—48no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—48no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C22×#12111yescharacteristic
631C22C22×C22⋊Q811yescharacteristic
632C22C2×#53211yescharacteristic
633C22C2×#53511yescharacteristic
634C22C2×#53711yescharacteristic
635C22C2×#54611yescharacteristic
636C22C2×#54612yes—
637C22C2×D4×Q811yescharacteristic
638C22C2×#55211yescharacteristic
639C22C2×#55412yes—
640C22C2×#55411yescharacteristic
641C22#347512yes—
642C22C2×#55912yes—
643C22#348812yes—
644C22#349811yescharacteristic
645C22#350011yescharacteristic
646C22#350112yes—
647C22#350612yes—
648C22#350612yes—
649C22#351112yes—
650C22#351912yes—
651C22#354712yes—
652C22#354912yes—
653C4—28no—
654C4—24no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—84no—
660C4—84no—
661C4—82no—
662C4—84no—
663C4#241718yes—
664C4#254418yes—
665C2—48no—
666C2—84no—
667C2C2×#241711yescharacteristic
668C2C2×#254411yescharacteristic
669C2#3553311yescharacteristic
670C2#3615412yes—
671C2#3641511yescharacteristic
672C2#3655911yescharacteristic
673C2#3895311yescharacteristic
674C2#3946912yes—
675C2#3961811yescharacteristic
676C2#5944212yes—
677C2#5986412yes—
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×#2563C2semidirect
2. C2×#2563C2semidirect
3. C2×D4×C4⋊C4C2semidirect
5. #15331C2semidirect
6. #15231C2semidirect
7. #15222C2semidirect
8. #15181C2semidirect
9. #14714C2semidirect
11. #13941C2semidirect
12. #13906C2semidirect
13. #13872C2semidirect
14. C4×#389C2semidirect
79. C2×#127Q8semidirect
147. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×D4×C4⋊C42
C2×C4×#1272
#153312
#152312
#152222
#151812
#147142
#140252
#139412
#139062
#138722
C4×#3892