Finite Group Databasebeta

← #636738

Subgroups — #636738

Order 512.

3694 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic, complemented
3C2×#2541C211yesmaximal, characteristic
4C2×#2295C211yesmaximal, characteristic, complemented
5#16289C218yesmaximal, complemented
6#15222C212yesmaximal, complemented
7#15215C214yesmaximal, complemented
8#15127C212yesmaximal, complemented
9#15126C214yesmaximal, complemented
10#14713C211yesmaximal, characteristic, complemented
11#14272C212yesmaximal, complemented
12#14260C212yesmaximal, complemented
13#13911C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C22×#127C2212yes—
17C22×#124C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2214yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2548—28no—
23#2541—28no—
24#2538—28no—
25C2×#396C2214yes—
26C2×#394C2214yes—
27C2×#394C2218yes—
28C2×#392C2214yes—
29#2295—28no—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2214yes—
40C2×#390C2212yes—
41C2×#390C2214yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#387C2214yes—
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2214yes—
52C2×#376C2212yes—
53C2×C43C2211yescharacteristic
54#1442C2214yes—
55#1440C2218yes—
56#1439C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1438C2218yes—
60#1438C2218yes—
61#1438C2218yes—
62#1437C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2211yescharacteristic
67C23×Q8C2311yescharacteristic
68C23×D4C2312yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2318yes—
73C24×C4C2312yes—
74C2×#127—28no—
75C2×#127—28no—
76C2×#127—24no—
77C2×#127—24no—
78C2×#127—24no—
79C2×#127—44no—
80C2×#124—28no—
81C2×#124—48no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—48no—
84C2×#121—216no—
85C2×#121—48no—
86C2×#121—48no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8D418yescomplemented
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2318yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#396—216no—
125#394—216no—
126#394—416no—
127#392—216no—
128#391—28no—
129#390—24no—
130#390—28no—
131#390—28no—
132#390—28no—
133#390—28no—
134#390—28no—
135#390—24no—
136#390—28no—
137#390—48no—
138#390—48no—
139#390—44no—
140#389—28no—
141#389—28no—
142#387—216no—
143#384—24no—
144#382—44no—
145#380—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4D418yes—
149C4×C22⋊C4—48no—
150#376—28no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2312yes—
161C43—24no—
162C25C2414yes—
163C25C2412yes—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8C2×D414yes—
169C22×D4—216no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—216no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—48no—
193C22×D4—48no—
194C22×D4—48no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4#4114yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#127—432no—
218#127—416no—
219#124—432no—
220C22⋊Q8—432no—
221#121—464no—
222C4×Q8—232no—
223C4×D4—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D418yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
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—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—42no—
301C2×C42C2×D418yes—
302C2×C42C2×D418yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306C2×C42#4114yes—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—48no—
315#95—48no—
316#95C2×D418yes—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—28no—
334C24—28no—
335C24—24no—
336C24—24no—
337C24—28no—
338C24C2×#4114yes—
339C24C2511yescharacteristic
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—432no—
349C2×D4—432no—
350C2×D4—416no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—88no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4—816no—
372C2×D4—816no—
373C2×D4—816no—
374C2×D4—816no—
375C2×D4—816no—
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—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
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—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4C22×D418yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—84no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—432no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—816no—
471C22⋊C4—816no—
472C42—216no—
473C42—216no—
474C42—28no—
475C42—48no—
476C42—48no—
477C42—48no—
478C42—44no—
479C42#121116yes—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—48no—
507C23—44no—
508C23—48no—
509C23—44no—
510C23—48no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—48no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C2×H(5,𝔽2)12yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+411yescharacteristic
536C23#58512yes—
537Q8—416no—
538D4—416no—
539D4—816no—
540D4—864no—
541D4—816no—
542D4—816no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
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—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—816no—
599C2×C4C2×#12114yes—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C2×C4#52914yes—
603C2×C4#53514yes—
604C2×C4#53518yes—
605C2×C4#53514yes—
606C2×C4#54018yes—
607C2×C4#54214yes—
608C2×C4#54614yes—
609C2×C4#54814yes—
610C2×C4#54814yes—
611C2×C4#55014yes—
612C22—44no—
613C22—44no—
614C22—48no—
615C22—44no—
616C22—48no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—88no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—88no—
630C22—84no—
631C22—84no—
632C22—88no—
633C22—88no—
634C22C22×#12111yescharacteristic
635C22C2×#52911yescharacteristic
636C22C2×#53511yescharacteristic
637C22C2×#53511yescharacteristic
638C22C2×#53512yes—
639C22C2×#54012yes—
640C22C2×#54211yescharacteristic
641C22#346012yes—
642C22C2×#54611yescharacteristic
643C22C2×#54811yescharacteristic
644C22C2×#54811yescharacteristic
645C22C2×#55011yescharacteristic
646C22#348812yes—
647C22#349811yescharacteristic
648C22#350011yescharacteristic
649C22#350612yes—
650C22#350812yes—
651C22#351412yes—
652C22#351912yes—
653C22#354312yes—
654C22#354612yes—
655C22#354812yes—
656C22#355412yes—
657C4—24no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—48no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4#242618yes—
668C4#261918yes—
669C2—84no—
670C2—164no—
671C2C2×#261911yescharacteristic
672C2C2×#242611yescharacteristic
673C2#3519712yes—
674C2#3525611yescharacteristic
675C2#3762512yes—
676C2#3891012yes—
677C2#3916911yescharacteristic
678C2#3917111yescharacteristic
679C2#3972512yes—
680C2#3974511yescharacteristic
681C2#3974711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×#2548C2semidirect
2. C2×#2538C2semidirect
4. C2×#2295C2semidirect
5. #16289C2semidirect
6. #15222C2semidirect
7. #15215C2semidirect
8. #15127C2semidirect
9. #15126C2semidirect
10. #14713C2semidirect
11. #14272C2semidirect
12. #14260C2semidirect
13. #13911C2semidirect
14. C4×#390C2semidirect
89. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#25382
C2×#25412
C2×#22952
#162892
#152222
#152152
#151272
#151262
#147132
#142722
#142602
#139112
C4×#3902