Finite Group Databasebeta

← #6496352

Subgroups — #6496352

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3116C211yesmaximal, characteristic
2C2×#3076C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic, complemented
4#34127C211yesmaximal, characteristic
5#33201C211yesmaximal, characteristic, complemented
6#33197C211yesmaximal, characteristic, complemented
7#33032C211yesmaximal, characteristic, complemented
8#33026C211yesmaximal, characteristic, complemented
9#29869C211yesmaximal, characteristic, complemented
10#29867C211yesmaximal, characteristic, complemented
11#29831C211yesmaximal, characteristic, complemented
12#29830C211yesmaximal, characteristic, complemented
13#24825C214yesmaximal, complemented
14#24691C214yesmaximal, complemented
15#24584C214yesmaximal, complemented
16#24536C214yesmaximal
17#21976C211yesmaximal, characteristic, complemented
18#21885C211yesmaximal, characteristic, complemented
19#21839C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23#3116—28no—
24C2×#485C2212yes—
25#3076—28no—
26C2×#479C2212yes—
27C2×#477C2211yescharacteristic
28C2×#477C2211yescharacteristic
29C2×#475C2211yescharacteristic
30C2×#478C2212yes—
31C2×#476C2211yescharacteristic
32C2×#476C2211yescharacteristic
33C2×#474C2211yescharacteristic
34C2×#462C2211yescharacteristic
35C2×#461C2211yescharacteristic
36C2×#452C2212yes—
37C2×#441C2212yes—
38C2×#440C2212yes—
39C2×#432C2211yescharacteristic
40C2×#427C2212yes—
41C22×C4⋊C8C2211yescharacteristic
42C2×#416C2212yes—
43#2868C2211yescharacteristic
44#2862C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2621—22no—
48#2565—21no—
49#2565—21no—
50#2565C2214yes—
51#2499—21no—
52#2499—21no—
53#2499C2214yes—
54#2355C2212yes—
55#2352—22no—
56D4×C4⋊C4C2212yes—
57#2347C2212yes—
58#2343—22no—
59#2339C2212yes—
60#2327—22no—
61#2295C2211yescharacteristic
62#2295C2211yescharacteristic
63#2290—21no—
64#2290—21no—
65#2285C2211yescharacteristic
66#2283—21no—
67C42×D4C2211yescharacteristic
68C42×D4C2211yescharacteristic
69C4×#117C2211yescharacteristic
70C2×C4×C4⋊C4—21no—
71#2085C2214yes—
72#2083C2214yes—
73#2074C2214yes—
74#2072C2214yes—
75#2068C2214yes—
76#2066C2214yes—
77#2056C2214yes—
78#2056C2214yes—
79#1992C2214yes—
80#1988C2211yescharacteristic
81#1988C2211yescharacteristic
82#1987C2211yescharacteristic
83#1987C2211yescharacteristic
84#1987C2211yescharacteristic
85#1987C2211yescharacteristic
86#1981C2211yescharacteristic
87#1981C2211yescharacteristic
88#1949C2214yes—
89#1948C2214yes—
90#1940C2214yes—
91#1928C2214yes—
92#1927C2214yes—
93#1892C2214yes—
94#1877C2214yes—
95#1877C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1842C2211yescharacteristic
99#1831C2211yescharacteristic
100#1831C2211yescharacteristic
101#1825C2214yes—
102#1804C2214yes—
103#1803C2214yes—
104#1802C2214yes—
105C22×#41C2311yescharacteristic
106C22×Q16C2312yes—
107C22×SD16C2312yes—
108C23×C8C2312yes—
109#533—24no—
110#533—24no—
111#533—24no—
112#533—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×#125—21no—
119C2×#125C2312yes—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C2312yes—
124C4×#41—24no—
125C4×#41D418yes—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2311yescharacteristic
137C2×C4×D4C2311yescharacteristic
138C2×#117—21no—
139C2×#117—21no—
140C2×#117—21no—
141C2×#117D412yes—
142C2×#117C2312yes—
143C2×#117C2311yescharacteristic
144C22×C4⋊C4—22no—
145C22×C42—21no—
146C22×C42—21no—
147C22×C42—21no—
148C22×C42C2311yescharacteristic
149#485—28no—
150#479—28no—
151#478—28no—
152#477—24no—
153#477—24no—
154#476—24no—
155#476—24no—
156#475—24no—
157#474—24no—
158#462—24no—
159#461—24no—
160#452—28no—
161#441—28no—
162#440—28no—
163#432—24no—
164#427—28no—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×C4⋊C8—24no—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2314yes—
178C2×C4⋊C8C2314yes—
179#416—28no—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×C22⋊C8C2312yes—
191C2×#97C2312yes—
192#398—24no—
193#398—24no—
194#397—24no—
195#397—24no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—22no—
201#391—22no—
202#391C2314yes—
203#391C2314yes—
204#391C2312yes—
205#391C2312yes—
206#389C2312yes—
207#389C2312yes—
208#388—22no—
209#388—22no—
210#387—22no—
211#387—22no—
212#387—24no—
213#387—24no—
214#387—24no—
215#387—24no—
216#387—22no—
217#387—22no—
218#386—22no—
219#386—22no—
220#385—22no—
221#385—22no—
222#384—21no—
223#384—21no—
224#384C2314yes—
225#384C2312yes—
226#383—22no—
227#382—22no—
228#382—21no—
229#382—21no—
230#382—22no—
231#382—21no—
232#382C2314yes—
233#381—22no—
234#380—21no—
235#380—22no—
236#380C2312yes—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—22no—
251C4×C22⋊C4—21no—
252C4×C22⋊C4—21no—
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2312yes—
256#376—21no—
257#376—21no—
258#376D412yes—
259C2×#95—22no—
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2314yes—
269#336C2314yes—
270C2×#41—22no—
271C2×#41—24no—
272C2×#41—42no—
273C2×#41C2×D414yes—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8C2412yes—
278C22×Q8C2411yescharacteristic
279C22×D4—21no—
280C22×D4C2411yescharacteristic
281C23×C4—21no—
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—22no—
285C23×C4—22no—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2411yescharacteristic
289C2×Q16—24no—
290C2×Q16—44no—
291C2×SD16—24no—
292C2×SD16—28no—
293C22×C8—24no—
294C22×C8—44no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—48no—
308#125—24no—
309#125—24no—
310#125—44no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8C2×D414yes—
324C4×Q8C2×D414yes—
325C4×D4—28no—
326C4×D4—28no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4C2×D414yes—
334C4×D4C2×D414yes—
335#117—22no—
336#117—28no—
337#117—24no—
338#117—24no—
339#117—44no—
340#117—44no—
341#117—82no—
342#117C2×D414yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—48no—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—28no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—82no—
391C2×C42—82no—
392C2×C42C2×D412yes—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D412yes—
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D414yes—
401C2×C42#4114yes—
402C2×C42C2412yes—
403C2×C42C2412yes—
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—24no—
407#107—24no—
408#107—24no—
409#106—24no—
410#106—24no—
411#106—24no—
412C4⋊C8—22no—
413C4⋊C8—22no—
414C4⋊C8—24no—
415C4⋊C8—22no—
416C4⋊C8—22no—
417C4⋊C8—44no—
418C4⋊C8—44no—
419#103—24no—
420#103—24no—
421#103—24no—
422#103—24no—
423#103—24no—
424#103—44no—
425#103—44no—
426#102—24no—
427#102—24no—
428#102—24no—
429C22⋊C8—24no—
430#97—24no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—21no—
439#95—21no—
440#95—22no—
441#95—22no—
442#95—24no—
443#95—24no—
444#95—24no—
445#95—24no—
446#95—24no—
447#95—24no—
448#95C2×D412yes—
449#95C2×D412yes—
450#95C2×D412yes—
451#95C2×D412yes—
452C24—21no—
453C24C22×D411yescharacteristic
454#41—28no—
455#41—44no—
456#41—48no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—24no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8C22×D412yes—
472C2×D4—24no—
473C2×D4—21no—
474C2×D4—22no—
475C2×D4—44no—
476C2×D4C22×D412yes—
477C2×D42−1+414yes—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4#12114yes—
506C22×C4#12114yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C2×#4112yes—
518C22×C4C2511yescharacteristic
519Q16—48no—
520SD16—48no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—48no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—48no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—44no—
559C4⋊C4—48no—
560C4⋊C4—82no—
561C4⋊C4C22×D412yes—
562C4⋊C4C22×D412yes—
563C22⋊C4—24no—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—42no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—82no—
573C42—24no—
574C42—24no—
575C42—24no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—42no—
586C42—44no—
587C42—44no—
588C42—42no—
589C42—42no—
590C42—82no—
591C42—82no—
592C42#12114yes—
593C42#12114yes—
594C42C22×D412yes—
595C42C22×D412yes—
596C42C2×#4112yes—
597C42C2×#4112yes—
598C23—22no—
599C23—24no—
600C23—22no—
601C23—22no—
602C23—21no—
603C23—21no—
604C23—42no—
605C23—41no—
606C23—41no—
607C23C2×#12112yes—
608C23C23×D411yescharacteristic
609Q8—22no—
610Q8—22no—
611Q8—42no—
612Q8—42no—
613Q8—42no—
614Q8—44no—
615Q8—84no—
616Q8—84no—
617Q8—84no—
618D4—22no—
619D4—24no—
620D4—48no—
621D4#53614yes—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—44no—
646C2×C4—48no—
647C2×C4—48no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—88no—
669C2×C4—88no—
670C2×C4C2×#12112yes—
671C2×C4C2×#12112yes—
672C2×C4C2×#12112yes—
673C2×C4C2×#12112yes—
674C2×C4C2×#12112yes—
675C2×C4#57514yes—
676C2×C4#57712yes—
677C2×C4#57812yes—
678C2×C4C23×D411yescharacteristic
679C2×C4C22×#4111yescharacteristic
680C2×C4C2×2−1+412yes—
681C2×C4#58512yes—
682C8—42no—
683C8—42no—
684C8—42no—
685C8—42no—
686C22—22no—
687C22—22no—
688C22—24no—
689C22—22no—
690C22—42no—
691C22—44no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22—42no—
696C22C22×#12111yescharacteristic
697C22#348912yes—
698C22C2×#57512yes—
699C22C2×#57711yescharacteristic
700C22C2×#57811yescharacteristic
701C4—22no—
702C4—24no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4#304614yes—
719C4C2×#53611yescharacteristic
720C4#344911yescharacteristic
721C4#349512yes—
722C2—42no—
723C2—44no—
724C2#3530611yescharacteristic
725C2#6041511yescharacteristic
726C2C2×#304611yescharacteristic
727C2#6212612yes—
728C2#6213312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #35322C2semidirect
5. #33201C2semidirect
6. #33197C2semidirect
7. #33032C2semidirect
8. #33026C2semidirect
9. #29869C2semidirect
10. #29867C2semidirect
11. #29831C2semidirect
12. #29830C2semidirect
13. #24825C2semidirect
14. #24691C2semidirect
15. #24584C2semidirect
17. #21976C2semidirect
18. #21885C2semidirect
19. #21839C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31162
C2×#30762
#353222
#341272
#332012
#331972
#330322
#330262
#298692
#298672
#298312
#298302
#248252
#246912
#245842
#245362
#219762
#218852
#218392