Finite Group Databasebeta

← #6496391

Subgroups — #6496391

Order 512.

1867 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3102C211yesmaximal, characteristic
2C2×#3079C211yesmaximal, characteristic, complemented
3#37632C211yesmaximal, characteristic, complemented
4#34191C211yesmaximal, characteristic, complemented
5#33203C212yesmaximal, complemented
6#33036C212yesmaximal, complemented
7#29873C212yesmaximal, complemented
8#29841C212yesmaximal, complemented
9#24815C212yesmaximal, complemented
10#24812C212yesmaximal, complemented
11#24682C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#24595C212yesmaximal, complemented
14#24594C212yesmaximal, complemented
15#24545C212yesmaximal, complemented
16#24544C212yesmaximal
17#21975C211yesmaximal, characteristic, complemented
18#21890C211yesmaximal, characteristic, complemented
19#21834C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C2×#529C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23#3102—28no—
24C2×#485C2211yescharacteristic
25C2×#482C2211yescharacteristic
26#3079—28no—
27C2×#479C2211yescharacteristic
28C2×#477C2212yes—
29C2×#475C2212yes—
30C2×#478C2211yescharacteristic
31C2×#476C2212yes—
32C2×#474C2212yes—
33C2×#464C2211yescharacteristic
34C2×#463C2211yescharacteristic
35C2×#452C2211yescharacteristic
36C2×#449C2211yescharacteristic
37C2×C4×Q16C2211yescharacteristic
38C2×C4×SD16C2211yescharacteristic
39C2×C4×SD16C2211yescharacteristic
40C2×C4×D8C2211yescharacteristic
41C2×#431C2211yescharacteristic
42C2×#428C2212yes—
43C2×#423C2211yescharacteristic
44C2×#417C2212yes—
45#2821—22no—
46#2660C2212yes—
47#2640C2212yes—
48#2628—21no—
49#2628—21no—
50#2614—22no—
51#2567C2212yes—
52#2563C2212yes—
53#2503C2212yes—
54#2501C2212yes—
55#2486—22no—
56#2355C2212yes—
57#2352—22no—
58D4×C4⋊C4C2212yes—
59#2346C2212yes—
60#2343—22no—
61#2339C2212yes—
62#2320C2212yes—
63#2318—22no—
64#2285C2211yescharacteristic
65#2283—21no—
66#2283C2211yescharacteristic
67C2×#382—21no—
68#2275C2212yes—
69#2085C2214yes—
70#2084C2214yes—
71#2074C2214yes—
72#2072C2214yes—
73#2066C2214yes—
74#2065C2214yes—
75#2056C2214yes—
76#2055C2214yes—
77#1990C2212yes—
78#1990C2212yes—
79#1989C2212yes—
80#1987C2212yes—
81#1987C2212yes—
82#1983C2212yes—
83#1949C2214yes—
84#1947C2214yes—
85#1939C2212yes—
86#1939C2212yes—
87#1928C2214yes—
88#1925C2214yes—
89#1895C2212yes—
90#1894C2212yes—
91#1879C2214yes—
92#1877C2214yes—
93#1851C2212yes—
94#1851C2212yes—
95#1843C2212yes—
96#1832C2212yes—
97#1824C2212yes—
98#1824C2212yes—
99#1804C2214yes—
100#1803C2212yes—
101#1802C2212yes—
102#1802C2214yes—
103C22×#41C2311yescharacteristic
104C23×Q8—21no—
105C22×Q16C2311yescharacteristic
106C22×SD16C2311yescharacteristic
107C22×SD16C2311yescharacteristic
108C22×D8C2311yescharacteristic
109C22×M16C2312yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116#529—24no—
117#529—24no—
118C2×#126—21no—
119C2×#125C2311yescharacteristic
120C2×#124C2311yescharacteristic
121C2×#123—21no—
122C2×#123—24no—
123C2×C22⋊Q8C2311yescharacteristic
124C2×#121C2311yescharacteristic
125C4×#41—24no—
126C4×#41D418yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2311yescharacteristic
134C2×C4×D4C2314yes—
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117C2311yescharacteristic
141C22×C4⋊C4—21no—
142C22×C4⋊C4—22no—
143C22×C4⋊C4C2312yes—
144C22×C42—21no—
145C22×C42C2311yescharacteristic
146#485—24no—
147#482—24no—
148#479—24no—
149#478—24no—
150#477—28no—
151#476—28no—
152#475—28no—
153#474—28no—
154#464—24no—
155#463—24no—
156#452—24no—
157#449—24no—
158C4×Q16—24no—
159C4×SD16—24no—
160C4×SD16—24no—
161C4×D8—24no—
162#431—24no—
163#428—28no—
164C2×#107C2311yescharacteristic
165C2×#107C2314yes—
166C2×#107C2311yescharacteristic
167C2×#106C2311yescharacteristic
168C2×#106C2314yes—
169C2×#106C2311yescharacteristic
170#423—24no—
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2314yes—
176C2×C4⋊C8C2314yes—
177#417—28no—
178C2×#103C2314yes—
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2311yescharacteristic
182C2×#103C2311yescharacteristic
183C2×#103C2314yes—
184C2×#102C2311yescharacteristic
185C2×#102C2311yescharacteristic
186C2×#102C2314yes—
187C2×#102C2311yescharacteristic
188C2×#102C2311yescharacteristic
189C2×#102C2314yes—
190C2×C22⋊C8C2311yescharacteristic
191C2×C22⋊C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194#400—22no—
195#400—22no—
196#399—22no—
197#399—22no—
198#398—24no—
199#397—24no—
200#395—24no—
201#395—22no—
202#395—22no—
203#393—24no—
204#393—22no—
205#393—22no—
206#391C2314yes—
207#391C2314yes—
208#390C2314yes—
209#389—22no—
210#389—22no—
211#389C2314yes—
212#388—24no—
213#387—24no—
214#387—24no—
215#386—24no—
216#385—24no—
217#384C2314yes—
218#384C2312yes—
219#383—22no—
220#383D412yes—
221#382—22no—
222#382—21no—
223#382—22no—
224#382—22no—
225#382D411yescharacteristic
226#382D411yescharacteristic
227#382C2314yes—
228#381—24no—
229#381—22no—
230#381D412yes—
231#380—21no—
232#380—22no—
233#380—22no—
234#380—22no—
235#380D412yes—
236#380C2312yes—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2314yes—
241C4×C4⋊C4C2312yes—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—24no—
244C4×C22⋊C4C2312yes—
245#376—22no—
246#376C2312yes—
247C2×#95—22no—
248C2×#95—22no—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2314yes—
256#336C2314yes—
257C2×#41—22no—
258C2×#41—24no—
259C2×#41—42no—
260C2×#41C2×D414yes—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—44no—
264C22×Q8—84no—
265C22×Q8C2411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—21no—
276C23×C4—24no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×Q16—22no—
281C2×Q16—42no—
282C2×SD16—22no—
283C2×SD16—22no—
284C2×SD16—24no—
285C2×SD16—42no—
286C2×D8—22no—
287C2×D8—24no—
288C2×M16—24no—
289C2×M16—44no—
290C22×C8C2414yes—
291C22×C8C2411yescharacteristic
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—24no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301#126—44no—
302#125—22no—
303#125—22no—
304#124—22no—
305#124—22no—
306#123—44no—
307#123—88no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310#121—22no—
311#121—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8C2×D414yes—
317C4×Q8C2×D414yes—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—28no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4C2×D414yes—
328C4×D4C2×D414yes—
329#117—22no—
330#117—28no—
331#117—44no—
332#117—82no—
333#117C2×D414yes—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—84no—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2414yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4C2×D414yes—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—28no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—24no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D414yes—
383C2×C42#4114yes—
384C2×C42C2414yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—22no—
389#107—22no—
390#106—28no—
391#106—22no—
392#106—22no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—44no—
398C4⋊C8—44no—
399#103—28no—
400#103—22no—
401#103—22no—
402#103—22no—
403#103—22no—
404#103—44no—
405#102—28no—
406#102—22no—
407#102—22no—
408#102—22no—
409#102—22no—
410#102—44no—
411C22⋊C8—22no—
412C22⋊C8—22no—
413C4×C8—22no—
414C4×C8—22no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95C2×D412yes—
426#95C2×D412yes—
427#95C2×D412yes—
428#95C2×D414yes—
429#95C2×D412yes—
430#95C2×D414yes—
431C24—21no—
432C24—21no—
433C24C22×D411yescharacteristic
434#41—28no—
435#41—44no—
436#41—48no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—84no—
449C2×Q8—816no—
450C2×Q8C22×D412yes—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4C22×D412yes—
462C2×D42−1+414yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—82no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4#12114yes—
492C22×C4#12114yes—
493C22×C4C22×D414yes—
494C22×C4C22×D414yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4112yes—
502C22×C4C2511yescharacteristic
503Q16—44no—
504SD16—44no—
505SD16—44no—
506D8—44no—
507M16—48no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—24no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—22no—
518C2×C8—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4C22×D412yes—
546C4⋊C4C22×D412yes—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C22⋊C4—82no—
560C42—24no—
561C42—24no—
562C42—24no—
563C42—24no—
564C42—24no—
565C42—42no—
566C42—84no—
567C42—82no—
568C42#12114yes—
569C42#12114yes—
570C42C22×D412yes—
571C42C22×D412yes—
572C42C2×#4112yes—
573C42C2×#4112yes—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—22no—
578C23—21no—
579C23—21no—
580C23—42no—
581C23—41no—
582C23—41no—
583C23—42no—
584C23—41no—
585C23—42no—
586C23—41no—
587C23—41no—
588C23C2×#12112yes—
589C23C23×D411yescharacteristic
590Q8—22no—
591Q8—22no—
592Q8—42no—
593Q8—41no—
594Q8—41no—
595Q8—42no—
596Q8—88no—
597Q8—88no—
598D4—22no—
599D4—24no—
600D4—48no—
601D4—41no—
602D4—41no—
603D4—42no—
604D4—84no—
605D4#53614yes—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—84no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4—168no—
648C2×C4C2×#12112yes—
649C2×C4C2×#12112yes—
650C2×C4C2×#12112yes—
651C2×C4C2×#12112yes—
652C2×C4C2×#12112yes—
653C2×C4#57712yes—
654C2×C4#57812yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×2−1+412yes—
658C2×C4#58512yes—
659C8—41no—
660C8—41no—
661C8—44no—
662C8—41no—
663C8—41no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—24no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—81no—
675C22—81no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22C22×#12111yescharacteristic
680C22#348912yes—
681C22C2×#57711yescharacteristic
682C22C2×#57811yescharacteristic
683C22#358512yes—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—81no—
694C4—84no—
695C4—81no—
696C4—84no—
697C4—81no—
698C4#304614yes—
699C4C2×#53611yescharacteristic
700C4#344911yescharacteristic
701C4#349512yes—
702C2—42no—
703C2—44no—
704C2—161no—
705C2#3530611yescharacteristic
706C2C2×#304611yescharacteristic
707C2#6042611yescharacteristic
708C2#6217912yes—
709C2#6218112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#449C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3079C2semidirect
3. #37632C2semidirect
4. #34191C2semidirect
5. #33203C2semidirect
6. #33036C2semidirect
7. #29873C2semidirect
8. #29841C2semidirect
9. #24815C2semidirect
10. #24812C2semidirect
11. #24682C2semidirect
12. #24679C2semidirect
13. #24595C2semidirect
14. #24594C2semidirect
15. #24545C2semidirect
17. #21975C2semidirect
18. #21890C2semidirect
19. #21834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31022
C2×#30792
#376322
#341912
#332032
#330362
#298732
#298412
#248152
#248122
#246822
#246792
#245952
#245942
#245452
#245442
#219752
#218902
#218342