Finite Group Databasebeta

← #6497535

Subgroups — #6497535

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3238C211yesmaximal, characteristic
2C2×#3233C211yesmaximal, characteristic
3#35281C211yesmaximal, characteristic, complemented
4#34084C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#33201C211yesmaximal, characteristic, complemented
7#33198C211yesmaximal, characteristic, complemented
8#33197C211yesmaximal, characteristic, complemented
9#29869C211yesmaximal, characteristic, complemented
10#29868C211yesmaximal, characteristic, complemented
11#29867C211yesmaximal, characteristic, complemented
12#29865C211yesmaximal, characteristic, complemented
13#28730C214yesmaximal, complemented
14#28698C214yesmaximal, complemented
15#27657C214yesmaximal, complemented
16#27655C214yesmaximal
17#21839C211yesmaximal, characteristic, complemented
18#21834C211yesmaximal, characteristic, complemented
19#20815C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#529C2211yescharacteristic
23#3238—28no—
24#3233—28no—
25C2×#501C2212yes—
26C2×#499C2212yes—
27C2×#489C2212yes—
28C2×#488C2212yes—
29C2×#485C2212yes—
30C2×#482C2212yes—
31C2×#483C2212yes—
32C2×#480C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#477C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#476C2211yescharacteristic
38C2×#476C2211yescharacteristic
39C2×#476C2211yescharacteristic
40C2×#474C2211yescharacteristic
41C2×#432C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#2868C2211yescharacteristic
44#2868C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2660C2214yes—
48#2567—21no—
49#2567—21no—
50#2546—22no—
51#2502—21no—
52#2502—21no—
53#2434—22no—
54#2335—22no—
55#2334—22no—
56#2330C2212yes—
57#2326C2212yes—
58#2324C2212yes—
59#2323C2212yes—
60#2317C2214yes—
61#2298—21no—
62#2296—21no—
63#2295C2211yescharacteristic
64#2293—21no—
65#2293C2211yescharacteristic
66#2290—21no—
67#2285C2211yescharacteristic
68C42×Q8C2211yescharacteristic
69C42×D4C2211yescharacteristic
70C4×#117C2211yescharacteristic
71#2085C2214yes—
72#2085C2214yes—
73#2084C2214yes—
74#2083C2214yes—
75#2074C2214yes—
76#2073C2214yes—
77#2072C2214yes—
78#2071C2214yes—
79#1992C2214yes—
80#1990C2214yes—
81#1988C2211yescharacteristic
82#1988C2211yescharacteristic
83#1987C2211yescharacteristic
84#1987C2211yescharacteristic
85#1987C2211yescharacteristic
86#1987C2211yescharacteristic
87#1987C2211yescharacteristic
88#1987C2211yescharacteristic
89#1949C2214yes—
90#1949C2214yes—
91#1948C2214yes—
92#1947C2214yes—
93#1842C2211yescharacteristic
94#1842C2211yescharacteristic
95#1831C2211yescharacteristic
96#1831C2211yescharacteristic
97#1825C2214yes—
98#1824C2214yes—
99#1805C2214yes—
100#1804C2214yes—
101#1803C2214yes—
102#1802C2214yes—
103#1798C2214yes—
104#1768C2214yes—
105C23×Q8—21no—
106#533—24no—
107#533—24no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—22no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119#529—24no—
120#529D418yes—
121C2×#126—21no—
122C2×#125C2312yes—
123C2×#125C2311yescharacteristic
124C2×#124C2311yescharacteristic
125C2×#123—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8C2311yescharacteristic
129C2×#121C2311yescharacteristic
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2311yescharacteristic
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117C2312yes—
141C2×#117C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2312yes—
144C22×C42—21no—
145C22×C42C2311yescharacteristic
146#501—28no—
147#499—28no—
148#489—28no—
149#488—28no—
150#485—28no—
151#483—28no—
152#482—28no—
153#480—28no—
154#477—24no—
155#477—24no—
156#477—24no—
157#476—24no—
158#476—24no—
159#476—24no—
160#475—24no—
161#474—24no—
162#432—24no—
163#431—24no—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2314yes—
177C2×C4⋊C8C2314yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#102C2312yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×C22⋊C8C2312yes—
187C2×C22⋊C8C2312yes—
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190#398—24no—
191#398—24no—
192#395—24no—
193#395—24no—
194#391—21no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—22no—
199#391—22no—
200#391C2312yes—
201#390—22no—
202#390C2312yes—
203#389—24no—
204#389—24no—
205#389—22no—
206#389—22no—
207#389—22no—
208#389C2312yes—
209#389C2312yes—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#386—22no—
221#385—22no—
222#385—24no—
223#385—24no—
224#385—22no—
225#385—22no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384—21no—
230#384C2314yes—
231#384C2314yes—
232#384C2312yes—
233#382—21no—
234#382C2314yes—
235#382C2314yes—
236#380—21no—
237#380C2312yes—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D412yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4—22no—
250C4×C22⋊C4—21no—
251C4×C22⋊C4—21no—
252C4×C22⋊C4D412yes—
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255#376—21no—
256#376C2312yes—
257C43—21no—
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2314yes—
269#336C2314yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8—44no—
276C22×Q8—84no—
277C22×Q8C2411yescharacteristic
278C22×D4—21no—
279C22×D4C2411yescharacteristic
280C23×C4—21no—
281C23×C4—22no—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—24no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—24no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4⋊Q8—84no—
303C4⋊Q8—84no—
304#126—28no—
305#125—24no—
306#125—22no—
307#125—24no—
308#125C2×D414yes—
309#124—22no—
310#124C2×D414yes—
311#123—28no—
312C22⋊Q8—22no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8C2×D414yes—
316#121—22no—
317#121C2×D414yes—
318C4×Q8—22no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8C2×D414yes—
324C4×D4—28no—
325C4×D4—22no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4C2×D414yes—
329#117—28no—
330#117—24no—
331#117—24no—
332#117—24no—
333#117—24no—
334#117—82no—
335#117—82no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
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—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2412yes—
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—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—82no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D412yes—
395C2×C42C2×D412yes—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D414yes—
399C2×C42#4114yes—
400C2×C42C2412yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2412yes—
404#107—24no—
405#107—24no—
406#107—24no—
407#107—24no—
408#106—24no—
409#106—24no—
410#106—24no—
411#106—24no—
412C4⋊C8—22no—
413C4⋊C8—22no—
414C4⋊C8—22no—
415C4⋊C8—22no—
416C4⋊C8—28no—
417C4⋊C8—28no—
418#103—24no—
419#103—24no—
420#103—24no—
421#103—24no—
422#102—24no—
423#102—24no—
424#102—24no—
425#102—24no—
426C22⋊C8—24no—
427C22⋊C8—24no—
428#97—24no—
429C4×C8—24no—
430#95—22no—
431#95—22no—
432#95—21no—
433#95—22no—
434#95—22no—
435#95—21no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—24no—
440#95—24no—
441#95—24no—
442#95—24no—
443#95C2×D412yes—
444#95C2×D412yes—
445#95C2×D412yes—
446#95C2×D412yes—
447C24—21no—
448C24C22×D411yescharacteristic
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—84no—
463C2×Q8—816no—
464C2×Q8C22×D412yes—
465C2×D4—24no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—82no—
470C2×D4C22×D412yes—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—84no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—84no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4—84no—
552C4⋊C4C22×D412yes—
553C4⋊C4C22×D412yes—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—42no—
574C42—42no—
575C42—44no—
576C42—44no—
577C42—42no—
578C42—42no—
579C42—82no—
580C42—82no—
581C42#12114yes—
582C42#12114yes—
583C42C22×D412yes—
584C42C22×D412yes—
585C42C2×#4112yes—
586C42C2×#4112yes—
587C422−1+414yes—
588C422−1+414yes—
589C23—22no—
590C23—22no—
591C23—21no—
592C23—21no—
593C23—22no—
594C23—42no—
595C23—42no—
596C23—41no—
597C23—41no—
598C23C2×#12112yes—
599C23C23×D411yescharacteristic
600Q8—22no—
601Q8—22no—
602Q8—42no—
603Q8—84no—
604Q8—84no—
605Q8—84no—
606Q8—88no—
607Q8—84no—
608D4—22no—
609D4—22no—
610D4—42no—
611D4—84no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—88no—
656C2×C4—88no—
657C2×C4C2×#12112yes—
658C2×C4C2×#12112yes—
659C2×C4C2×#12112yes—
660C2×C4C2×#12112yes—
661C2×C4C2×#12112yes—
662C2×C4#53614yes—
663C2×C4#53614yes—
664C2×C4#57712yes—
665C2×C4#57812yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×2−1+412yes—
669C2×C4C2×2−1+412yes—
670C8—42no—
671C8—42no—
672C8—42no—
673C8—42no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—82no—
683C22—82no—
684C22C22×#12111yescharacteristic
685C22C2×#53612yes—
686C22C2×#57711yescharacteristic
687C22C2×#57811yescharacteristic
688C22#358512yes—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—24no—
693C4—24no—
694C4—22no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4#304614yes—
707C4#349911yescharacteristic
708C4#349911yescharacteristic
709C4#350912yes—
710C2—42no—
711C2—82no—
712C2#3531611yescharacteristic
713C2C2×#304611yescharacteristic
714C2#6042611yescharacteristic
715C2#6089212yes—
716C2#6089312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. #35281C2semidirect
5. #33202C2semidirect
6. #33201C2semidirect
7. #33198C2semidirect
8. #33197C2semidirect
9. #29869C2semidirect
10. #29868C2semidirect
11. #29867C2semidirect
12. #29865C2semidirect
13. #28730C2semidirect
14. #28698C2semidirect
15. #27657C2semidirect
17. #21839C2semidirect
18. #21834C2semidirect
19. #20815C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
C2×#32332
#352812
#340842
#332022
#332012
#331982
#331972
#298692
#298682
#298672
#298652
#287302
#286982
#276572
#276552
#218392
#218342
#208152