Finite Group Databasebeta

← #6354849

Subgroups — #6354849

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60969C211yesmaximal, characteristic
2#60941C211yesmaximal, characteristic
3C2×#3231C211yesmaximal, characteristic, complemented
4#60236C211yesmaximal, characteristic
5#19518C214yesmaximal, complemented
6#19517C214yesmaximal, complemented
7#19431C214yesmaximal, complemented
8#19395C214yesmaximal, complemented
9#19199C212yesmaximal, complemented
10#19197C212yesmaximal, complemented
11#19192C212yesmaximal, complemented
12#19191C212yesmaximal, complemented
13#17803C211yesmaximal, characteristic, complemented
14#17747C211yesmaximal, characteristic, complemented
15#17715C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19#3254C2214yes—
20#3246C2214yes—
21#3231—22no—
22#3231—22no—
23#3231C2212yes—
24#3231C2212yes—
25#3231C2212yescomplemented
26#3231C2212yescomplemented
27#3230—21no—
28#3229—21no—
29#3227C2212yes—
30#3226C2212yes—
31#3208—21no—
32#3208—21no—
33#3206C2212yes—
34#3206C2212yes—
35#3193—22no—
36#3187—22no—
37#3182—22no—
38#3167—22no—
39C2×#500C2211yescharacteristic, complemented
40C2×#498C2211yescharacteristic, complemented
41C2×#487C2211yescharacteristic, complemented
42C2×#487C2211yescharacteristic, complemented
43#3112—22no—
44#3110C2214yes—
45#3103—22no—
46#3101C2214yes—
47C2×#485—22no—
48C2×#479—22no—
49#2981C2212yes—
50#2981C2212yes—
51#2979—22no—
52#2975C2212yes—
53#2974—22no—
54#2972C2212yes—
55#2935—22no—
56#2931C2212yes—
57#2931C2212yes—
58#2920C2211yescharacteristic
59#2920C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C2×#431C2211yescharacteristic
62#2880—22no—
63#1703C2218yescomplemented
64#1699C2218yescomplemented
65#1693C2218yescomplemented
66#1691C2218yescomplemented
67#1683C2218yescomplemented
68#1679C2218yescomplemented
69#1673C2218yescomplemented
70#1671C2218yescomplemented
71#1572C2212yes—
72#1572C2212yes—
73#1571C2212yescomplemented
74#1570C2212yescomplemented
75#1564C2212yescomplemented
76#1564C2212yescomplemented
77#1559C2214yescomplemented
78#1554C2214yes—
79#1552C2214yescomplemented
80#1549C2212yescomplemented
81#1549C2212yescomplemented
82#1548C2212yescomplemented
83#1546C2212yescomplemented
84#1544C2214yes—
85C22×#41—21no—
86C2×#131—22no—
87C2×#131—22no—
88#533—21no—
89#533—21no—
90#533—21no—
91#533—21no—
92#533—21no—
93#533—21no—
94#533—21no—
95#533—21no—
96#533—21no—
97#533—21no—
98#533—21no—
99#533—21no—
100#533C2312yes—
101#533C2312yes—
102#533C2312yes—
103#533C2312yes—
104#532—21no—
105#532—21no—
106#532—21no—
107#532—21no—
108#532D412yes—
109#532D412yes—
110#532C2312yes—
111#532C2312yes—
112C2×C4⋊Q8D411yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#127D411yescharacteristic
118C2×#125—21no—
119C2×#125—21no—
120C2×#124—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×#121—21no—
126C2×#121D412yes—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C22×C42C2311yescharacteristic
136#500—22no—
137#500—22no—
138#500C2312yes—
139#500C2312yes—
140#498—22no—
141#498—22no—
142#498C2312yes—
143#498C2312yes—
144#492—22no—
145#492—22no—
146#487—22no—
147#487—22no—
148#487—22no—
149#487—22no—
150#487C2312yes—
151#487C2312yes—
152#487C2312yes—
153#487C2312yes—
154#486—22no—
155#486—22no—
156#485—24no—
157#485—24no—
158#485—24no—
159#485—24no—
160#484—24no—
161#483—24no—
162#481—24no—
163#480—24no—
164#479—24no—
165#479—24no—
166#479—24no—
167#479—24no—
168#477—24no—
169#476C2318yes—
170#475—24no—
171#474C2318yes—
172#468—28no—
173#465—28no—
174#461C2318yes—
175#461C2318yes—
176#460—24no—
177#460—24no—
178#450—24no—
179#450—24no—
180#448—24no—
181#448—24no—
182#440—28no—
183C4×SD16—28no—
184#436—24no—
185#436—22no—
186#436—22no—
187#436C2314yes—
188#436C2314yes—
189#435—24no—
190#435—24no—
191C8×D4—24no—
192C8×D4—22no—
193C8×D4—22no—
194C8×D4C2314yes—
195C8×D4C2314yes—
196#432—21no—
197#432—21no—
198#432C2312yes—
199#432C2312yes—
200#431—21no—
201#431—21no—
202#431—21no—
203#431—21no—
204#431D412yes—
205#431D412yes—
206#431C2312yes—
207#431C2312yes—
208#431C2312yes—
209#431C2312yes—
210#431C2311yescharacteristic
211#431C2311yescharacteristic
212#431C2311yescharacteristic
213#431C2311yescharacteristic
214C2×#107—22no—
215C2×#106—22no—
216#423C2312yes—
217#423C2312yes—
218C2×C4⋊C8D412yes—
219C2×C4⋊C8C2312yes—
220C2×#103—22no—
221C2×#103—22no—
222#408—22no—
223#408—22no—
224#408—24no—
225#408—24no—
226#407—21no—
227#407—21no—
228#407—24no—
229C2×C22⋊C8—21no—
230C2×C22⋊C8—24no—
231C2×C22⋊C8D412yes—
232C4×M16—21no—
233C4×M16—21no—
234C2×C4×C8—21no—
235C2×C4×C8C2311yescharacteristic
236C2×C4×C8C2311yescharacteristic
237#335C2314yes—
238#335C2314yes—
239#334C2314yes—
240#334C2314yes—
241#333C2318yes—
242#332C2318yes—
243#332C2318yes—
244#332C2318yes—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—84no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C2×D412yes—
259C23×C4—22no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4#4111yescharacteristic
263C2×SD16—28no—
264C2×SD16—28no—
265#131—84no—
266#131—84no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—24no—
270C2×M16—24no—
271C22×C8—22no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8C2×D412yes—
275C22×C8C2×D412yes—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8C2×D412yes—
291C4⋊Q8C2×D412yes—
292C4⋊Q8C2412yes—
293C4⋊Q8C2412yes—
294C4⋊Q8C2412yes—
295C4⋊Q8C2412yes—
296#127—21no—
297#127—21no—
298#127C2×D412yes—
299#127C2×D412yes—
300#125—22no—
301#125—22no—
302#125—22no—
303#125—22no—
304#125—41no—
305#125—41no—
306#125—41no—
307#125—41no—
308#124—22no—
309#124—22no—
310#124—22no—
311#124—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—41no—
321C22⋊Q8—41no—
322C22⋊Q8—41no—
323C22⋊Q8—41no—
324C22⋊Q8—41no—
325C22⋊Q8—41no—
326C22⋊Q8—41no—
327C22⋊Q8—41no—
328#121—22no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—22no—
333#121—22no—
334#121C2×D414yes—
335#121C2×D414yes—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—41no—
345C4×Q8—41no—
346C4×Q8—41no—
347C4×Q8—41no—
348C4×Q8—41no—
349C4×Q8—41no—
350C4×Q8—41no—
351C4×Q8—41no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4C2×D414yes—
359C4×D4C2414yes—
360#117—22no—
361#117—22no—
362#117—22no—
363#117—22no—
364#117—41no—
365#117—41no—
366#117—41no—
367#117—41no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4C2×D412yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4—41no—
386C2×C42—41no—
387C2×C42—41no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42#4111yescharacteristic
395C2×C42#4111yescharacteristic
396C2×C42#4111yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—24no—
401#107—24no—
402#107—28no—
403#106—24no—
404#106—24no—
405#106—28no—
406C4⋊C8—22no—
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8C2×D412yes—
410C4⋊C8C2×D412yes—
411C4⋊C8C2×D412yes—
412C4⋊C8C2×D412yes—
413C4⋊C8C2414yes—
414C4⋊C8C2414yes—
415C4⋊C8C2414yes—
416#103—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—28no—
421#103—28no—
422#102—28no—
423#102—28no—
424C22⋊C8—22no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8—22no—
428C22⋊C8—22no—
429C22⋊C8—22no—
430C22⋊C8—24no—
431C22⋊C8—24no—
432C22⋊C8—24no—
433C22⋊C8—24no—
434C22⋊C8C2×D412yes—
435C22⋊C8C2×D412yes—
436C22⋊C8C2×D412yes—
437C22⋊C8C2×D412yes—
438#97—22no—
439#97—22no—
440C4×C8—22no—
441C4×C8—22no—
442C4×C8—22no—
443C4×C8—22no—
444C4×C8C2×D412yes—
445C4×C8C2×D412yes—
446C4×C8C2411yescharacteristic
447C4×C8C2411yescharacteristic
448C4×C8C2411yescharacteristic
449C4×C8C2411yescharacteristic
450C24—22no—
451C24#12111yescharacteristic
452#41—44no—
453#41—44no—
454#41—84no—
455#41—84no—
456#41—84no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—41no—
464C2×Q8—41no—
465C2×Q8—41no—
466C2×Q8—41no—
467C2×Q8—44no—
468C2×Q8—82no—
469C2×Q8—82no—
470C2×Q8—82no—
471C2×Q8—82no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—44no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4—84no—
483C2×D4C22×D414yes—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—84no—
495C22×C4#12111yescharacteristic
496C22×C4#12111yescharacteristic
497C22×C4#12111yescharacteristic
498C22×C4#12111yescharacteristic
499C22×C4#12111yescharacteristic
500C22×C4#12111yescharacteristic
501C22×C4#12111yescharacteristic
502C22×C4#12111yescharacteristic
503C22×C4#12111yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C2×#4111yescharacteristic
511C22×C4C2×#4111yescharacteristic
512C22×C4C2×#4111yescharacteristic
513SD16—88no—
514SD16—88no—
515M16—44no—
516M16—84no—
517M16—84no—
518M16—82no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—84no—
525C2×C8—84no—
526C2×C8—82no—
527C2×C8C22×D412yes—
528C2×C8C22×D412yes—
529C2×C8C22×D412yes—
530C2×C8C22×D412yes—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4C22×D414yes—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42—42no—
571C42#12111yescharacteristic
572C42#12111yescharacteristic
573C42#12111yescharacteristic
574C42#12111yescharacteristic
575C42#12111yescharacteristic
576C42#12111yescharacteristic
577C42C22×D411yescharacteristic
578C42C22×D411yescharacteristic
579C42C22×D411yescharacteristic
580C42C22×D411yescharacteristic
581C42C22×D411yescharacteristic
582C42C22×D411yescharacteristic
583C42C2×#4111yescharacteristic
584C42C2×#4111yescharacteristic
585C42C2×#4111yescharacteristic
586C42C2511yescharacteristic
587C23—22no—
588C23—24no—
589C23—41no—
590C23—41no—
591C23—44no—
592C23—82no—
593C23—82no—
594C23C2×#12111yescharacteristic
595C23C2×#12111yescharacteristic
596C23C2×#12111yescharacteristic
597Q8—44no—
598Q8—44no—
599Q8—44no—
600Q8—44no—
601Q8—82no—
602Q8—82no—
603Q8—82no—
604Q8—82no—
605Q8—82no—
606Q8—82no—
607D4—28no—
608D4—44no—
609D4—48no—
610D4—44no—
611D4—44no—
612D4—82no—
613D4—82no—
614D4—84no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4C2×#12111yescharacteristic
646C2×C4C2×#12111yescharacteristic
647C2×C4C2×#12111yescharacteristic
648C2×C4C2×#12111yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×#12111yescharacteristic
651C2×C4C2×#12111yescharacteristic
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4#57714yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C23×D411yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C8—22no—
659C8—22no—
660C8—22no—
661C8—22no—
662C8—44no—
663C8—44no—
664C8—44no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—82no—
677C22—164no—
678C22#304212yes—
679C22C22×#12111yescharacteristic
680C4—21no—
681C4—21no—
682C4—21no—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4#304212yes—
694C4#306912yes—
695C4C2×#57712yes—
696C4#358412yes—
697C2—22no—
698C2—41no—
699C2—41no—
700C2—84no—
701C2C2×#304211yescharacteristic
702C2#6045011yescharacteristic
703C2C2×#306911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#3231C2semidirect
5. #19518C2semidirect
6. #19517C2semidirect
7. #19431C2semidirect
8. #19395C2semidirect
9. #19199C2semidirect
10. #19197C2semidirect
11. #19192C2semidirect
12. #19191C2semidirect
13. #17803C2semidirect
14. #17747C2semidirect
15. #17715C2semidirect
25. #3231C22semidirect
26. #3231C22semidirect
39. C2×#500C22semidirect
40. C2×#498C22semidirect
41. C2×#487C22semidirect
42. C2×#487C22semidirect
63. #1703C22semidirect
64. #1699C22semidirect
65. #1693C22semidirect
66. #1691C22semidirect
67. #1683C22semidirect
68. #1679C22semidirect
69. #1673C22semidirect
70. #1671C22semidirect
73. #1571C22semidirect
74. #1570C22semidirect
75. #1564C22semidirect
76. #1564C22semidirect
77. #1559C22semidirect
79. #1552C22semidirect
80. #1549C22semidirect
81. #1549C22semidirect
82. #1548C22semidirect
83. #1546C22semidirect

Maximal subgroups

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

subgroupindex
#609692
#609412
C2×#32312
#602362
#195182
#195172
#194312
#193952
#191992
#191972
#191922
#191912
#178032
#177472
#177152