Finite Group Databasebeta

← #6508652

Subgroups — #6508652

Order 512.

3250 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3536C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3C22×#480C211yesmaximal, characteristic, complemented
4#35227C211yesmaximal, characteristic, complemented
5#24098C214yesmaximal, complemented
6#24088C214yesmaximal, complemented
7#24073C214yesmaximal, complemented
8#24039C214yesmaximal, complemented
9#24035C214yesmaximal, complemented
10#24011C214yesmaximal, complemented
11#21762C211yesmaximal, characteristic, complemented
12#21761C211yesmaximal, characteristic, complemented
13#21739C211yesmaximal, characteristic, complemented
14#3536—28no—
15#3536—28no—
16C2×#562—22no—
17C2×#554C2212yes—
18C2×#548C2212yes—
19C2×#547C2212yes—
20C2×#546—22no—
21C2×#546C2212yes—
22C2×#543—21no—
23C2×#542C2212yes—
24C2×#539—21no—
25C2×#538—21no—
26C2×#537—21no—
27C2×#537C2212yes—
28C2×#525C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×#121C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C2×#484—28no—
33C2×#484C2214yes—
34C2×#484C2214yescomplemented
35C2×#480—28no—
36C2×#480C2214yes—
37C2×#480C2214yes—
38C22×#107C2212yes—
39C22×#107C2212yescomplemented
40C22×#103C2212yescomplemented
41C22×#102C2212yes—
42C22×C22⋊C8C2211yescharacteristic
43C22×C22⋊C8C2211yescharacteristic
44#2849—22no—
45#2623—22no—
46#2504—21no—
47#2503—21no—
48#2502—21no—
49#2501—21no—
50#2487C2214yes—
51#2486C2214yescomplemented
52#2317C2212yescomplemented
53#2311C2212yescomplemented
54#2280C2211yescharacteristic
55#2057C2218yescomplemented
56#2042C2218yescomplemented
57#2036C2218yescomplemented
58#2033C2218yescomplemented
59#1893C2214yes—
60#1892C2214yes—
61#1875C2214yes—
62#1873C2214yescomplemented
63#1869C2214yes—
64#1869C2214yescomplemented
65#1863C2218yescomplemented
66#1862C2218yescomplemented
67#1852C2218yescomplemented
68#1824C2212yes—
69#1823C2212yescomplemented
70#1818C2212yescomplemented
71#1788C2218yescomplemented
72#1787C2212yescomplemented
73#1786C2212yes—
74#1780C2212yescomplemented
75C22×#41—22no—
76C22×#41C2312yes—
77C22×#41C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C23×C8C2312yes—
84C23×C8C2312yes—
85#562—232no—
86#554—28no—
87#554—28no—
88#548—28no—
89#548—28no—
90#547—28no—
91#547—28no—
92#546—232no—
93#546—28no—
94#546—28no—
95#543—216no—
96#542—28no—
97#542—28no—
98#539—216no—
99#538—216no—
100#537—216no—
101#537—28no—
102#537—28no—
103C2×C4⋊Q8C2312yes—
104C2×#126—22no—
105C2×#125—22no—
106C2×#124—22no—
107C2×#124—22no—
108C2×#124C2312yes—
109#525—24no—
110#525—24no—
111C2×#123—24no—
112C2×#123—22no—
113C2×#123—22no—
114C2×#123—22no—
115C2×#123—22no—
116C2×#123D412yes—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8D418yes—
127C2×C22⋊Q8D411yescharacteristic
128C2×C22⋊Q8D411yescharacteristic
129C2×C22⋊Q8C2312yes—
130C2×C22⋊Q8C2312yes—
131C2×C22⋊Q8C2312yes—
132C2×#121—22no—
133C2×#121—22no—
134C2×#121—22no—
135C2×#121—22no—
136C2×#121—21no—
137C2×#121—21no—
138C2×#121—24no—
139C2×#121—22no—
140C2×#121D418yes—
141C2×#121D411yescharacteristic
142C2×#121D411yescharacteristic
143C2×#121C2312yes—
144C2×#121C2312yes—
145C2×#121C2312yes—
146C2×#120—22no—
147C2×#120D412yes—
148C2×C4×Q8C2312yes—
149C2×C4×D4—22no—
150C2×C4×D4—22no—
151C2×C4×D4C2312yes—
152C2×C4×D4C2312yes—
153C2×C4×D4C2312yes—
154C2×#117C2312yes—
155C2×#117C2311yescharacteristic
156C22×C4⋊C4—21no—
157C22×C4⋊C4—24no—
158C22×C4⋊C4—24no—
159C22×C4⋊C4—21no—
160C22×C4⋊C4—21no—
161C22×C4⋊C4—21no—
162C22×C4⋊C4C2312yes—
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2312yes—
166C22×C4⋊C4C2311yescharacteristic
167C22×C4⋊C4C2311yescharacteristic
168C22×C22⋊C4—21no—
169C22×C22⋊C4—21no—
170C22×C22⋊C4—22no—
171#484—216no—
172#484—216no—
173#480—216no—
174#480—216no—
175C2×#107—28no—
176C2×#107—28no—
177C2×#107C2318yes—
178C2×#107C2318yes—
179C2×#103—28no—
180C2×#103C2318yes—
181C2×#103C2318yes—
182C2×#102—28no—
183C2×#102C2318yes—
184C2×#102C2318yes—
185C2×C22⋊C8—24no—
186C2×C22⋊C8—24no—
187C2×C22⋊C8C2312yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2318yes—
192#400—24no—
193#399—24no—
194#398—24no—
195#397—24no—
196#391—22no—
197#390—22no—
198#389C2318yes—
199#386—24no—
200#386—24no—
201#385—24no—
202#385—24no—
203#384—22no—
204#382—22no—
205#381—21no—
206#381—24no—
207#380C2318yes—
208#380C2312yes—
209#380C2312yes—
210#379—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213#340C2318yes—
214#340C2318yes—
215#336C2318yes—
216C25—21no—
217C25C2×D411yescharacteristic
218C2×#41—216no—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—44no—
224C2×#41—48no—
225C2×#41—88no—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—44no—
230C22×Q8C2×D414yes—
231C22×Q8C2×D411yescharacteristic
232C22×Q8C2412yes—
233C22×D4—22no—
234C22×D4—28no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—44no—
241C22×D4—84no—
242C22×D4C2×D412yes—
243C22×D4C2×D412yes—
244C22×D4C2×D412yes—
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D414yes—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—84no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D414yes—
264C23×C4C2×D414yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—44no—
272C22×C8—44no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277#126—216no—
278#125—216no—
279#124—216no—
280#124—216no—
281#124—24no—
282#124—24no—
283#123—216no—
284#123—216no—
285#123—216no—
286#123—216no—
287#123—28no—
288#123—88no—
289C22⋊Q8—216no—
290C22⋊Q8—216no—
291C22⋊Q8—216no—
292C22⋊Q8—216no—
293C22⋊Q8—216no—
294C22⋊Q8—216no—
295C22⋊Q8—24no—
296C22⋊Q8—28no—
297C22⋊Q8—28no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8C2×D418yes—
304C22⋊Q8C2×D418yes—
305#121—216no—
306#121—216no—
307#121—216no—
308#121—216no—
309#121—216no—
310#121—24no—
311#121—28no—
312#121—28no—
313#121—24no—
314#121—24no—
315#121—24no—
316#121—24no—
317#121—24no—
318#121C2×D418yes—
319#121C2×D418yes—
320#120—216no—
321#120—28no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×D4—216no—
325C4×D4—216no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—24no—
332#117—22no—
333#117—22no—
334#117—44no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—28no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2×D412yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2×D414yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—216no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—28no—
387C2×C22⋊C4C2×D412yes—
388C2×C22⋊C4C2×D412yes—
389C2×C22⋊C4C2×D412yes—
390C2×C22⋊C4C2×D412yes—
391C2×C22⋊C4C2×D412yes—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C2412yes—
395#107—216no—
396#107—216no—
397#103—216no—
398#103—48no—
399#102—216no—
400#102—48no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—48no—
406#95—22no—
407#95—22no—
408#95—28no—
409#95—28no—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—24no—
414C24—42no—
415C24—44no—
416C24—82no—
417C24—82no—
418C24#12014yes—
419C24C22×D412yes—
420C24C22×D411yescharacteristic
421C24C22×D411yescharacteristic
422C24C22×D411yescharacteristic
423#41—84no—
424#41—84no—
425#41—88no—
426#41—832no—
427#41—88no—
428C2×Q8—28no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—28no—
433C2×Q8—42no—
434C2×Q8—48no—
435C2×Q8—82no—
436C2×Q8—84no—
437C2×Q8—84no—
438C2×Q8C22×D414yes—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—28no—
452C2×D4—48no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—48no—
456C2×D4—84no—
457C2×D4—82no—
458C2×D4—84no—
459C2×D4—84no—
460C2×D4—84no—
461C2×D4—816no—
462C2×D4C22×D414yes—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—84no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—84no—
497C22×C4—816no—
498C22×C4#12014yes—
499C22×C4#12014yes—
500C22×C4#12014yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C2511yescharacteristic
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—28no—
517C2×C8—28no—
518C2×C8—48no—
519C2×C8—48no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—28no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—88no—
541C4⋊C4C22×D414yes—
542C4⋊C4C22×D414yes—
543C22⋊C4—28no—
544C22⋊C4—28no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—28no—
549C22⋊C4—24no—
550C22⋊C4—28no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—28no—
554C22⋊C4—28no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—88no—
561C42—28no—
562C42—22no—
563C42—22no—
564C42—44no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—28no—
571C23—28no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—22no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—44no—
580C23—416no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—84no—
586C23—84no—
587C23—168no—
588C23C2×#12012yes—
589C23C2×#12012yes—
590C23C2×#12012yes—
591C23#57614yes—
592C23#57814yes—
593C23C23×D411yescharacteristic
594Q8—28no—
595Q8—44no—
596Q8—84no—
597Q8—82no—
598Q8—88no—
599D4—28no—
600D4—48no—
601D4—416no—
602D4—48no—
603D4—44no—
604D4—84no—
605D4—82no—
606D4—84no—
607D4—88no—
608D4—88no—
609D4—88no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—48no—
641C2×C4—44no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4—88no—
650C2×C4C2×#12012yes—
651C2×C4C2×#12012yes—
652C2×C4C2×#12012yes—
653C2×C4C23×D411yescharacteristic
654C2×C4C23×D411yescharacteristic
655C2×C4#58514yes—
656C8—44no—
657C8—44no—
658C22—28no—
659C22—28no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—416no—
668C22—44no—
669C22—48no—
670C22—42no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—44no—
676C22—88no—
677C22—168no—
678C22#300814yes—
679C22#301114yes—
680C22C22×#12011yescharacteristic
681C22#348512yes—
682C22C2×#57612yes—
683C22C2×#57812yes—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—42no—
688C4—44no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—84no—
694C4—84no—
695C4#348512yes—
696C4#348512yes—
697C2—24no—
698C2—24no—
699C2—42no—
700C2—82no—
701C2—84no—
702C2#3544611yescharacteristic
703C2C2×#300811yescharacteristic
704C2C2×#301111yescharacteristic
705C2#6176414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3536C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#484C2semidirect
3. C22×#480C2semidirect
4. #35227C2semidirect
5. #24098C2semidirect
6. #24088C2semidirect
7. #24073C2semidirect
8. #24039C2semidirect
9. #24035C2semidirect
10. #24011C2semidirect
11. #21762C2semidirect
12. #21761C2semidirect
13. #21739C2semidirect
34. C2×#484C22semidirect
39. C22×#107C22semidirect
40. C22×#103C22semidirect
51. #2486C22semidirect
52. #2317C22semidirect
53. #2311C22semidirect
55. #2057C22semidirect
56. #2042C22semidirect
57. #2036C22semidirect
58. #2033C22semidirect
62. #1873C22semidirect
64. #1869C22semidirect
65. #1863C22semidirect
66. #1862C22semidirect
67. #1852C22semidirect
69. #1823C22semidirect
70. #1818C22semidirect
71. #1788C22semidirect
72. #1787C22semidirect
74. #1780C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35362
C22×#4842
C22×#4802
#352272
#240982
#240882
#240732
#240392
#240352
#240112
#217622
#217612
#217392