Finite Group Databasebeta

← #6481378

Subgroups — #6481378

Order 512.

3238 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
1C22×#552C211yesmaximal, characteristic, complemented
2C2×#3234C211yesmaximal, characteristic, complemented
3C2×#3103C211yesmaximal, characteristic, complemented
4#35237C211yesmaximal, characteristic
5#32980C212yesmaximal, complemented
6#32979C212yesmaximal, complemented
7#29595C212yesmaximal, complemented
8#29593C212yesmaximal, complemented
9#25715C214yesmaximal, complemented
10#25583C214yesmaximal, complemented
11#24125C218yesmaximal, complemented
12#21836C211yesmaximal, characteristic, complemented
13#21788C211yesmaximal, characteristic, complemented
14#20741C211yesmaximal, characteristic, complemented
15C2×#552—28no—
16C2×#552—28no—
17C2×#552—22no—
18C2×#552—22no—
19C2×#552C2214yes—
20C2×#552C2214yes—
21C2×#532C2211yescharacteristic
22C22×#126—21no—
23C22×#125—21no—
24C22×#125C2211yescharacteristic
25C22×#123—21no—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117—21no—
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3234—28no—
34C2×#501C2212yescomplemented
35C2×#490C2212yescomplemented
36#3103—28no—
37C2×#481C2212yes—
38C2×#481C2212yes—
39C2×#480C2212yes—
40C2×#475C2212yescomplemented
41C2×#478C2214yescomplemented
42C2×#460C2212yescomplemented
43C2×#448C2212yes—
44C2×C4×SD16C2214yescomplemented
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2847C2212yes—
48#2847C2212yes—
49#2631—22no—
50#2563C2212yes—
51#2541C2212yes—
52#2535C2214yes—
53#2504—21no—
54#2433—21no—
55#2404—21no—
56#2404—21no—
57#2328—22no—
58#2283C2211yescharacteristic
59#2036C2218yescomplemented
60#2035C2218yescomplemented
61#2000C2218yescomplemented
62#2000C2218yescomplemented
63#1991C2214yescomplemented
64#1989C2214yescomplemented
65#1955C2214yes—
66#1954C2214yes—
67#1875C2218yescomplemented
68#1874C2218yescomplemented
69#1854C2218yescomplemented
70#1853C2218yescomplemented
71#1824C2212yes—
72#1824C2212yes—
73#1823C2212yes—
74#1805C2214yescomplemented
75#1787C2212yes—
76#1779C2212yes—
77#1779C2212yes—
78#1763C2214yescomplemented
79C22×#41—21no—
80C23×Q8C2311yescharacteristic
81C23×D4D411yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×SD16C2314yes—
86#552—232no—
87#552—232no—
88#552—216no—
89#552—216no—
90#552—216no—
91#552—216no—
92#532—24no—
93#532—24no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127—22no—
96C2×#127C2311yescharacteristic
97C2×#126—216no—
98C2×#126—24no—
99C2×#126—24no—
100C2×#125—216no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—21no—
104C2×#125—21no—
105C2×#125—24no—
106C2×#125D418yescomplemented
107C2×#125C2312yes—
108C2×#125C2312yes—
109C2×#124—21no—
110C2×#123—216no—
111C2×#123—24no—
112C2×#123—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8C2314yes—
120C2×C22⋊Q8C2314yes—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×#121C2314yes—
124C2×#121C2314yes—
125C2×#120—22no—
126C2×#120—22no—
127C2×C4×Q8—24no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8D418yescomplemented
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—21no—
136C2×C4×D4D412yes—
137C2×C4×D4D412yes—
138C2×C4×D4C2312yes—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×#117—216no—
142C2×#117—22no—
143C2×#117—24no—
144C2×#117—24no—
145C2×#117—24no—
146C2×#117—24no—
147C2×#117C2314yes—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—22no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—24no—
156C22×C4⋊C4—24no—
157C22×C4⋊C4D411yescharacteristic
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2311yescharacteristic
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4D411yescharacteristic
166C22×C22⋊C4D411yescharacteristic
167C22×C42—21no—
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170#501—28no—
171#490—28no—
172#481—28no—
173#481—28no—
174#480—28no—
175#478—216no—
176#475—28no—
177#460—28no—
178#448—28no—
179C4×SD16—216no—
180#432—24no—
181#431—24no—
182C2×#107C2314yes—
183C2×#106C2314yes—
184C2×#106C2314yes—
185C2×C4⋊C8C2312yes—
186C2×C4⋊C8C2312yes—
187C2×#103C2314yes—
188C2×#103C2318yes—
189C2×#102C2314yes—
190C2×#102C2314yes—
191C2×#102C2314yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2318yes—
195C2×C22⋊C8C2318yes—
196C2×#97C2312yes—
197C2×C4×C8C2312yes—
198#397—24no—
199#393—22no—
200#393—22no—
201#392—24no—
202#392—22no—
203#392—22no—
204#390—22no—
205#390—22no—
206#386—24no—
207#386—24no—
208#386—22no—
209#386—22no—
210#386—24no—
211#386—24no—
212#383—21no—
213#382—22no—
214#382C2312yes—
215#381—21no—
216#380C2314yes—
217#380C2314yes—
218#380C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221#376—21no—
222#340C2318yes—
223#340C2318yes—
224#340C2318yes—
225C25C2×D411yescharacteristic
226C25C2×D411yescharacteristic
227C2×#41—44no—
228C2×#41—84no—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—44no—
233C22×Q8C2×D414yes—
234C22×Q8C2412yes—
235C22×D4—22no—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—44no—
242C22×D4C2×D412yes—
243C22×D4C2×D412yes—
244C22×D4C2412yes—
245C23×C4—28no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—48no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C2×SD16—28no—
269C2×SD16—48no—
270C22×C8C2414yes—
271C22×C8C2414yes—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274#127—22no—
275#127—22no—
276#127—48no—
277#126—232no—
278#126—232no—
279#125—232no—
280#125—232no—
281#125—28no—
282#125—28no—
283#125—24no—
284#125—24no—
285#125C2×D418yes—
286#125C2×D418yes—
287#124—44no—
288#123—232no—
289#123—232no—
290C22⋊Q8—216no—
291C22⋊Q8—216no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296#121—24no—
297#121—24no—
298#121—44no—
299#121—48no—
300#121—48no—
301#120—84no—
302#120—84no—
303C4×Q8—28no—
304C4×Q8—28no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8C2×D418yes—
308C4×Q8C2×D418yes—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—44no—
318#117—216no—
319#117—232no—
320#117—232no—
321#117—28no—
322#117—28no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—216no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—28no—
379C2×C42—28no—
380C2×C42—24no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D414yes—
387C2×C42C2×D414yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—28no—
391#106—28no—
392#106—28no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395#103—28no—
396#103—48no—
397#102—28no—
398#102—28no—
399#102—28no—
400C22⋊C8—24no—
401C22⋊C8—24no—
402C22⋊C8—48no—
403C22⋊C8—48no—
404#97—24no—
405C4×C8—24no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412C24—22no—
413C24—24no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—22no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24C22×D412yes—
422C24C22×D412yes—
423C24C22×D411yescharacteristic
424#41—816no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—48no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—42no—
433C2×Q8—82no—
434C2×Q8C22×D414yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—28no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—416no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—48no—
450C2×D4—48no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—84no—
455C2×D4—84no—
456C2×D4—84no—
457C2×D4—84no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—216no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—432no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—44no—
495C22×C4—48no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4#12014yes—
501C22×C4#12014yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C2511yescharacteristic
512SD16—416no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—28no—
516C2×C8—28no—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—28no—
537C4⋊C4—42no—
538C4⋊C4C22×D414yes—
539C4⋊C4C22×D414yes—
540C4⋊C4C22×D414yes—
541C22⋊C4—216no—
542C22⋊C4—216no—
543C22⋊C4—216no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—216no—
556C42—28no—
557C42—28no—
558C42—44no—
559C42#12014yes—
560C42#12014yes—
561C42C22×D412yes—
562C42C22×D412yes—
563C42C22×D412yes—
564C42C22×D412yes—
565C23—22no—
566C23—22no—
567C23—28no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—24no—
572C23—22no—
573C23—24no—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—416no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—44no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23—88no—
589C23C2×#12012yes—
590C23#57514yes—
591C23#57714yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—48no—
596Q8—44no—
597Q8—84no—
598D4—48no—
599D4—48no—
600D4—44no—
601D4—44no—
602D4—84no—
603D4—84no—
604D4—88no—
605D4—168no—
606D4—168no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—432no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—88no—
648C2×C4C2×#12012yes—
649C2×C4C2×#12012yes—
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—22no—
660C22—22no—
661C22—28no—
662C22—24no—
663C22—416no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—88no—
673C22—84no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22#301014yes—
679C22C22×#12011yescharacteristic
680C22#348512yes—
681C22C2×#57512yes—
682C22C2×#57712yes—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—84no—
695C4—82no—
696C4#344912yes—
697C4#348712yes—
698C2—24no—
699C2—44no—
700C2—82no—
701C2—162no—
702C2#3514911yescharacteristic
703C2#6035911yescharacteristic
704C2C2×#301011yescharacteristic
705C2#6172914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35237C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#552C2semidirect
2. C2×#3234C2semidirect
3. C2×#3103C2semidirect
5. #32980C2semidirect
6. #32979C2semidirect
7. #29595C2semidirect
8. #29593C2semidirect
9. #25715C2semidirect
10. #25583C2semidirect
11. #24125C2semidirect
12. #21836C2semidirect
13. #21788C2semidirect
14. #20741C2semidirect
34. C2×#501C22semidirect
35. C2×#490C22semidirect
40. C2×#475C22semidirect
41. C2×#478C22semidirect
42. C2×#460C22semidirect
44. C2×C4×SD16C22semidirect
59. #2036C22semidirect
60. #2035C22semidirect
61. #2000C22semidirect
62. #2000C22semidirect
63. #1991C22semidirect
64. #1989C22semidirect
67. #1875C22semidirect
68. #1874C22semidirect
69. #1854C22semidirect
70. #1853C22semidirect
74. #1805C22semidirect
78. #1763C22semidirect
106. C2×#125D4semidirect
130. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#32342
C2×#31032
#352372
#329802
#329792
#295952
#295932
#257152
#255832
#241252
#218362
#217882
#207412