Finite Group Databasebeta

← #6477475

Subgroups — #6477475

Order 512.

3370 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2C2×#3187C211yesmaximal, characteristic, complemented
3C2×#3095C211yesmaximal, characteristic, complemented
4#35084C211yesmaximal, characteristic, complemented
5#33007C212yesmaximal, complemented
6#32982C212yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#29598C212yesmaximal, complemented
9#25532C218yesmaximal, complemented
10#24101C214yesmaximal, complemented
11#23974C214yesmaximal, complemented
12#21767C211yesmaximal, characteristic, complemented
13#21740C211yesmaximal, characteristic, complemented
14#20699C211yesmaximal, characteristic, complemented
15C2×#553—28no—
16C2×#553—22no—
17C2×#553—22no—
18C2×#553—28no—
19C2×#553C2214yes—
20C2×#553C2214yes—
21C22×#126—21no—
22C22×#125C2211yescharacteristic
23C22×#123—21no—
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121—21no—
26C22×#121C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3187—28no—
34#3095—28no—
35C2×#484C2212yes—
36C2×#481C2212yescomplemented
37C2×#485C2214yescomplemented
38C2×#483C2212yes—
39C2×#483C2212yes—
40C2×#465C2214yescomplemented
41C2×#448C2212yescomplemented
42C2×#450C2212yes—
43C2×#428C2212yescomplemented
44C2×#419C2212yescomplemented
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2638—22no—
50#2583—21no—
51#2561—21no—
52#2486C2212yescomplemented
53#2483—21no—
54#2415C2214yescomplemented
55#2409C2212yescomplemented
56#2405—21no—
57#2322—22no—
58#2280C2211yescharacteristic
59#2059C2218yescomplemented
60#2033C2218yescomplemented
61#2026C2218yescomplemented
62#2000C2218yescomplemented
63#1895C2214yescomplemented
64#1877C2218yescomplemented
65#1875C2214yescomplemented
66#1872C2218yescomplemented
67#1870C2214yescomplemented
68#1869C2214yescomplemented
69#1864C2218yescomplemented
70#1853C2218yescomplemented
71#1825C2214yescomplemented
72#1824C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1817C2212yes—
76#1786C2212yes—
77#1779C2212yes—
78#1735C2214yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4—21no—
82C23×D4D411yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4—21no—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C22×SD16C2314yes—
89C22×M16C2312yes—
90C23×C8C2312yes—
91#553—232no—
92#553—232no—
93#553—216no—
94#553—216no—
95#553—216no—
96#553—216no—
97C2×#126—216no—
98C2×#126—24no—
99C2×#126—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—22no—
103C2×#125C2314yes—
104C2×#123—216no—
105C2×#123—22no—
106C2×#123—24no—
107C2×#123—24no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2314yes—
115C2×#121—216no—
116C2×#121—24no—
117C2×#121—24no—
118C2×#121—24no—
119C2×#121—24no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121—22no—
123C2×#121—24no—
124C2×#121D418yes—
125C2×#121C2314yes—
126C2×#121C2312yes—
127C2×#121C2312yes—
128C2×C4×D4—216no—
129C2×C4×D4—24no—
130C2×C4×D4—24no—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—22no—
136C2×C4×D4D412yes—
137C2×C4×D4D412yes—
138C2×C4×D4C2314yes—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×#117—24no—
142C2×#117—21no—
143C2×#117—21no—
144C2×#117D418yes—
145C2×#117C2312yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—22no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2314yes—
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4D411yescharacteristic
167C22×C22⋊C4D411yescharacteristic
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170#485—216no—
171#484—28no—
172#483—28no—
173#483—28no—
174#481—28no—
175#465—216no—
176#450—28no—
177#448—28no—
178#428—28no—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×#106C2314yes—
182#419—28no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#102C2314yes—
187C2×#102C2318yes—
188#407—24no—
189C2×C22⋊C8—24no—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2318yes—
195#398—22no—
196#397—22no—
197#397—24no—
198#395—22no—
199#394—24no—
200#393—22no—
201#391C2314yes—
202#389—24no—
203#389C2314yes—
204#388—24no—
205#387—24no—
206#386—22no—
207#385—24no—
208#385—22no—
209#385—22no—
210#385—22no—
211#381—22no—
212#381—21no—
213#381—22no—
214#380C2312yes—
215#380C2314yes—
216#380C2312yes—
217#379—22no—
218#379—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C25C2×D411yescharacteristic
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—44no—
232C22×Q8C2412yes—
233C22×D4—28no—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4—28no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—48no—
244C22×D4—84no—
245C22×D4C2×D412yes—
246C22×D4C2×D412yes—
247C22×D4C2×D414yes—
248C22×D4C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—28no—
253C23×C4—28no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—24no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—84no—
263C23×C4—84no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D414yes—
272C23×C4C2×D414yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C2×SD16—28no—
278C2×SD16—48no—
279C2×M16—24no—
280C2×M16—44no—
281C22×C8—24no—
282C22×C8—44no—
283C22×C8C2414yes—
284C22×C8C2414yes—
285#126—232no—
286#126—232no—
287#125—216no—
288#125—28no—
289#125—28no—
290#123—232no—
291#123—232no—
292#123—48no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C22⋊Q8—48no—
300#121—232no—
301#121—232no—
302#121—216no—
303#121—28no—
304#121—28no—
305#121—28no—
306#121—24no—
307#121—24no—
308#121—28no—
309#121C2×D418yes—
310#121C2×D418yes—
311C4×D4—232no—
312C4×D4—232no—
313C4×D4—216no—
314C4×D4—28no—
315C4×D4—28no—
316C4×D4—28no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—28no—
322#117—28no—
323#117—28no—
324#117—24no—
325#117—24no—
326#117—44no—
327#117C2×D418yes—
328#117C2×D418yes—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—216no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—28no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—28no—
364C2×C22⋊C4—28no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—44no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C42—28no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42C2×D412yes—
394C2×C42C2412yes—
395#107—28no—
396#106—28no—
397#106—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#102—28no—
402#102—48no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—48no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—24no—
419C24—22no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—44no—
424C24—44no—
425C24—44no—
426C24#12014yes—
427C24C22×D411yescharacteristic
428C24C22×D411yescharacteristic
429C24C22×D412yes—
430C24C22×D411yescharacteristic
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—28no—
434C2×Q8—48no—
435C2×Q8—42no—
436C2×Q8—82no—
437C2×Q8—84no—
438C2×D4—216no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—28no—
448C2×D4—48no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—48no—
453C2×D4—416no—
454C2×D4—432no—
455C2×D4—48no—
456C2×D4—48no—
457C2×D4—816no—
458C2×D4—84no—
459C2×D4—84no—
460C2×D4C22×D414yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—216no—
475C22×C4—216no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—88no—
500C22×C4—88no—
501C22×C4—82no—
502C22×C4—816no—
503C22×C4—816no—
504C22×C4#12014yes—
505C22×C4#12014yes—
506C22×C4#12014yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C2511yescharacteristic
520SD16—88no—
521M16—48no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—28no—
525C2×C8—28no—
526C2×C8—84no—
527C4⋊C4—216no—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—28no—
541C4⋊C4—48no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—84no—
546C4⋊C4C22×D414yes—
547C4⋊C4C22×D414yes—
548C4⋊C4C22×D414yes—
549C22⋊C4—216no—
550C22⋊C4—216no—
551C22⋊C4—28no—
552C22⋊C4—28no—
553C22⋊C4—24no—
554C22⋊C4—24no—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—82no—
565C22⋊C4—84no—
566C42—28no—
567C42—28no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—82no—
572C23—22no—
573C23—22no—
574C23—28no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—22no—
584C23—22no—
585C23—28no—
586C23—24no—
587C23—44no—
588C23—44no—
589C23—416no—
590C23—44no—
591C23—48no—
592C23—48no—
593C23—44no—
594C23—44no—
595C23—48no—
596C23—44no—
597C23—44no—
598C23—88no—
599C23—816no—
600C23C2×#12012yes—
601C23C2×#12012yes—
602C23C2×#12012yes—
603C23#57514yes—
604C23#57714yes—
605C23C23×D411yescharacteristic
606Q8—44no—
607Q8—44no—
608Q8—84no—
609D4—28no—
610D4—48no—
611D4—48no—
612D4—48no—
613D4—432no—
614D4—44no—
615D4—88no—
616D4—88no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—48no—
643C2×C4—44no—
644C2×C4—48no—
645C2×C4—48no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—816no—
659C2×C4—816no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4C2×#12012yes—
663C2×C4C2×#12012yes—
664C2×C4C2×#12012yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C23×D411yescharacteristic
667C2×C4#58514yes—
668C8—44no—
669C8—44no—
670C22—22no—
671C22—28no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—28no—
676C22—22no—
677C22—24no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—416no—
682C22—42no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—88no—
691C22—816no—
692C22—88no—
693C22#300914yes—
694C22C22×#12011yescharacteristic
695C22#348512yes—
696C22C2×#57512yes—
697C22C2×#57712yes—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—44no—
706C4—44no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4#344912yes—
711C4#348512yes—
712C2—24no—
713C2—44no—
714C2—42no—
715C2—42no—
716C2—84no—
717C2#3508811yescharacteristic
718C2#6035911yescharacteristic
719C2C2×#300911yescharacteristic
720C2#6173614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#553C2
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
2. C2×#3187C2semidirect
3. C2×#3095C2semidirect
4. #35084C2semidirect
5. #33007C2semidirect
6. #32982C2semidirect
7. #29643C2semidirect
8. #29598C2semidirect
9. #25532C2semidirect
10. #24101C2semidirect
11. #23974C2semidirect
12. #21767C2semidirect
13. #21740C2semidirect
14. #20699C2semidirect
36. C2×#481C22semidirect
37. C2×#485C22semidirect
40. C2×#465C22semidirect
41. C2×#448C22semidirect
43. C2×#428C22semidirect
44. C2×#419C22semidirect
52. #2486C22semidirect
54. #2415C22semidirect
55. #2409C22semidirect
59. #2059C22semidirect
60. #2033C22semidirect
61. #2026C22semidirect
62. #2000C22semidirect
63. #1895C22semidirect
64. #1877C22semidirect
65. #1875C22semidirect
66. #1872C22semidirect
67. #1870C22semidirect
68. #1869C22semidirect
69. #1864C22semidirect
70. #1853C22semidirect
71. #1825C22semidirect
78. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
C2×#31872
C2×#30952
#350842
#330072
#329822
#296432
#295982
#255322
#241012
#239742
#217672
#217402
#206992