Finite Group Databasebeta

← #6526703

Subgroups — #6526703

Order 512.

1894 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C211yesmaximal, characteristic
2C2×#3117C211yesmaximal, characteristic
3C2×#2567C211yesmaximal, characteristic, complemented
4#34224C211yesmaximal, characteristic
5#28121C212yesmaximal, complemented
6#28115C212yesmaximal
7#27648C212yesmaximal, complemented
8#27645C212yesmaximal
9#25467C214yesmaximal, complemented
10#25336C214yesmaximal, complemented
11#24833C212yesmaximal, complemented
12#24831C212yesmaximal, complemented
13#24698C212yesmaximal, complemented
14#24696C212yesmaximal, complemented
15#21955C211yesmaximal, characteristic, complemented
16#21841C211yesmaximal, characteristic, complemented
17#20741C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3239—28no—
22C2×#501C2212yes—
23C2×#489C2212yes—
24#3117—28no—
25C2×#484C2211yescharacteristic
26C2×#484C2212yes—
27C2×#481C2211yescharacteristic
28C2×#481C2212yes—
29C2×#479C2212yes—
30C2×#475C2212yes—
31C2×#474C2212yes—
32C2×#464C2212yes—
33C2×#451C2211yescharacteristic
34C2×#448C2211yescharacteristic
35C2×#441C2212yes—
36C2×#440C2212yes—
37C2×#432C2211yescharacteristic
38C2×#432C2211yescharacteristic
39#2567—21no—
40#2567—21no—
41#2567—22no—
42#2567C2212yes—
43#2567C2212yes—
44#2567C2214yes—
45C2×#395—21no—
46C2×#395—21no—
47#2373—22no—
48#2370C2212yes—
49#2367C2212yes—
50#2357C2211yescharacteristic
51#2356C2211yescharacteristic
52#2351C2211yescharacteristic
53#2349C2211yescharacteristic
54#2347—21no—
55#2343—21no—
56#2322—22no—
57#2318—21no—
58C4×C22⋊Q8C2212yes—
59C4×#121C2212yes—
60C2×#391C2212yes—
61C2×#389—22no—
62C2×#384C2212yes—
63#2283C2211yescharacteristic
64#2275—21no—
65C2×C4×C4⋊C4C2211yescharacteristic
66#2084C2214yes—
67#2083C2214yes—
68#2072C2214yes—
69#2071C2214yes—
70#2068C2214yes—
71#2066C2214yes—
72#2056C2214yes—
73#2055C2214yes—
74#1993C2212yes—
75#1993C2212yes—
76#1988C2214yes—
77#1948C2214yes—
78#1947C2214yes—
79#1940C2212yes—
80#1940C2212yes—
81#1940C2214yes—
82#1928C2214yes—
83#1927C2214yes—
84#1878C2214yes—
85#1877C2214yes—
86#1824C2212yes—
87#1824C2212yes—
88#1824C2212yes—
89#1805C2212yes—
90#1805C2212yes—
91#1803C2214yes—
92#1802C2214yes—
93#1797C2212yes—
94#1797C2212yes—
95#1787C2212yes—
96#1763C2214yes—
97C23×Q8—21no—
98C23×Q8C2311yescharacteristic
99C22×Q16C2312yes—
100C22×SD16C2312yes—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107#529—24no—
108#529D418yes—
109C2×#126—21no—
110C2×#125C2311yescharacteristic
111C2×#124C2311yescharacteristic
112C2×#123—21no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121C2311yescharacteristic
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4D412yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#501—28no—
139#489—28no—
140#484—24no—
141#484—28no—
142#481—24no—
143#481—28no—
144#479—28no—
145#475—28no—
146#474—28no—
147#464—28no—
148#451—24no—
149#448—24no—
150#441—28no—
151#440—28no—
152#432—24no—
153#432—24no—
154C2×#107C2312yes—
155C2×#107C2314yes—
156C2×#106C2314yes—
157C2×#106C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2311yescharacteristic
172C2×C22⋊C8C2311yescharacteristic
173C2×#97C2312yes—
174C2×#97C2312yes—
175#395—22no—
176#395—22no—
177#395—24no—
178#395—22no—
179#395—22no—
180#395—24no—
181#391—22no—
182#391—22no—
183#391C2314yes—
184#391C2314yes—
185#391C2312yes—
186#390C2312yes—
187#389—24no—
188#389—24no—
189#389—24no—
190#389—24no—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#386—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384—22no—
208#384—22no—
209#384C2314yes—
210#384C2314yes—
211#383—21no—
212#383—21no—
213#382—21no—
214#382C2311yescharacteristic
215#382C2311yescharacteristic
216#381—21no—
217#381—21no—
218#380—21no—
219#380C2312yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D412yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2312yes—
232C4×C22⋊C4—21no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—22no—
237#376—21no—
238#376—21no—
239C2×#95—22no—
240C2×#95—22no—
241C2×#95—21no—
242C2×#95D411yescharacteristic
243C2×#95D411yescharacteristic
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C22×Q8—24no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—42no—
259C22×Q8—44no—
260C22×Q8—84no—
261C22×Q8C2411yescharacteristic
262C22×Q8C2412yes—
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D412yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D412yes—
274C23×C4C2411yescharacteristic
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—42no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—42no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2414yes—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290#126—28no—
291#125—22no—
292#125C2×D414yes—
293#124—22no—
294#124C2×D414yes—
295#123—28no—
296C22⋊Q8—22no—
297C22⋊Q8C2×D414yes—
298#121—22no—
299#121C2×D414yes—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8C2×D414yes—
309C4×D4—28no—
310C4×D4—22no—
311C4×D4—44no—
312C4×D4—48no—
313C4×D4C2×D414yes—
314#117—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—84no—
340C2×C4⋊C4—84no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—44no—
368C2×C42—44no—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D414yes—
374C2×C42#4114yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—24no—
380#107—28no—
381#106—24no—
382#106—28no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C4⋊C8—44no—
386C4⋊C8—44no—
387#103—24no—
388#103—28no—
389#103—28no—
390#103—24no—
391#103—24no—
392#103—44no—
393#102—24no—
394#102—28no—
395C22⋊C8—22no—
396C22⋊C8—22no—
397C22⋊C8—24no—
398#97—24no—
399#97—24no—
400#95—21no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—21no—
410#95—21no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418#95C2×D412yes—
419#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D412yes—
422C24—21no—
423C24C22×D411yescharacteristic
424C2×Q8—24no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—84no—
440C2×Q8—84no—
441C2×Q8—816no—
442C2×Q8C22×D412yes—
443C2×Q82−1+414yes—
444C2×D4—24no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—44no—
449C2×D4C22×D412yes—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4#12114yes—
481C22×C4#12114yes—
482C22×C4#13718yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C2×#4112yes—
494C22×C4C2511yescharacteristic
495Q16—48no—
496SD16—48no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—82no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—82no—
535C4⋊C4C22×D412yes—
536C4⋊C4C22×D412yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—42no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—82no—
555C42—82no—
556C42—82no—
557C42#12114yes—
558C42#12114yes—
559C42C22×D412yes—
560C42C22×D412yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23C2×#12112yes—
572C23C2×#13714yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—24no—
576Q8—24no—
577Q8—48no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—88no—
582Q8—88no—
583D4—24no—
584D4—48no—
585D4—42no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—164no—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4C2×#13714yes—
640C2×C4C2×#13714yes—
641C2×C4C23×D411yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×2−1+412yes—
644C2×C4#58512yes—
645C8—44no—
646C8—42no—
647C8—42no—
648C22—22no—
649C22—24no—
650C22—22no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—82no—
656C22—82no—
657C22#303914yes—
658C22C22×#12111yescharacteristic
659C22#348511yescharacteristic
660C22#348611yescharacteristic
661C22C22×#13712yes—
662C22#358512yes—
663C4—22no—
664C4—22no—
665C4—24no—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—84no—
678C4—82no—
679C4—82no—
680C4#348912yes—
681C4#349512yes—
682C2—24no—
683C2—82no—
684C2#3563311yescharacteristic
685C2C2×#303911yescharacteristic
686C2#6042611yescharacteristic
687C2#6170912yes—
688C2#6183012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2567C2semidirect
5. #28121C2semidirect
7. #27648C2semidirect
9. #25467C2semidirect
10. #25336C2semidirect
11. #24833C2semidirect
12. #24831C2semidirect
13. #24698C2semidirect
14. #24696C2semidirect
15. #21955C2semidirect
16. #21841C2semidirect
17. #20741C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
C2×#31172
C2×#25672
#342242
#281212
#281152
#276482
#276452
#254672
#253362
#248332
#248312
#246982
#246962
#219552
#218412
#207412