Finite Group Databasebeta

← #6529715

Subgroups — #6529715

Order 512.

1873 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic, complemented
3#34972C211yesmaximal, characteristic
4#34216C211yesmaximal, characteristic, complemented
5#28283C212yesmaximal, complemented
6#28281C212yesmaximal, complemented
7#28205C212yesmaximal, complemented
8#28203C212yesmaximal, complemented
9#25464C214yesmaximal, complemented
10#25287C214yesmaximal, complemented
11#24579C212yesmaximal, complemented
12#24578C212yesmaximal
13#24483C212yesmaximal
14#24482C212yesmaximal, complemented
15#21976C211yesmaximal, characteristic, complemented
16#21887C211yesmaximal, characteristic
17#20740C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3239—28no—
22C2×#501C2212yes—
23C2×#489C2212yes—
24C2×#484C2212yes—
25C2×#481C2212yes—
26C2×#475—28no—
27C2×#475C2212yes—
28C2×#475C2211yescharacteristic
29C2×#475C2212yes—
30C2×#475C2214yes—
31C2×#475C2211yescharacteristic
32C2×#474C2211yescharacteristic
33C2×#474C2211yescharacteristic
34C2×#432C2211yescharacteristic
35C22×#106C2212yes—
36C22×C4⋊C8C2211yescharacteristic
37C22×#103C2212yes—
38#2604—22no—
39#2565C2212yes—
40#2545C2212yes—
41#2503C2211yescharacteristic
42#2501C2211yescharacteristic
43#2436C2211yescharacteristic
44#2434C2211yescharacteristic
45#2413—21no—
46#2413—21no—
47#2396—22no—
48#2369C2214yes—
49#2365—22no—
50#2345C2212yes—
51#2345C2212yes—
52Q8×C22⋊C4—21no—
53Q8×C22⋊C4—21no—
54#2331—22no—
55#2316—21no—
56C4×C4⋊Q8C2212yes—
57C4×C4⋊Q8C2212yes—
58C4×C4⋊Q8C2212yes—
59C4×#124C2212yes—
60C4×#123—21no—
61C4×#123—21no—
62#2285C2211yescharacteristic
63#2274—21no—
64C4×#117C2211yescharacteristic
65#2085C2214yes—
66#2084C2214yes—
67#2068C2214yes—
68#2066C2214yes—
69#2066C2214yes—
70#2065C2214yes—
71#2043C2214yes—
72#2042C2214yes—
73#1989C2212yes—
74#1983C2214yes—
75#1983C2212yes—
76#1981C2214yes—
77#1949C2214yes—
78#1948C2214yes—
79#1940C2214yes—
80#1928C2214yes—
81#1927C2214yes—
82#1927C2214yes—
83#1926C2214yes—
84#1893C2212yes—
85#1892C2212yes—
86#1850C2212yes—
87#1850C2212yes—
88#1842C2212yes—
89#1831C2212yes—
90#1823C2212yes—
91#1803C2212yes—
92#1803C2212yes—
93#1803C2212yes—
94#1802C2212yes—
95#1787C2212yes—
96#1763C2214yes—
97C23×Q8—21no—
98C23×Q8C2311yescharacteristic
99C23×C8C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8D418yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108#529—24no—
109#529—24no—
110C2×#126—21no—
111C2×#125C2311yescharacteristic
112C2×#124C2311yescharacteristic
113C2×#124C2312yes—
114C2×#123—21no—
115C2×#123—21no—
116C2×#123—21no—
117C2×C22⋊Q8C2311yescharacteristic
118C2×#121C2311yescharacteristic
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117—21no—
132C2×#117D411yescharacteristic
133C2×#117D411yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#501—28no—
141#489—28no—
142#484—28no—
143#481—28no—
144#475—28no—
145#475—28no—
146#475—216no—
147#475—24no—
148#475—24no—
149#474—24no—
150#474—24no—
151#432—24no—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#106—28no—
155C2×#106C2314yes—
156C2×#106C2314yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×C4⋊C8—24no—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2314yes—
165C2×C4⋊C8C2314yes—
166C2×#103—28no—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2314yes—
177C2×C22⋊C8C2312yes—
178C2×#97C2312yes—
179#400—22no—
180#399—22no—
181#398—22no—
182#398—22no—
183#396—22no—
184#396—22no—
185#395—22no—
186#393—22no—
187#391—22no—
188#391—22no—
189#391—24no—
190#391—24no—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#390C2312yes—
195#389—24no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389C2314yes—
201#389C2314yes—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—24no—
209#387—24no—
210#386—21no—
211#386—21no—
212#385—21no—
213#385—21no—
214#384C2312yes—
215#384C2312yes—
216#382—21no—
217#382C2312yes—
218#381—21no—
219#381—21no—
220#380—21no—
221#380—22no—
222#380—22no—
223#380C2312yes—
224C4×C4⋊C4—24no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4D412yes—
231C4×C4⋊C4C2314yes—
232C4×C4⋊C4C2314yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4D411yescharacteristic
240C4×C22⋊C4D411yescharacteristic
241C4×C22⋊C4C2311yescharacteristic
242C4×C22⋊C4C2311yescharacteristic
243#376D412yes—
244C43C2312yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251C22×Q8—28no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—21no—
256C22×Q8—24no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—42no—
260C22×Q8—84no—
261C22×Q8C2411yescharacteristic
262C22×Q8C2412yes—
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C22×C8—24no—
273C22×C8—44no—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—24no—
278C4⋊Q8—28no—
279C4⋊Q8—28no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—24no—
283C4⋊Q8—42no—
284C4⋊Q8C2×D414yes—
285C4⋊Q8C2×D414yes—
286C4⋊Q8C2×D418yes—
287#126—44no—
288#125—22no—
289#125—22no—
290#124—24no—
291#124—22no—
292#124—22no—
293#124—42no—
294#123—44no—
295#123—44no—
296#123—44no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299#121—22no—
300#121—22no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—22no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—22no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—84no—
310C4×Q8—84no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—44no—
314#117—24no—
315#117—42no—
316#117—44no—
317#117—44no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
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—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D418yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—44no—
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D414yes—
379C2×C42#4114yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—24no—
385#107—24no—
386#106—28no—
387#106—28no—
388#106—24no—
389#106—24no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394C4⋊C8—44no—
395C4⋊C8—44no—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—24no—
400#103—24no—
401#103—44no—
402#103—44no—
403#102—24no—
404#102—24no—
405#102—44no—
406C22⋊C8—24no—
407#97—24no—
408#95—24no—
409#95—24no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422#95C2×D412yes—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426C24—21no—
427C24C22×D411yescharacteristic
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—28no—
431C2×Q8—22no—
432C2×Q8—28no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—84no—
448C2×Q8—816no—
449C2×Q82−1+414yes—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—84no—
479C22×C4#12114yes—
480C22×C4#12114yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C4C2511yescharacteristic
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—48no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×D414yes—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4#13714yes—
548C42—24no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—84no—
561C42—84no—
562C42#12114yes—
563C42#12114yes—
564C42C22×D412yes—
565C42C22×D412yes—
566C42C2×#4112yes—
567C42C2×#4112yes—
568C23—22no—
569C23—22no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23—41no—
574C23—42no—
575C23C2×#12112yes—
576C23C2×#13712yes—
577C23#57512yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—24no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—48no—
585Q8—48no—
586Q8—41no—
587Q8—41no—
588Q8—42no—
589Q8—88no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—84no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—88no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4C2×#12112yes—
641C2×C4C2×#12112yes—
642C2×C4C2×#12112yes—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4C2×#13714yes—
646C2×C4#57714yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×2−1+412yes—
650C2×C4#58512yes—
651C8—42no—
652C8—42no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—24no—
657C22—24no—
658C22—81no—
659C22—81no—
660C22—82no—
661C22—82no—
662C22—81no—
663C22#303714yes—
664C22C22×#12111yescharacteristic
665C22#348912yes—
666C22C22×#13711yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57712yes—
669C4—22no—
670C4—22no—
671C4—24no—
672C4—22no—
673C4—22no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—81no—
681C4—81no—
682C4—84no—
683C4—82no—
684C4—81no—
685C4#304214yes—
686C4#348611yescharacteristic
687C4#348711yescharacteristic
688C4#349512yes—
689C2—24no—
690C2—161no—
691C2#3566111yescharacteristic
692C2C2×#303711yescharacteristic
693C2C2×#304211yescharacteristic
694C2#6212912yes—
695C2#6213112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#475C2semidirect
4. #34216C2semidirect
5. #28283C2semidirect
6. #28281C2semidirect
7. #28205C2semidirect
8. #28203C2semidirect
9. #25464C2semidirect
10. #25287C2semidirect
11. #24579C2semidirect
14. #24482C2semidirect
15. #21976C2semidirect
17. #20740C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
C22×#4752
#349722
#342162
#282832
#282812
#282052
#282032
#254642
#252872
#245792
#245782
#244832
#244822
#219762
#218872
#207402