Finite Group Databasebeta

← #6529819

Subgroups — #6529819

Order 512.

1881 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic, complemented
3#35247C211yesmaximal, characteristic
4C2×#2582C211yesmaximal, characteristic, complemented
5#28272C212yesmaximal, complemented
6#28265C212yesmaximal, complemented
7#28196C212yesmaximal, complemented
8#28186C212yesmaximal, complemented
9#25484C214yesmaximal, complemented
10#25307C214yesmaximal, complemented
11#24609C212yesmaximal, complemented
12#24600C212yesmaximal
13#24509C212yesmaximal
14#24502C212yesmaximal, complemented
15#21982C211yesmaximal, characteristic, complemented
16#21867C211yesmaximal, characteristic
17#20740C211yesmaximal, characteristic, complemented
18C2×#529C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3108—28no—
22C2×#483C2211yescharacteristic
23C2×#480C2211yescharacteristic
24C2×#479C2212yes—
25C2×#464C2212yes—
26C2×#451C2211yescharacteristic
27C2×#448C2211yescharacteristic
28#2984—28no—
29C2×#446C2212yes—
30C2×#446C2214yes—
31C2×C8×Q8C2212yes—
32C2×C4×Q16C2212yes—
33C2×C4×SD16C2212yes—
34C2×#432C2211yescharacteristic
35C2×#432C2212yes—
36C2×#431C2212yes—
37C2×#431C2211yescharacteristic
38#2855C2212yes—
39#2854C2212yes—
40#2637—22no—
41#2592C2212yes—
42#2584C2212yes—
43#2582—21no—
44#2582—21no—
45#2582—22no—
46#2582C2212yes—
47#2582C2211yescharacteristic
48#2582C2212yes—
49#2582C2211yescharacteristic
50#2582C2214yes—
51#2538C2211yescharacteristic
52#2538C2211yescharacteristic
53#2486—21no—
54#2413—22no—
55#2409—21no—
56C2×#398—22no—
57#2323—21no—
58#2323—21no—
59C2×#391—21no—
60C2×#391C2212yes—
61C2×#389—21no—
62C2×#389C2212yes—
63#2285C2211yescharacteristic
64C2×#384C2211yescharacteristic
65#2088C2214yes—
66#2086C2214yes—
67#2068C2214yes—
68#2068C2214yes—
69#2068C2214yes—
70#2067C2214yes—
71#2056C2214yes—
72#2055C2214yes—
73#1992C2212yes—
74#1973C2214yes—
75#1969C2214yes—
76#1959C2212yes—
77#1947C2214yes—
78#1943C2214yes—
79#1942C2214yes—
80#1935C2214yes—
81#1935C2214yes—
82#1934C2214yes—
83#1933C2214yes—
84#1874C2212yes—
85#1872C2212yes—
86#1852C2212yes—
87#1852C2212yes—
88#1845C2212yes—
89#1834C2212yes—
90#1823C2212yes—
91#1803C2212yes—
92#1803C2212yes—
93#1803C2212yes—
94#1802C2212yes—
95#1787C2212yes—
96#1763C2214yes—
97C23×Q8D411yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×Q16C2312yes—
100C22×SD16C2312yes—
101#529—24no—
102#529—24no—
103C2×#126—21no—
104C2×#125C2311yescharacteristic
105C2×#124C2311yescharacteristic
106C2×#123—21no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×#121C2311yescharacteristic
112C2×C4×Q8—24no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—24no—
116C2×C4×Q8—24no—
117C2×C4×Q8—24no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D412yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×#117—21no—
130C2×#117—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4D412yes—
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C42—21no—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#483—24no—
143#480—24no—
144#479—28no—
145#464—28no—
146#451—24no—
147#448—24no—
148#446—216no—
149#446—28no—
150C8×Q8—28no—
151C4×Q16—28no—
152C4×SD16—28no—
153#432—24no—
154#432—28no—
155#431—28no—
156#431D418yes—
157C2×#107C2312yes—
158C2×#107C2314yes—
159C2×#107C2314yes—
160C2×#106C2312yes—
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2314yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#102C2312yes—
172C2×#102C2314yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2311yescharacteristic
175C2×C22⋊C8C2311yescharacteristic
176C2×#97C2314yes—
177C2×#97C2312yes—
178C2×C4×C8C2314yes—
179C2×C4×C8C2312yes—
180#398—24no—
181#398—24no—
182#398—24no—
183#398—24no—
184#398—22no—
185#398—22no—
186#398—22no—
187#397—22no—
188#396—22no—
189#396—22no—
190#395—22no—
191#393—22no—
192#391—21no—
193#391—21no—
194#391—22no—
195#391—24no—
196#391—22no—
197#391—22no—
198#391—22no—
199#391C2314yes—
200#391C2312yes—
201#391C2314yes—
202#391C2312yes—
203#390C2312yes—
204#390C2312yes—
205#389—21no—
206#389—21no—
207#389—22no—
208#389—22no—
209#389—22no—
210#389—22no—
211#389—24no—
212#389—22no—
213#389—22no—
214#389C2314yes—
215#389C2312yes—
216#389C2312yes—
217#389C2314yes—
218#389C2312yes—
219#389C2312yes—
220#388—22no—
221#388—22no—
222#388—22no—
223#388—22no—
224#387—22no—
225#387—22no—
226#386—22no—
227#385—22no—
228#384—21no—
229#384—21no—
230#384C2312yes—
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2312yes—
234#382—21no—
235#381—22no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2312yes—
242C2×#95—22no—
243C2×#95—22no—
244C2×#95—22no—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8—21no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—42no—
258C22×Q8—42no—
259C22×Q8C2×D411yescharacteristic
260C22×Q8C2×D412yes—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2411yescharacteristic
263C22×Q8C2412yes—
264C22×D4—21no—
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C2×D412yes—
271C23×C4C2×D412yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2411yescharacteristic
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—42no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—42no—
281C22×C8C2414yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284#126—44no—
285#125—22no—
286#125—22no—
287#124—22no—
288#124—22no—
289#123—44no—
290#123—84no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295#121—22no—
296#121—22no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—28no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—22no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—44no—
317#117—44no—
318#117—82no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
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—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—82no—
349C2×C4⋊C4—84no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—44no—
380C2×C42C2×D412yes—
381C2×C42C2×D412yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D414yes—
385C2×C42#4114yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—24no—
391#107—44no—
392#107—44no—
393#106—24no—
394C4⋊C8—28no—
395C4⋊C8—28no—
396C4⋊C8—24no—
397C4⋊C8C2×D418yes—
398#103—28no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—44no—
403#103—44no—
404#102—24no—
405#102—44no—
406C22⋊C8—24no—
407C22⋊C8C2×D414yes—
408C22⋊C8C2×D414yes—
409#97—28no—
410#97—24no—
411C4×C8—28no—
412C4×C8C2×D418yes—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429C24—21no—
430C24C22×D411yescharacteristic
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—42no—
449C2×Q8—48no—
450C2×Q8—48no—
451C2×Q8—416no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8#13714yes—
455C2×Q82−1+414yes—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4#12114yes—
496C22×C4#12114yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C2×#4112yes—
508C22×C4C2511yescharacteristic
509Q16—48no—
510SD16—48no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8C22×D414yes—
519C2×C8C22×D414yes—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—84no—
554C4⋊C4—84no—
555C4⋊C4—81no—
556C4⋊C4—81no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C22⋊C4—44no—
560C22⋊C4—42no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—44no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C42—24no—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—44no—
576C42—44no—
577C42—42no—
578C42—82no—
579C42#12114yes—
580C42#12114yes—
581C42C22×D412yes—
582C42C22×D412yes—
583C42C2×#4112yes—
584C42C2×#4112yes—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23C2×#12112yes—
597C23#57612yes—
598C23#57812yes—
599C23C23×D411yescharacteristic
600Q8—28no—
601Q8—24no—
602Q8—24no—
603Q8—48no—
604Q8—48no—
605Q8—42no—
606Q8—42no—
607Q8—44no—
608Q8—42no—
609Q8—42no—
610D4—42no—
611D4—42no—
612D4—84no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—28no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—48no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—88no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#12112yes—
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4C2×#13714yes—
672C2×C4C2×#13712yes—
673C2×C4#57512yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×2−1+412yes—
677C2×C4#58512yes—
678C8—24no—
679C8—24no—
680C8—44no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22#306114yes—
695C22C22×#12111yescharacteristic
696C22#348611yescharacteristic
697C22#348711yescharacteristic
698C22C22×#13711yescharacteristic
699C22C2×#57511yescharacteristic
700C22C2×#57611yescharacteristic
701C22C2×#57811yescharacteristic
702C4—22no—
703C4—24no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—44no—
711C4—81no—
712C4—82no—
713C4—81no—
714C4—81no—
715C4—82no—
716C4—82no—
717C4—84no—
718C4#305614yes—
719C4#348912yes—
720C4#349512yes—
721C2—24no—
722C2—161no—
723C2#3566111yescharacteristic
724C2C2×#305611yescharacteristic
725C2C2×#306111yescharacteristic
726C2#6184211yescharacteristic
727C2#6184411yescharacteristic
728C2#6189811yescharacteristic
729C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2984C2semidirect
4. C2×#2582C2semidirect
5. #28272C2semidirect
6. #28265C2semidirect
7. #28196C2semidirect
8. #28186C2semidirect
9. #25484C2semidirect
10. #25307C2semidirect
11. #24609C2semidirect
14. #24502C2semidirect
15. #21982C2semidirect
17. #20740C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31082
C2×#29842
#352472
C2×#25822
#282722
#282652
#281962
#281862
#254842
#253072
#246092
#246002
#245092
#245022
#219822
#218672
#207402