Finite Group Databasebeta

← #7501924

Subgroups — #7501924

Order 512.

1907 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#3098C211yesmaximal, characteristic
3#35639C211yesmaximal, characteristic, complemented
4#35592C211yesmaximal, characteristic
5#30795C214yesmaximal, complemented
6#30794C214yesmaximal, complemented
7#30792C214yesmaximal, complemented
8#30789C214yesmaximal, complemented
9#23738C211yesmaximal, characteristic, complemented
10#23640C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic, complemented
12#23552C211yesmaximal, characteristic, complemented
13#23489C211yesmaximal, characteristic
14#23453C211yesmaximal, characteristic, complemented
15#21174C211yesmaximal, characteristic, complemented
16#21086C211yesmaximal, characteristic, complemented
17#21047C211yesmaximal, characteristic, complemented
18#21037C211yesmaximal, characteristic, complemented
19#20847C211yesmaximal, characteristic, complemented
20C2×#537C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#501—28no—
25C2×#501C2214yes—
26C2×#501C2214yes—
27#3098—28no—
28C2×#484C2212yes—
29C2×#485C2212yes—
30C2×#482C2212yes—
31C2×#480C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#408C2211yescharacteristic
44C2×#408C2211yescharacteristic
45C22×#97C2211yescharacteristic
46#2591—22no—
47#2579—22no—
48#2492—22no—
49#2492C2214yes—
50#2491—22no—
51#2491C2214yes—
52#2487—22no—
53#2454—22no—
54#2453—22no—
55#2442C2214yes—
56#2441C2214yes—
57#2436—22no—
58#2317C2211yescharacteristic
59#2316C2211yescharacteristic
60#2311C2211yescharacteristic
61#2311C2211yescharacteristic
62#2311C2211yescharacteristic
63#2297C2211yescharacteristic
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2097C2212yes—
68#2096C2212yes—
69#2094C2212yes—
70#2092C2212yes—
71#2077C2214yes—
72#2077C2214yes—
73#2068C2214yes—
74#2067C2214yes—
75#2058C2214yes—
76#2057C2214yes—
77#2035C2214yes—
78#2034C2214yes—
79#2026C2214yes—
80#2025C2214yes—
81#1949C2214yes—
82#1948C2214yes—
83#1936C2212yes—
84#1936C2212yes—
85#1935C2212yes—
86#1935C2212yes—
87#1924C2214yes—
88#1924C2214yes—
89#1921C2214yes—
90#1918C2214yes—
91#1854C2214yes—
92#1852C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C22×#41C2311yescharacteristic
103C23×Q8—21no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#537—24no—
109#537—42no—
110C2×C4⋊Q8—24no—
111C2×C4⋊Q8—22no—
112C2×C4⋊Q8—42no—
113C2×C4⋊Q8C2314yes—
114C2×#125—24no—
115C2×#125—22no—
116C2×#125—42no—
117C2×#125C2314yes—
118C2×#124—24no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2312yes—
121C2×#121—22no—
122C2×#121—21no—
123C2×#121C2312yes—
124#520—24no—
125#520—42no—
126C2×C4×Q8—21no—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#501—216no—
144#501—216no—
145#485—28no—
146#484—28no—
147#482—28no—
148#480—28no—
149#428—24no—
150#428—42no—
151C2×#107—24no—
152C2×#107D418yes—
153C2×#107C2312yes—
154C2×#107C2314yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106—24no—
159C2×#106—24no—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2314yes—
164C2×#106C2312yes—
165#424—24no—
166C2×C4⋊C8C2314yes—
167#417—24no—
168#417—42no—
169#416—24no—
170#416—24no—
171C2×#103C2311yescharacteristic
172C2×#103C2312yes—
173C2×#103C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#102C2311yescharacteristic
177C2×#102C2312yes—
178C2×#102C2311yescharacteristic
179C2×#102C2312yes—
180C2×#102C2312yes—
181#408—24no—
182#408—24no—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2314yes—
186C2×#97—24no—
187C2×#97C2312yes—
188C2×#97C2312yes—
189#400—24no—
190#400—24no—
191#399—24no—
192#398—24no—
193#396—24no—
194#394—24no—
195#393—24no—
196#393—24no—
197#391—21no—
198#391—21no—
199#391—24no—
200#391D412yes—
201#391C2312yes—
202#391C2312yes—
203#390—21no—
204#390—21no—
205#390D412yes—
206#390C2312yes—
207#390C2312yes—
208#389—22no—
209#389—22no—
210#389—24no—
211#389C2312yes—
212#389C2312yes—
213#389C2314yes—
214#388—24no—
215#388—24no—
216#387—24no—
217#386—22no—
218#386—22no—
219#386—21no—
220#386—21no—
221#386—22no—
222#386D412yes—
223#386D412yes—
224#385—21no—
225#385—21no—
226#385—22no—
227#385—24no—
228#384C2312yes—
229#382C2312yes—
230#380—22no—
231#380—22no—
232#380—24no—
233#380C2314yes—
234#380C2314yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4C2314yes—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2314yes—
246#340C2312yes—
247#340C2314yes—
248#340C2312yes—
249#336C2314yes—
250#336C2312yes—
251#336C2312yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C22×Q8—24no—
265C22×Q8—44no—
266C22×Q8—84no—
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2411yescharacteristic
269C22×D4—21no—
270C22×D4—22no—
271C22×D4C2×D412yes—
272C22×D4C2×D411yescharacteristic
273C22×D4C2411yescharacteristic
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—22no—
279C23×C4—21no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2412yes—
283C23×C4C2411yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8C2×D414yes—
295C22×C8#4114yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2412yes—
301C4⋊Q8—28no—
302C4⋊Q8—44no—
303C4⋊Q8—48no—
304#125—28no—
305#125—44no—
306#125—48no—
307#124—88no—
308C22⋊Q8—24no—
309C22⋊Q8—42no—
310C22⋊Q8—44no—
311#121—24no—
312#121—42no—
313#121—44no—
314#121—48no—
315C4×Q8—44no—
316C4×D4—44no—
317C4×D4—48no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—24no—
327#117—42no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D414yes—
376C2×C22⋊C4C2×D414yes—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—24no—
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387#107—28no—
388#107—24no—
389#107—42no—
390#107C2×D418yes—
391#107C2×D418yes—
392#106—24no—
393#106—24no—
394#106—28no—
395#106—24no—
396#106—42no—
397C4⋊C8—28no—
398#103—22no—
399#103—22no—
400#103—24no—
401#103—24no—
402#103—42no—
403#102—22no—
404#102—22no—
405#102—24no—
406#102—24no—
407#102—42no—
408C22⋊C8—24no—
409C22⋊C8—24no—
410C22⋊C8—28no—
411#97—24no—
412#97—24no—
413#95—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24—21no—
426C24C22×D412yes—
427C24C22×D411yescharacteristic
428#41—44no—
429#41—44no—
430#41—44no—
431#41—44no—
432#41—44no—
433#41—44no—
434#41—44no—
435#41—44no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—48no—
446C2×Q8—48no—
447C2×Q8—84no—
448C2×Q8—816no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—84no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—84no—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C22×C42−1+412yes—
508C22×C42−1+412yes—
509C22×C4C2511yescharacteristic
510M16—44no—
511M16—44no—
512C2×C8—28no—
513C2×C8—28no—
514C2×C8—22no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8C22×D412yes—
527C2×C8C22×D412yes—
528C2×C8C2×#4114yes—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—44no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—84no—
554C4⋊C4—82no—
555C4⋊C4—84no—
556C4⋊C4—82no—
557C4⋊C4C22×D414yes—
558C4⋊C4C22×D414yes—
559C22⋊C4—24no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C22⋊C4—84no—
573C42—24no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—44no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42—84no—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—24no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—42no—
591C23—42no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23#53612yes—
596C23D4212yes—
597C23#54712yes—
598C23C23×D411yescharacteristic
599Q8—44no—
600Q8—41no—
601Q8—41no—
602Q8—41no—
603Q8—41no—
604Q8—816no—
605D4—48no—
606D4—44no—
607D4—41no—
608D4—41no—
609D4—41no—
610D4—41no—
611D4—84no—
612D4—164no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—164no—
658C2×C4#53612yes—
659C2×C4#53614yes—
660C2×C4#54214yes—
661C2×C4D4212yes—
662C2×C4#54712yes—
663C2×C4#54814yes—
664C2×C4#56314yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—22no—
672C8—22no—
673C8—24no—
674C8—42no—
675C8—42no—
676C22—24no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—82no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—82no—
690C22#267414yes—
691C22C2×#53611yescharacteristic
692C22C2×D4211yescharacteristic
693C22C2×#54711yescharacteristic
694C22#358512yes—
695C22#358612yes—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—82no—
703C4—81no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—82no—
708C4—82no—
709C4#267414yes—
710C4C2×#53611yescharacteristic
711C4C2×#54211yescharacteristic
712C4C2×#54811yescharacteristic
713C4C2×#56311yescharacteristic
714C2—24no—
715C2—82no—
716C2—161no—
717C2C2×#267411yescharacteristic
718C2#6088612yes—
719C2#6105811yescharacteristic
720C2#6106111yescharacteristic
721C2#6129412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
3. #35639C2semidirect
5. #30795C2semidirect
6. #30794C2semidirect
7. #30792C2semidirect
8. #30789C2semidirect
9. #23738C2semidirect
10. #23640C2semidirect
11. #23611C2semidirect
12. #23552C2semidirect
14. #23453C2semidirect
15. #21174C2semidirect
16. #21086C2semidirect
17. #21047C2semidirect
18. #21037C2semidirect
19. #20847C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#30982
#356392
#355922
#307952
#307942
#307922
#307892
#237382
#236402
#236112
#235522
#234892
#234532
#211742
#210862
#210472
#210372
#208472