Finite Group Databasebeta

← #7505381

Subgroups — #7505381

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35674C211yesmaximal, characteristic, complemented
2#35620C211yesmaximal, characteristic
3#31664C214yesmaximal, complemented
4#31663C214yesmaximal, complemented
5#31660C214yesmaximal, complemented
6#31659C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23545C211yesmaximal, characteristic
10#23457C211yesmaximal, characteristic, complemented
11#23457C211yesmaximal, characteristic, complemented
12#23432C211yesmaximal, characteristic
13C2×#1960C211yesmaximal, characteristic, complemented
14#21086C211yesmaximal, characteristic
15#21021C211yesmaximal, characteristic, complemented
16#20931C211yesmaximal, characteristic
17#20931C211yesmaximal, characteristic
18#20930C211yesmaximal, characteristic
19#20651C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C22×#137C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#519C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2568C2212yes—
37#2566C2212yes—
38#2565C2212yes—
39#2563C2212yes—
40#2484—24no—
41#2450—24no—
42#2446—22no—
43#2446—22no—
44#2446C2214yes—
45#2446C2214yes—
46#2442—22no—
47#2439—22no—
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2310C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2101C2214yes—
58#2101C2214yes—
59#2074C2214yes—
60#2072C2214yes—
61#2056C2214yes—
62#2055C2214yes—
63#2026C2214yes—
64#2026C2214yes—
65#2024C2214yes—
66#2024C2214yes—
67#2016C2212yes—
68#2016C2212yes—
69#2013C2212yes—
70#2013C2212yes—
71#1960C2214yes—
72#1960C2214yes—
73#1936C2212yes—
74#1936C2212yes—
75#1935C2212yes—
76#1935C2212yes—
77#1917C2214yes—
78#1917C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1805C2214yes—
86#1803C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—24no—
103C2×#137—42no—
104C2×#137—42no—
105C2×#137—42no—
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#544—42no—
115#544—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—24no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8—42no—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2312yes—
123C2×#125—21no—
124C2×#125C2312yes—
125C2×C22⋊Q8—22no—
126#520—42no—
127#520—42no—
128#519—24no—
129#519—42no—
130C2×C4×Q8—22no—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—22no—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117—22no—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4—22no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#424—42no—
148C2×C4⋊C8C2314yes—
149#419—42no—
150#419—42no—
151#416—24no—
152#416—42no—
153C2×#103—42no—
154C2×#103—42no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#97—42no—
170C2×#97C2312yes—
171C2×#97C2312yes—
172#400—24no—
173#400—24no—
174#400—24no—
175#398—24no—
176#398—24no—
177#398—24no—
178#396—24no—
179#393—24no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—22no—
185#391—24no—
186#391C2311yescharacteristic
187#391C2312yes—
188#391C2314yes—
189#391C2314yes—
190#391C2314yes—
191#391C2311yescharacteristic
192#390C2311yescharacteristic
193#390C2312yes—
194#390C2311yescharacteristic
195#389—22no—
196#389D412yes—
197#389D412yes—
198#389C2312yes—
199#389C2312yes—
200#387—24no—
201#387—24no—
202#387—24no—
203#387—24no—
204#387—24no—
205#386—22no—
206#386—22no—
207#385—22no—
208#385—22no—
209#385—24no—
210#385D414yes—
211#384C2312yes—
212#382—22no—
213#382C2312yes—
214#382C2312yes—
215#382C2312yes—
216#381—22no—
217#381—22no—
218#381—22no—
219#380—24no—
220#380—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—22no—
225C2×#95—22no—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41#4114yes—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8#4114yes—
260C22×Q8C2412yes—
261C22×Q8C2412yes—
262C22×Q8C2411yescharacteristic
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4C2×D412yes—
272C23×C4C2411yescharacteristic
273#137—416no—
274#137—416no—
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—42no—
279C2×Q16—42no—
280C2×Q16—42no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—42no—
285C2×SD16—42no—
286C2×SD16—42no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—24no—
304C4⋊Q8—24no—
305C4⋊Q8—42no—
306C4⋊Q8—42no—
307C4⋊Q8—42no—
308C4⋊Q8—42no—
309C4⋊Q8—48no—
310C4⋊Q8—44no—
311#125—42no—
312#125—42no—
313#125—44no—
314C22⋊Q8—48no—
315C4×Q8—28no—
316C4×Q8—44no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—48no—
320C4×D4—42no—
321C4×D4—42no—
322C4×D4—48no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—48no—
334#117—44no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—48no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—24no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42C2×D412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C4⋊C8—44no—
386#103—24no—
387#103—44no—
388#103—44no—
389#103—44no—
390#103—44no—
391#103—42no—
392#103—42no—
393#103—42no—
394#103—42no—
395#103—42no—
396#102—24no—
397#102—42no—
398#97—42no—
399#97—42no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95C2×D414yes—
415#95C2×D414yes—
416C24—21no—
417C24C22×D411yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—84no—
454C2×Q8C2×#4112yes—
455C2×Q8C2×#4114yes—
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4C2×#4112yes—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—84no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+412yes—
504C22×C4C2511yescharacteristic
505Q16—48no—
506Q16—48no—
507SD16—48no—
508SD16—48no—
509M16—44no—
510M16—44no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—44no—
524C2×C8—44no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—84no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—44no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—84no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—24no—
564C42—21no—
565C42—21no—
566C42—22no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—48no—
573C42—44no—
574C42—42no—
575C42—42no—
576C42—42no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23#53512yes—
585C23#54612yes—
586C23#54812yes—
587C23C23×D411yescharacteristic
588Q8—22no—
589Q8—24no—
590Q8—24no—
591Q8—22no—
592Q8—42no—
593Q8—41no—
594Q8—48no—
595Q8—42no—
596Q8—41no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—84no—
602Q8—164no—
603D4—22no—
604D4—22no—
605D4—48no—
606D4—41no—
607D4—41no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—48no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—164no—
649C2×C4—162no—
650C2×C4—162no—
651C2×C4#53512yes—
652C2×C4#53814yes—
653C2×C4#53914yes—
654C2×C4#54612yes—
655C2×C4#54812yes—
656C2×C4#55114yes—
657C2×C4#56214yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×2−1+411yescharacteristic
664C8—42no—
665C8—42no—
666C8—42no—
667C8—42no—
668C22—22no—
669C22—22no—
670C22—24no—
671C22—24no—
672C22—42no—
673C22—44no—
674C22—81no—
675C22—81no—
676C22—82no—
677C22#269614yes—
678C22C2×#53511yescharacteristic
679C22C2×#54611yescharacteristic
680C22C2×#54811yescharacteristic
681C22#358612yes—
682C22#358612yes—
683C4—42no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—42no—
689C4—42no—
690C4—82no—
691C4—81no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—82no—
696C4—82no—
697C4#269614yes—
698C4C2×#53811yescharacteristic
699C4C2×#53911yescharacteristic
700C4C2×#55111yescharacteristic
701C4C2×#56211yescharacteristic
702C2—24no—
703C2—82no—
704C2C2×#269611yescharacteristic
705C2#6082611yescharacteristic
706C2#6082611yescharacteristic
707C2#6117612yes—
708C2#6133812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35674C2semidirect
3. #31664C2semidirect
4. #31663C2semidirect
5. #31660C2semidirect
6. #31659C2semidirect
7. #23746C2semidirect
8. #23633C2semidirect
10. #23457C2semidirect
11. #23457C2semidirect
13. C2×#1960C2semidirect
15. #21021C2semidirect
19. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#356742
#356202
#316642
#316632
#316602
#316592
#237462
#236332
#235452
#234572
#234322
C2×#19602
#210862
#210212
#209312
#209302
#206512