Finite Group Databasebeta

← #6660645

Subgroups — #6660645

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35310C211yesmaximal, characteristic
2#34634C211yesmaximal, characteristic
3#32795C212yesmaximal, complemented
4#32784C212yesmaximal, complemented
5#30375C212yesmaximal, complemented
6#30371C212yesmaximal, complemented
7#29947C212yesmaximal, complemented
8#29897C212yesmaximal
9#29897C212yesmaximal
10#29699C212yesmaximal, complemented
11#29338C211yesmaximal, characteristic, complemented
12#29332C211yesmaximal, characteristic, complemented
13#29305C211yesmaximal, characteristic, complemented
14#29300C211yesmaximal, characteristic, complemented
15#27685C212yesmaximal, complemented
16#24718C212yesmaximal, complemented
17#24714C212yesmaximal, complemented
18#24303C212yesmaximal, complemented
19#20716C211yesmaximal, characteristic
20C2×#541C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#477C2212yescomplemented
24C2×#475C2212yescomplemented
25C2×#433C2211yescharacteristic
26C2×#433C2211yescharacteristic
27#2857C2211yescharacteristic
28#2851C2211yescharacteristic
29#2848C2212yes—
30#2629C2212yes—
31#2629C2211yescharacteristic
32#2629C2212yes—
33#2628C2212yes—
34#2628C2212yes—
35#2627—22no—
36#2626—22no—
37#2625C2212yes—
38#2621C2211yescharacteristic
39#2480—22no—
40#2478—22no—
41#2475—22no—
42#2473—22no—
43#2396—22no—
44#2396—22no—
45#2323C2212yes—
46#2314C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2296C2211yescharacteristic
49#2296C2211yescharacteristic
50#2285C2211yescharacteristic
51#2280C2211yescharacteristic
52#2280C2211yescharacteristic
53#2100C2212yescomplemented
54#2100C2212yescomplemented
55#2099C2212yescomplemented
56#2099C2212yescomplemented
57#2096C2212yescomplemented
58#2095C2212yescomplemented
59#2085C2212yescomplemented
60#2085C2212yescomplemented
61#2077C2212yes—
62#2077C2212yes—
63#2077C2212yes—
64#2077C2212yes—
65#2076C2212yes—
66#2076C2212yes—
67#2075C2212yes—
68#2075C2212yes—
69#2059C2212yescomplemented
70#2059C2212yescomplemented
71#2059C2212yescomplemented
72#2059C2212yescomplemented
73#2043C2212yescomplemented
74#2043C2212yescomplemented
75#2036C2212yes—
76#2036C2212yes—
77#2036C2212yes—
78#2036C2212yes—
79#2035C2212yes—
80#2035C2212yes—
81#2035C2212yes—
82#2035C2212yes—
83#2026C2212yescomplemented
84#2026C2212yescomplemented
85#1988C2212yescomplemented
86#1987C2212yescomplemented
87#1949C2212yescomplemented
88#1949C2212yescomplemented
89#1944C2212yescomplemented
90#1943C2212yescomplemented
91#1942C2212yescomplemented
92#1941C2212yescomplemented
93#1935C2212yescomplemented
94#1935C2212yescomplemented
95#1933C2212yescomplemented
96#1933C2212yescomplemented
97#1873C2212yes—
98#1873C2212yes—
99#1873C2212yes—
100#1873C2212yes—
101#1854C2212yes—
102#1854C2212yes—
103#1854C2212yes—
104#1854C2212yes—
105#1818C2212yes—
106#1805C2212yescomplemented
107#1805C2212yescomplemented
108#1798C2212yes—
109#1797C2212yes—
110#1779C2212yes—
111#1750C2211yescharacteristic
112#1750C2211yescharacteristic
113#1749C2211yescharacteristic
114C4×C22⋊C8C2211yescharacteristic
115C24×C4C2311yescharacteristic
116#541—42no—
117#541—42no—
118#533—42no—
119#533—42no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×#126—22no—
123C2×#125—22no—
124C2×#125C2311yescharacteristic
125C2×#125C2312yes—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8C2312yes—
128C2×C22⋊Q8C2312yes—
129C2×C22⋊Q8C2312yes—
130C2×C4×Q8—22no—
131C2×C4×Q8C2312yes—
132C2×#117—42no—
133C2×#117—42no—
134C2×#117D412yes—
135C2×#117D412yes—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C4⋊C4D412yes—
145C22×C22⋊C4D412yes—
146C22×C42C2311yescharacteristic
147C22×C42C2311yescharacteristic
148#477—44no—
149#475—44no—
150#433—42no—
151#433—42no—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2312yes—
172C2×C22⋊C8C2312yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2312yes—
176C2×#97C2311yescharacteristic
177C2×#97C2311yescharacteristic
178C2×C4×C8C2311yescharacteristic
179C2×C4×C8C2311yescharacteristic
180#400—22no—
181#400—22no—
182#400—22no—
183#399—22no—
184#399—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#397—22no—
193#397—22no—
194#397—22no—
195#396—22no—
196#396—22no—
197#396—22no—
198#396—22no—
199#396—22no—
200#395—22no—
201#395—22no—
202#393—22no—
203#393—22no—
204#391C2312yes—
205#391C2312yes—
206#389—22no—
207#389—22no—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#388—22no—
215#388—22no—
216#388—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#385—22no—
222#385—22no—
223#385—22no—
224#385—22no—
225#385—22no—
226#385—22no—
227#385—22no—
228#384C2311yescharacteristic
229#384C2312yes—
230#382C2311yescharacteristic
231#382C2312yes—
232#381—21no—
233#381—21no—
234#380—22no—
235#380—22no—
236#380C2312yes—
237#380C2312yes—
238#380C2312yes—
239#380C2312yes—
240#380C2312yes—
241#380C2312yes—
242#380C2312yes—
243#379—22no—
244#379—21no—
245#379—21no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4—22no—
248C4×C4⋊C4—22no—
249C4×C4⋊C4—22no—
250C4×C4⋊C4C2312yes—
251C4×C22⋊C4—21no—
252C4×C22⋊C4—21no—
253C4×C22⋊C4—21no—
254C4×C22⋊C4—21no—
255C4×C22⋊C4—22no—
256#340C2312yes—
257#340C2312yes—
258#340C2312yes—
259#340C2312yes—
260#340C2312yes—
261#340C2312yes—
262#340C2312yes—
263#340C2312yes—
264#340C2312yes—
265#340C2312yes—
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276C25C2×D411yescharacteristic
277C22×Q8—22no—
278C22×Q8C2412yes—
279C23×C4—24no—
280C23×C4—41no—
281C23×C4—41no—
282C23×C4—42no—
283C23×C4C2×D412yes—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D412yes—
288C23×C4C2411yescharacteristic
289C23×C4C2411yescharacteristic
290C23×C4C2411yescharacteristic
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—41no—
298C4⋊Q8—41no—
299#126—48no—
300#125—42no—
301#125—42no—
302#125—41no—
303#125—41no—
304#125—48no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307C22⋊Q8—42no—
308C22⋊Q8—42no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311C22⋊Q8—48no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—48no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—41no—
320#117—41no—
321#117—41no—
322#117—41no—
323#117—44no—
324#117—44no—
325#117—42no—
326#117—42no—
327#117—41no—
328#117—41no—
329#117—41no—
330#117—41no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—41no—
389C2×C42—41no—
390C2×C42C2×D412yes—
391C2×C42C2×D412yes—
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#107—42no—
397#107—42no—
398#107—42no—
399#107—42no—
400#106—42no—
401#106—42no—
402#106—42no—
403#106—42no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406#103—42no—
407#103—42no—
408#103—42no—
409#103—42no—
410#103—42no—
411#103—42no—
412#103—42no—
413#103—42no—
414C22⋊C8—24no—
415C22⋊C8—24no—
416C22⋊C8—42no—
417C22⋊C8—42no—
418C22⋊C8—42no—
419C22⋊C8—42no—
420#97—41no—
421#97—41no—
422C4×C8—41no—
423C4×C8—41no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445C24—22no—
446C24—41no—
447C24—41no—
448C24—42no—
449C24—42no—
450C24C22×D411yescharacteristic
451C24C22×D411yescharacteristic
452C24C22×D411yescharacteristic
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—44no—
460C2×Q8—82no—
461C2×Q8—82no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C42−1+414yes—
506C22×C4C2511yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—44no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C42—24no—
574C42—21no—
575C42—21no—
576C42—21no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—42no—
581C42—42no—
582C42—42no—
583C42—42no—
584C42—42no—
585C42—44no—
586C42—82no—
587C42H(5,𝔽2)12yes—
588C42H(5,𝔽2)12yes—
589C23—22no—
590C23—22no—
591C23—21no—
592C23—21no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—44no—
604C23—82no—
605C23—82no—
606C23—84no—
607C23—84no—
608C23#53612yes—
609C23C23×D411yescharacteristic
610Q8—42no—
611Q8—42no—
612Q8—42no—
613Q8—42no—
614Q8—88no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4#53414yes—
670C2×C4#53612yes—
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×2−1+412yes—
674C2×C4#58512yes—
675C8—42no—
676C8—42no—
677C8—42no—
678C8—42no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—41no—
687C22—41no—
688C22—42no—
689C22—42no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—88no—
695C22C2×#53411yescharacteristic
696C22C2×#53611yescharacteristic
697C22#344911yescharacteristic
698C22#358412yes—
699C22#358612yes—
700C4—22no—
701C4—24no—
702C4—41no—
703C4—41no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#354112yes—
716C4#354112yes—
717C2—42no—
718C2—81no—
719C2—81no—
720C2#3743811yescharacteristic
721C2#6076112yes—
722C2#6085911yescharacteristic
723C2#6086911yescharacteristic
724C2#6098712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32795C2semidirect
4. #32784C2semidirect
5. #30375C2semidirect
6. #30371C2semidirect
7. #29947C2semidirect
10. #29699C2semidirect
11. #29338C2semidirect
12. #29332C2semidirect
13. #29305C2semidirect
14. #29300C2semidirect
15. #27685C2semidirect
16. #24718C2semidirect
17. #24714C2semidirect
18. #24303C2semidirect
23. C2×#477C22semidirect
24. C2×#475C22semidirect
53. #2100C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
56. #2099C22semidirect
57. #2096C22semidirect
58. #2095C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
69. #2059C22semidirect
70. #2059C22semidirect
71. #2059C22semidirect
72. #2059C22semidirect
73. #2043C22semidirect
74. #2043C22semidirect
83. #2026C22semidirect
84. #2026C22semidirect
85. #1988C22semidirect
86. #1987C22semidirect
87. #1949C22semidirect
88. #1949C22semidirect
89. #1944C22semidirect
90. #1943C22semidirect
91. #1942C22semidirect
92. #1941C22semidirect
93. #1935C22semidirect
94. #1935C22semidirect
95. #1933C22semidirect
96. #1933C22semidirect
106. #1805C22semidirect
107. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#353102
#346342
#327952
#327842
#303752
#303712
#299472
#298972
#296992
#293382
#293322
#293052
#293002
#276852
#247182
#247142
#243032
#207162