Finite Group Databasebeta

← #8926556

Subgroups — #8926556

Order 512.

4142 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#58565C218yesmaximal, complemented
2#53921C218yesmaximal, complemented
3#53916C218yesmaximal, complemented
4#47660C214yesmaximal, complemented
5#47578C218yesmaximal, complemented
6#47196C212yesmaximal, complemented
7#47195C211yesmaximal, characteristic
8#46998C211yesmaximal, characteristic
9#46838C212yesmaximal, complemented
10#36970C212yesmaximal, complemented
11#36970C212yesmaximal, complemented
12#36966C212yesmaximal
13#36854C218yesmaximal, complemented
14#36829C214yesmaximal, complemented
15#35626C211yesmaximal, characteristic
16#35452C212yesmaximal
17C2×#543C2212yes—
18C22×#126C2212yes—
19C22×#124C2211yescharacteristic
20C2×#518C2211yescharacteristic
21#2855C2214yes—
22#2855C2214yescomplemented
23#2855C2214yescomplemented
24#2842C2211yescharacteristic
25#2840C2211yescharacteristic
26#2838C2218yescomplemented
27#2833C2214yescomplemented
28#2831C2218yescomplemented
29#2831C2218yescomplemented
30#2806C2218yescomplemented
31#2802C2218yescomplemented
32#2799C2214yescomplemented
33#2798C2218yescomplemented
34#2788C2218yescomplemented
35#2783C2218yescomplemented
36#2783C2218yescomplemented
37#2776C2218yescomplemented
38#2774C2218yescomplemented
39#2774C2218yescomplemented
40#2765C2218yescomplemented
41#2763C2218yescomplemented
42#2762C2218yescomplemented
43#2759C2218yescomplemented
44#2745C2218yes—
45#2745C2218yescomplemented
46#2745C2218yescomplemented
47#2744C2218yescomplemented
48#2739C2218yescomplemented
49#2738C2218yes—
50#2738C2218yescomplemented
51#2737C2218yes—
52#2736C2214yes—
53#2735C2214yes—
54#2734C2214yescomplemented
55#2713C2218yescomplemented
56#2707C2218yes—
57#2700C2218yes—
58#2700C2218yes—
59#2698C2218yes—
60#2698C2218yes—
61#2698C2218yescomplemented
62#2698C2218yescomplemented
63#2689C2218yes—
64#2689C2218yes—
65#2689C2218yescomplemented
66#2689C2218yescomplemented
67#2674C2218yes—
68#2674C2218yes—
69#2665C2218yescomplemented
70#2655C2218yescomplemented
71#2652C2218yescomplemented
72#2627C2214yescomplemented
73#2627C2214yescomplemented
74#2626C2214yes—
75#2612C2218yescomplemented
76#2611C2214yes—
77#2605C2214yescomplemented
78#2599C2214yes—
79#2588C2218yescomplemented
80#2586C2214yescomplemented
81#2584C2214yescomplemented
82#2583C2214yescomplemented
83#2583C2214yescomplemented
84#2579C2214yes—
85#2575C2218yes—
86#2573C2218yescomplemented
87#2561C2214yes—
88#2529C2218yescomplemented
89#2527C2218yes—
90#2527C2218yescomplemented
91#2527C2218yescomplemented
92#2527C2218yescomplemented
93#2526C2218yes—
94#2522C2218yescomplemented
95#2515C2218yescomplemented
96#2503C2214yescomplemented
97#2503C2214yescomplemented
98#2486C2214yes—
99#2483C2212yes—
100#2483C2212yes—
101#2483C2214yes—
102#2483C2211yescharacteristic
103#2483C2212yes—
104#2483C2212yes—
105#2483C2212yes—
106#2483C2211yescharacteristic
107#2483C2212yes—
108#2480C2218yes—
109#2478C2218yes—
110#2468C2218yes—
111#2466C2218yes—
112#2458C2218yes—
113#2451C2218yescomplemented
114#2447C2218yes—
115#2447C2218yes—
116#2434C2214yes—
117#2405C2212yes—
118#2405C2212yes—
119#2336C2214yescomplemented
120#2331C2218yescomplemented
121#2322C2214yes—
122#2320C2214yescomplemented
123#2312C2212yes—
124#2312C2212yes—
125#2312C2212yes—
126#2312C2212yes—
127#2310C2212yes—
128#2309C2211yescharacteristic
129C4×#123C2214yes—
130#2296C2212yes—
131C23×Q8C2311yescharacteristic
132C23×D4C2311yescharacteristic
133#543—48no—
134C2×C4⋊Q8C2318yes—
135C2×#126—48no—
136C2×#126C2318yes—
137C2×#126C2318yes—
138C2×#125C2314yes—
139C2×#124—44no—
140C2×#124C2318yes—
141C2×#123C2314yes—
142C2×#123C2314yes—
143C2×C22⋊Q8C2314yes—
144C2×C22⋊Q8C2314yes—
145C2×C22⋊Q8C2314yes—
146C2×C22⋊Q8C2314yes—
147C2×C22⋊Q8C2314yes—
148C2×C22⋊Q8C2314yes—
149#518—44no—
150C2×C4×Q8C2314yes—
151C2×C4×D4C2314yes—
152C2×C4×D4C2314yes—
153C2×#117C2312yes—
154C2×#117C2312yes—
155C22×C4⋊C4C2312yes—
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2312yes—
158C22×C22⋊C4C2312yes—
159C22×C22⋊C4C2312yes—
160C22×C22⋊C4C2312yes—
161C22×C22⋊C4C2312yes—
162C22×C22⋊C4C2312yes—
163C22×C22⋊C4C2312yes—
164C22×C42C2311yescharacteristic
165C22×C42C2312yes—
166#401C2318yes—
167#401C2318yes—
168#400C2318yes—
169#400C2318yes—
170#400C2318yes—
171#400C2318yes—
172#400C2318yes—
173#400C2318yes—
174#400C2318yes—
175#400C2318yes—
176#400C2314yes—
177#400C2318yes—
178#400C2318yes—
179#400C2318yes—
180#400C2318yes—
181#400C2318yes—
182#399C2314yes—
183#399C2318yes—
184#398C2318yes—
185#398C2318yes—
186#398C2318yes—
187#398C2318yes—
188#398C2318yes—
189#398C2314yes—
190#398C2318yes—
191#398C2318yes—
192#398C2314yes—
193#398C2318yes—
194#398C2318yes—
195#398C2318yes—
196#397C2318yes—
197#397C2318yes—
198#397C2318yes—
199#397C2318yes—
200#397C2318yes—
201#397C2318yes—
202#397C2318yes—
203#397C2314yes—
204#397C2314yes—
205#397C2318yes—
206#397C2318yes—
207#397C2318yes—
208#397C2318yes—
209#397C2318yes—
210#396C2318yes—
211#396C2318yes—
212#396C2318yes—
213#396C2318yes—
214#396C2318yes—
215#395C2318yes—
216#395C2318yes—
217#395C2318yes—
218#395C2318yes—
219#395C2318yes—
220#395C2318yes—
221#395C2314yes—
222#395C2318yes—
223#394C2318yes—
224#394C2318yes—
225#393C2318yes—
226#393C2318yes—
227#393C2318yes—
228#393C2314yes—
229#393C2318yes—
230#391C2314yes—
231#391C2318yes—
232#391C2314yes—
233#391C2318yes—
234#391C2318yes—
235#391C2318yes—
236#391C2318yes—
237#391C2318yes—
238#391C2318yes—
239#391C2318yes—
240#391C2318yes—
241#391C2318yes—
242#390C2318yes—
243#389C2318yes—
244#389C2318yes—
245#389C2318yes—
246#389C2318yes—
247#389C2318yes—
248#389C2318yes—
249#389C2318yes—
250#389C2318yes—
251#389C2318yes—
252#389C2318yes—
253#389C2318yes—
254#389C2318yes—
255#388C2318yes—
256#388C2318yes—
257#388C2318yes—
258#388C2318yes—
259#388C2318yes—
260#388C2318yes—
261#388C2318yes—
262#388C2318yes—
263#388C2318yes—
264#388C2318yes—
265#388C2318yes—
266#388C2318yes—
267#388C2318yes—
268#388C2318yes—
269#388C2318yes—
270#388C2318yes—
271#387C2318yes—
272#387C2318yes—
273#387C2318yes—
274#387C2318yes—
275#387C2318yes—
276#387C2318yes—
277#387C2318yes—
278#387C2318yes—
279#387C2318yes—
280#387C2318yes—
281#387C2318yes—
282#387C2318yes—
283#387C2318yes—
284#387C2318yes—
285#387C2318yes—
286#387C2318yes—
287#387C2318yes—
288#387C2318yes—
289#387C2318yes—
290#387C2318yes—
291#387C2318yes—
292#387C2318yes—
293#387C2318yes—
294#387C2318yes—
295#386C2314yes—
296#386C2314yes—
297#385C2314yes—
298#385C2314yes—
299#385C2314yes—
300#385C2318yes—
301#385C2318yes—
302#385C2314yes—
303#385C2314yes—
304#385C2314yes—
305#385C2314yes—
306#385C2314yes—
307#385C2314yes—
308#385C2314yes—
309#385C2314yes—
310#385C2314yes—
311#385C2314yes—
312#385C2314yes—
313#385C2318yes—
314#385C2314yes—
315#385C2314yes—
316#383C2318yes—
317#383C2314yes—
318#382C2318yes—
319#382C2318yes—
320#382C2318yes—
321#381C2314yes—
322#381C2314yes—
323#381C2314yes—
324#381C2314yes—
325#381C2314yes—
326#381C2314yes—
327#381C2314yes—
328#381C2314yes—
329#381C2314yes—
330#381C2314yes—
331#381C2314yes—
332#381C2314yes—
333#381C2314yes—
334#381C2318yes—
335#381C2314yes—
336#380C2314yes—
337#380C2314yes—
338#380C2314yes—
339#380C2314yes—
340#380C2318yes—
341#379C2314yes—
342#379C2314yes—
343#379C2311yescharacteristic
344#379C2311yescharacteristic
345#379C2312yes—
346#379C2312yes—
347#379C2312yes—
348C4×C4⋊C4C2318yes—
349C4×C4⋊C4C2314yes—
350C4×C4⋊C4C2314yes—
351C4×C4⋊C4C2318yes—
352C4×C22⋊C4C2314yes—
353C4×C22⋊C4C2314yes—
354C4×C22⋊C4C2314yes—
355C4×C22⋊C4C2314yes—
356C4×C22⋊C4C2318yes—
357#376C2314yes—
358#376C2318yes—
359C25C2411yescharacteristic
360C25C2411yescharacteristic
361C22×Q8—42no—
362C22×Q8—44no—
363C22×Q8C2414yes—
364C22×Q8C2414yes—
365C22×Q8C2414yes—
366C22×Q8C2414yes—
367C22×D4—24no—
368C22×D4—44no—
369C22×D4C2414yes—
370C23×C4C2411yescharacteristic
371C23×C4C2412yes—
372C23×C4C2412yes—
373C23×C4C2412yes—
374C23×C4C2412yes—
375C23×C4C2412yes—
376C23×C4C2412yes—
377C23×C4C2412yes—
378C23×C4C2412yes—
379C23×C4C2414yes—
380C4⋊Q8—416no—
381#126—416no—
382#126—416no—
383#125—48no—
384#124—416no—
385#123—48no—
386#123—48no—
387C22⋊Q8—48no—
388C22⋊Q8—48no—
389C22⋊Q8—48no—
390C22⋊Q8—48no—
391C22⋊Q8—48no—
392C22⋊Q8—48no—
393C4×Q8—48no—
394C4×D4—48no—
395C4×D4—48no—
396#117—44no—
397#117—44no—
398C2×C4⋊C4—44no—
399C2×C4⋊C4—44no—
400C2×C4⋊C4—44no—
401C2×C4⋊C4C2418yes—
402C2×C4⋊C4C2418yes—
403C2×C4⋊C4C2418yes—
404C2×C4⋊C4C2418yes—
405C2×C4⋊C4C2418yes—
406C2×C4⋊C4C2414yes—
407C2×C4⋊C4C2414yes—
408C2×C4⋊C4C2414yes—
409C2×C4⋊C4C2414yes—
410C2×C4⋊C4C2414yes—
411C2×C4⋊C4C2414yes—
412C2×C4⋊C4C2414yes—
413C2×C4⋊C4C2414yes—
414C2×C4⋊C4C2414yes—
415C2×C4⋊C4C2418yes—
416C2×C4⋊C4C2414yes—
417C2×C4⋊C4C2414yes—
418C2×C4⋊C4C2418yes—
419C2×C4⋊C4C2418yes—
420C2×C4⋊C4C2418yes—
421C2×C4⋊C4C2418yes—
422C2×C4⋊C4C2418yes—
423C2×C4⋊C4C2418yes—
424C2×C4⋊C4C2418yes—
425C2×C4⋊C4C2418yes—
426C2×C4⋊C4C2418yes—
427C2×C4⋊C4C2418yes—
428C2×C22⋊C4—28no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—28no—
431C2×C22⋊C4—28no—
432C2×C22⋊C4—44no—
433C2×C22⋊C4—44no—
434C2×C22⋊C4C2414yes—
435C2×C22⋊C4C2414yes—
436C2×C22⋊C4C2414yes—
437C2×C22⋊C4C2414yes—
438C2×C22⋊C4C2414yes—
439C2×C22⋊C4C2414yes—
440C2×C22⋊C4C2414yes—
441C2×C22⋊C4C2414yes—
442C2×C22⋊C4C2412yes—
443C2×C22⋊C4C2412yes—
444C2×C22⋊C4C2418yes—
445C2×C22⋊C4C2412yes—
446C2×C22⋊C4C2412yes—
447C2×C22⋊C4C2414yes—
448C2×C22⋊C4C2414yes—
449C2×C22⋊C4C2418yes—
450C2×C22⋊C4C2418yes—
451C2×C22⋊C4C2414yes—
452C2×C22⋊C4C2414yes—
453C2×C22⋊C4C2414yes—
454C2×C22⋊C4C2414yes—
455C2×C22⋊C4C2414yes—
456C2×C22⋊C4C2414yes—
457C2×C22⋊C4C2418yes—
458C2×C22⋊C4C2418yes—
459C2×C22⋊C4C2418yes—
460C2×C42—42no—
461C2×C42—44no—
462C2×C42C2414yes—
463C2×C42C2412yes—
464C2×C42C2412yes—
465C2×C42C2418yes—
466C2×C42C2414yes—
467C2×C42C2418yes—
468C2×C42C2418yes—
469C2×C42C2418yes—
470C2×C42C2418yes—
471#95C2414yes—
472#95C2418yes—
473#95C2418yes—
474#95C2418yes—
475#95C2414yes—
476#95C2418yes—
477#95C2418yes—
478#95C2418yes—
479#95C2418yes—
480#95C2418yes—
481#95C2418yes—
482#95C2418yes—
483#95C2418yes—
484#95C2418yes—
485#95C2418yes—
486#95C2418yes—
487#95C2418yes—
488#95C2418yes—
489#95C2418yes—
490#95C2418yes—
491#95C2418yes—
492#95C2418yes—
493#95C2418yes—
494#95C2418yes—
495#95C2418yes—
496#95C2418yes—
497#95C2414yes—
498#95C2418yes—
499#95C2418yes—
500#95C2418yes—
501#95C2418yes—
502#95C2418yes—
503#95C2418yes—
504#95C2418yes—
505#95C2414yes—
506C24—22no—
507C24—42no—
508C24—42no—
509C24—44no—
510C24—44no—
511C24H(5,𝔽2)14yes—
512C24C2512yes—
513C24C2511yescharacteristic
514C24C2512yes—
515C2×Q8—28no—
516C2×Q8—44no—
517C2×Q8—44no—
518C2×Q8—44no—
519C2×Q8—48no—
520C2×Q8—44no—
521C2×Q8—44no—
522C2×Q8—48no—
523C2×Q8—48no—
524C2×Q8—88no—
525C2×D4—28no—
526C2×D4—28no—
527C2×D4—44no—
528C2×D4—416no—
529C22×C4—22no—
530C22×C4—24no—
531C22×C4—24no—
532C22×C4—24no—
533C22×C4—24no—
534C22×C4—24no—
535C22×C4—24no—
536C22×C4—24no—
537C22×C4—24no—
538C22×C4—24no—
539C22×C4—44no—
540C22×C4—44no—
541C22×C4—44no—
542C22×C4—44no—
543C22×C4—44no—
544C22×C4—44no—
545C22×C4—44no—
546C22×C4—44no—
547C22×C4—44no—
548C22×C4—48no—
549C22×C4C2514yes—
550C22×C4C2518yes—
551C22×C4C2514yes—
552C22×C4C2514yes—
553C22×C4C2518yes—
554C22×C4C2514yes—
555C22×C4C2514yes—
556C22×C4C2514yes—
557C22×C4C2514yes—
558C22×C4C2514yes—
559C22×C4C2514yes—
560C22×C4C2514yes—
561C22×C4C2512yes—
562C4⋊C4—216no—
563C4⋊C4—216no—
564C4⋊C4—48no—
565C4⋊C4—48no—
566C4⋊C4—48no—
567C4⋊C4—48no—
568C4⋊C4—48no—
569C4⋊C4—48no—
570C4⋊C4—48no—
571C4⋊C4—48no—
572C4⋊C4—48no—
573C4⋊C4—48no—
574C4⋊C4—48no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4—44no—
579C4⋊C4—44no—
580C4⋊C4—44no—
581C4⋊C4—44no—
582C4⋊C4—44no—
583C4⋊C4—44no—
584C4⋊C4—44no—
585C4⋊C4—44no—
586C4⋊C4—48no—
587C4⋊C4—48no—
588C4⋊C4—48no—
589C22⋊C4—28no—
590C22⋊C4—28no—
591C22⋊C4—28no—
592C22⋊C4—28no—
593C22⋊C4—28no—
594C22⋊C4—28no—
595C22⋊C4—28no—
596C22⋊C4—28no—
597C22⋊C4—42no—
598C22⋊C4—42no—
599C22⋊C4—48no—
600C22⋊C4—44no—
601C22⋊C4—48no—
602C22⋊C4—44no—
603C22⋊C4—48no—
604C22⋊C4—44no—
605C22⋊C4—44no—
606C22⋊C4—44no—
607C22⋊C4—44no—
608C22⋊C4—48no—
609C22⋊C4—42no—
610C22⋊C4—42no—
611C22⋊C4—44no—
612C22⋊C4—48no—
613C22⋊C4—44no—
614C22⋊C4—48no—
615C42—28no—
616C42—44no—
617C42—48no—
618C42—48no—
619C42—48no—
620C42—42no—
621C42—42no—
622C42—48no—
623C42—48no—
624C23—22no—
625C23—24no—
626C23—24no—
627C23—24no—
628C23—24no—
629C23—22no—
630C23—24no—
631C23—28no—
632C23—44no—
633C23—48no—
634C23—44no—
635C23—84no—
636C23—164no—
637C23—164no—
638C23C2×H(5,𝔽2)12yes—
639C23C2×H(5,𝔽2)14yes—
640C23C2611yescharacteristic
641Q8—88no—
642Q8—88no—
643Q8—164no—
644Q8—164no—
645D4—416no—
646C2×C4—28no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—28no—
660C2×C4—28no—
661C2×C4—28no—
662C2×C4—28no—
663C2×C4—28no—
664C2×C4—28no—
665C2×C4—28no—
666C2×C4—28no—
667C2×C4—28no—
668C2×C4—28no—
669C2×C4—28no—
670C2×C4—28no—
671C2×C4—28no—
672C2×C4—28no—
673C2×C4—28no—
674C2×C4—28no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—164no—
684C2×C4#58514yes—
685C22—22no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—48no—
691C22—44no—
692C22—42no—
693C22—44no—
694C22—48no—
695C22—164no—
696C22—164no—
697C22C22×H(5,𝔽2)11yescharacteristic
698C22C2×#58512yes—
699C222−1+614yes—
700C4—24no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—48no—
711C4—44no—
712C4—48no—
713C2—44no—
714C2—42no—
715C2—82no—
716C2#6282011yescharacteristic
717C2#6287112yes—
718C2#6289114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #58565C2semidirect
2. #53921C2semidirect
3. #53916C2semidirect
4. #47660C2semidirect
5. #47578C2semidirect
6. #47196C2semidirect
9. #46838C2semidirect
10. #36970C2semidirect
11. #36970C2semidirect
13. #36854C2semidirect
14. #36829C2semidirect
22. #2855C22semidirect
23. #2855C22semidirect
26. #2838C22semidirect
27. #2833C22semidirect
28. #2831C22semidirect
29. #2831C22semidirect
30. #2806C22semidirect
31. #2802C22semidirect
32. #2799C22semidirect
33. #2798C22semidirect
34. #2788C22semidirect
35. #2783C22semidirect
36. #2783C22semidirect
37. #2776C22semidirect
38. #2774C22semidirect
39. #2774C22semidirect
40. #2765C22semidirect
41. #2763C22semidirect
42. #2762C22semidirect
43. #2759C22semidirect
45. #2745C22semidirect
46. #2745C22semidirect
47. #2744C22semidirect
48. #2739C22semidirect
50. #2738C22semidirect
54. #2734C22semidirect
55. #2713C22semidirect
61. #2698C22semidirect
62. #2698C22semidirect
65. #2689C22semidirect
66. #2689C22semidirect
69. #2665C22semidirect
70. #2655C22semidirect
71. #2652C22semidirect
72. #2627C22semidirect
73. #2627C22semidirect
75. #2612C22semidirect
77. #2605C22semidirect
79. #2588C22semidirect
80. #2586C22semidirect
81. #2584C22semidirect
82. #2583C22semidirect
83. #2583C22semidirect
86. #2573C22semidirect
88. #2529C22semidirect
90. #2527C22semidirect
91. #2527C22semidirect
92. #2527C22semidirect
94. #2522C22semidirect
95. #2515C22semidirect
96. #2503C22semidirect
97. #2503C22semidirect
113. #2451C22semidirect
119. #2336C22semidirect
120. #2331C22semidirect
122. #2320C22semidirect

Maximal subgroups

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

subgroupindex
#585652
#539212
#539162
#476602
#475782
#471962
#471952
#469982
#468382
#369702
#369662
#368542
#368292
#356262
#354522