Finite Group Databasebeta

← #7717448

Subgroups — #7717448

Order 512.

4292 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#51627C218yesmaximal, complemented
2#51616C218yesmaximal, complemented
3#44853C218yesmaximal, complemented
4#44851C218yesmaximal, complemented
5#39677C214yesmaximal, complemented
6#37958C214yesmaximal
7#37885C214yesmaximal
8#36977C212yesmaximal, complemented
9#36728C212yesmaximal
10#36199C214yesmaximal, complemented
11#35325C212yesmaximal
12#35120C211yesmaximal, characteristic, complemented
13#35120C212yesmaximal, complemented
14#35012C212yesmaximal
15#34642C211yesmaximal, characteristic, complemented
16#34399C212yesmaximal, complemented
17C2×#2346C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×#117C2211yescharacteristic
20#2855C2214yes—
21#2855C2214yes—
22#2855C2214yes—
23#2851C2214yes—
24#2848C2214yes—
25#2806C2218yescomplemented
26#2802C2218yescomplemented
27#2793C2218yes—
28#2783C2218yes—
29#2777C2218yes—
30#2775C2218yes—
31#2774C2218yescomplemented
32#2773C2218yes—
33#2766C2218yescomplemented
34#2762C2218yescomplemented
35#2759C2218yes—
36#2755C2218yescomplemented
37#2754C2218yes—
38#2747C2218yes—
39#2726C2218yescomplemented
40#2722C2218yes—
41#2721C2218yes—
42#2719C2218yescomplemented
43#2704C2218yes—
44#2690C2218yes—
45#2660C2214yescomplemented
46#2660C2214yescomplemented
47#2659C2212yes—
48#2659C2212yes—
49#2659C2214yes—
50#2659C2212yes—
51#2646C2212yes—
52#2646C2214yes—
53#2628C2214yes—
54#2626C2214yes—
55#2621C2214yes—
56#2612C2218yescomplemented
57#2612C2218yescomplemented
58#2612C2218yescomplemented
59#2611C2218yes—
60#2606C2214yescomplemented
61#2606C2218yescomplemented
62#2605C2214yescomplemented
63#2605C2218yescomplemented
64#2604C2218yes—
65#2599C2218yes—
66#2594C2218yes—
67#2590C2218yescomplemented
68#2584C2218yescomplemented
69#2583C2214yes—
70#2574C2218yes—
71#2570C2214yescomplemented
72#2565C2214yescomplemented
73#2561C2214yes—
74#2556C2218yes—
75#2532C2218yes—
76#2532C2218yes—
77#2529C2218yes—
78#2528C2218yescomplemented
79#2525C2218yes—
80#2524C2218yescomplemented
81#2524C2218yescomplemented
82#2522C2218yes—
83#2519C2218yes—
84#2514C2214yescomplemented
85#2511C2214yescomplemented
86#2508C2214yescomplemented
87#2506C2214yescomplemented
88#2493C2218yes—
89#2484C2214yes—
90#2482C2214yes—
91#2480C2218yes—
92#2473C2218yes—
93#2473C2218yes—
94#2469C2218yes—
95#2465C2218yes—
96#2462C2218yescomplemented
97#2453C2218yescomplemented
98#2399C2218yescomplemented
99#2382C2218yescomplemented
100#2371C2218yescomplemented
101#2370C2214yescomplemented
102#2360C2218yes—
103#2356C2214yescomplemented
104#2356C2214yescomplemented
105#2351C2214yescomplemented
106#2346C2218yes—
107#2346C2218yes—
108#2343C2218yes—
109#2331C2214yescomplemented
110#2329C2214yes—
111#2329C2214yes—
112#2329C2214yes—
113#2329C2214yes—
114#2327C2214yescomplemented
115#2323C2214yes—
116#2323C2214yes—
117#2320C2218yescomplemented
118#2312C2214yes—
119C4×C22⋊Q8C2214yes—
120C2×#389C2212yes—
121C2×#388C2212yes—
122C2×#388C2212yes—
123#2285C2214yes—
124#2283C2214yes—
125#2283C2212yes—
126C2×#382C2211yescharacteristic
127C2×#382C2211yescharacteristic
128C2×#380C2211yescharacteristic
129C2×C4×C4⋊C4C2212yes—
130C2×C4×C22⋊C4C2211yescharacteristic
131C2×#376C2211yescharacteristic
132C2×#376C2211yescharacteristic
133C24×C4C2311yescharacteristic
134#533—216no—
135C2×C4⋊Q8C2312yes—
136C2×C4⋊Q8C2312yes—
137C2×#126C2318yes—
138C2×#125C2314yes—
139C2×#125C2314yes—
140C2×#125C2318yes—
141C2×#125C2318yes—
142C2×C22⋊Q8C2314yes—
143C2×C22⋊Q8C2314yes—
144C2×C22⋊Q8C2314yes—
145C2×C22⋊Q8C2314yes—
146C2×C22⋊Q8C2314yes—
147C2×C4×Q8C2314yes—
148C2×C4×Q8C2314yes—
149C2×C4×Q8C2314yes—
150C2×C4×Q8C2318yes—
151C2×#117—28no—
152C2×#117C2314yes—
153C2×#117C2314yes—
154C2×#117C2314yes—
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2312yes—
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2312yes—
160C22×C22⋊C4C2311yescharacteristic
161C22×C22⋊C4C2312yes—
162C22×C22⋊C4C2311yescharacteristic
163C22×C22⋊C4C2312yes—
164C22×C42C2311yescharacteristic
165C22×C42C2312yes—
166C22×C42C2312yes—
167C22×C42C2312yes—
168C22×C42C2311yescharacteristic
169#401C2318yes—
170#400C2318yes—
171#400C2318yes—
172#400C2318yes—
173#400C2318yes—
174#400C2318yes—
175#400C2318yes—
176#400C2318yes—
177#400C2318yes—
178#400C2318yes—
179#399C2318yes—
180#399C2318yes—
181#398C2318yes—
182#398C2318yes—
183#398C2318yes—
184#398C2318yes—
185#398C2318yes—
186#398C2318yes—
187#398C2318yes—
188#398C2318yes—
189#398C2318yes—
190#398C2318yes—
191#398C2318yes—
192#398C2318yes—
193#398C2318yes—
194#398C2318yes—
195#398C2318yes—
196#398C2318yes—
197#397C2318yes—
198#397C2318yes—
199#397C2318yes—
200#397C2318yes—
201#397C2318yes—
202#396C2318yes—
203#396C2318yes—
204#396C2318yes—
205#395C2318yes—
206#395C2318yes—
207#395C2318yes—
208#395C2318yes—
209#395C2318yes—
210#395C2318yes—
211#393C2318yes—
212#393C2318yes—
213#391C2314yes—
214#391C2314yes—
215#391C2314yes—
216#391C2318yes—
217#391C2318yes—
218#391C2314yes—
219#391C2314yes—
220#391C2318yes—
221#391C2318yes—
222#391C2318yes—
223#391C2314yes—
224#391C2314yes—
225#389C2314yes—
226#389C2314yes—
227#389C2318yes—
228#389C2318yes—
229#389C2318yes—
230#389C2318yes—
231#389C2314yes—
232#389C2314yes—
233#389C2314yes—
234#389C2318yes—
235#389C2318yes—
236#389C2318yes—
237#389C2318yes—
238#389C2318yes—
239#389C2318yes—
240#389C2318yes—
241#389C2318yes—
242#389C2314yes—
243#389C2314yes—
244#389C2314yes—
245#389C2318yes—
246#389C2314yes—
247#388C2318yes—
248#388C2318yes—
249#388C2318yes—
250#388C2318yes—
251#388C2318yes—
252#388C2318yes—
253#388C2318yes—
254#388C2318yes—
255#388C2318yes—
256#388C2318yes—
257#388C2318yes—
258#388C2318yes—
259#388C2318yes—
260#388C2318yes—
261#388C2318yes—
262#387C2318yes—
263#387C2318yes—
264#387C2318yes—
265#387C2318yes—
266#387C2318yes—
267#387C2318yes—
268#387C2318yes—
269#387C2318yes—
270#387C2318yes—
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#385C2318yes—
295#385C2318yes—
296#385C2318yes—
297#385C2318yes—
298#385C2318yes—
299#385C2318yes—
300#385C2318yes—
301#384C2314yes—
302#384C2314yes—
303#383C2314yes—
304#383C2318yes—
305#383C2318yes—
306#383C2318yes—
307#382C2314yes—
308#382C2314yes—
309#382C2314yes—
310#382C2314yes—
311#382C2318yes—
312#382C2314yes—
313#382C2318yes—
314#382C2314yes—
315#382C2318yes—
316#381C2318yes—
317#381C2314yes—
318#380C2314yes—
319#380C2314yes—
320#380C2314yes—
321#380C2314yes—
322#380C2314yes—
323#380C2318yes—
324#380C2314yes—
325#380C2318yes—
326#379C2318yes—
327C4×C4⋊C4C2314yes—
328C4×C4⋊C4C2314yes—
329C4×C4⋊C4C2318yes—
330C4×C4⋊C4C2318yes—
331C4×C4⋊C4C2312yes—
332C4×C4⋊C4C2312yes—
333C4×C4⋊C4C2318yes—
334C4×C4⋊C4C2314yes—
335C4×C4⋊C4C2314yes—
336C4×C4⋊C4C2314yes—
337C4×C4⋊C4C2314yes—
338C4×C4⋊C4C2314yes—
339C4×C4⋊C4C2314yes—
340C4×C4⋊C4C2314yes—
341C4×C4⋊C4C2318yes—
342C4×C22⋊C4C2314yes—
343C4×C22⋊C4C2318yes—
344C4×C22⋊C4C2318yes—
345C4×C22⋊C4C2318yes—
346#376C2318yes—
347#376C2314yes—
348#376C2318yes—
349#376C2314yes—
350#376C2314yes—
351#376C2318yes—
352C2×#95C2311yescharacteristic
353C2×#95C2311yescharacteristic
354C2×#95C2312yes—
355C2×#95C2312yes—
356C2×#95C2312yes—
357C2×#95C2311yescharacteristic
358C2×#95C2311yescharacteristic
359C2×#95C2312yes—
360C25C2411yescharacteristic
361C22×Q8C2414yes—
362C22×Q8C2414yes—
363C22×Q8C2414yes—
364C23×C4—24no—
365C23×C4—44no—
366C23×C4C2411yescharacteristic
367C23×C4C2411yescharacteristic
368C23×C4C2414yes—
369C23×C4C2411yescharacteristic
370C23×C4C2411yescharacteristic
371C23×C4C2412yes—
372C23×C4C2412yes—
373C23×C4C2412yes—
374C23×C4C2411yescharacteristic
375C23×C4C2411yescharacteristic
376C23×C4C2412yes—
377C4⋊Q8—28no—
378C4⋊Q8—28no—
379#126—88no—
380#125—216no—
381#125—48no—
382#125—416no—
383#125—88no—
384C22⋊Q8—216no—
385C22⋊Q8—216no—
386C22⋊Q8—216no—
387C22⋊Q8—48no—
388C22⋊Q8—48no—
389C4×Q8—216no—
390C4×Q8—216no—
391C4×Q8—48no—
392C4×Q8—88no—
393#117—216no—
394#117—216no—
395#117—216no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—44no—
399C2×C4⋊C4—82no—
400C2×C4⋊C4—82no—
401C2×C4⋊C4C2412yes—
402C2×C4⋊C4C2412yes—
403C2×C4⋊C4C2414yes—
404C2×C4⋊C4C2414yes—
405C2×C4⋊C4C2414yes—
406C2×C4⋊C4C2414yes—
407C2×C4⋊C4C2418yes—
408C2×C4⋊C4C2418yes—
409C2×C4⋊C4C2418yes—
410C2×C4⋊C4C2414yes—
411C2×C4⋊C4C2414yes—
412C2×C4⋊C4C2418yes—
413C2×C4⋊C4C2418yes—
414C2×C4⋊C4C2414yes—
415C2×C4⋊C4C2418yes—
416C2×C4⋊C4C2418yes—
417C2×C4⋊C4C2414yes—
418C2×C4⋊C4C2418yes—
419C2×C4⋊C4C2418yes—
420C2×C4⋊C4C2414yes—
421C2×C4⋊C4C2414yes—
422C2×C4⋊C4C2414yes—
423C2×C4⋊C4C2418yes—
424C2×C4⋊C4C2418yes—
425C2×C4⋊C4C2418yes—
426C2×C4⋊C4C2414yes—
427C2×C4⋊C4C2414yes—
428C2×C4⋊C4C2418yes—
429C2×C4⋊C4C2418yes—
430C2×C4⋊C4C2418yes—
431C2×C4⋊C4C2418yes—
432C2×C4⋊C4C2414yes—
433C2×C4⋊C4C2414yes—
434C2×C4⋊C4C2414yes—
435C2×C4⋊C4C2418yes—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4—28no—
439C2×C22⋊C4—82no—
440C2×C22⋊C4C2412yes—
441C2×C22⋊C4C2414yes—
442C2×C22⋊C4C2412yes—
443C2×C22⋊C4C2418yes—
444C2×C22⋊C4C2418yes—
445C2×C22⋊C4C2418yes—
446C2×C22⋊C4C2418yes—
447C2×C22⋊C4C2418yes—
448C2×C22⋊C4C2418yes—
449C2×C22⋊C4C2418yes—
450C2×C22⋊C4C2418yes—
451C2×C42—24no—
452C2×C42—28no—
453C2×C42—28no—
454C2×C42—44no—
455C2×C42#4118yes—
456C2×C42C2414yes—
457C2×C42C2418yes—
458C2×C42C2414yes—
459C2×C42C2414yes—
460C2×C42C2414yes—
461C2×C42C2414yes—
462C2×C42C2414yes—
463C2×C42C2414yes—
464C2×C42C2418yes—
465C2×C42C2418yes—
466C2×C42C2418yes—
467C2×C42C2418yes—
468C2×C42C2414yes—
469C2×C42C2414yes—
470C2×C42C2412yes—
471C2×C42C2412yes—
472#95C2414yes—
473#95C2414yes—
474#95C2414yes—
475#95C2414yes—
476#95C2418yes—
477#95C2418yes—
478#95C2414yes—
479#95C2414yes—
480#95C2414yes—
481#95C2414yes—
482#95C2414yes—
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#95C2414yes—
497#95C2418yes—
498#95C2418yes—
499#95C2418yes—
500#95C2418yes—
501#95C2418yes—
502#95C2418yes—
503#95C2418yes—
504#95C2418yes—
505#95C2418yes—
506#95C2418yes—
507#95C2418yes—
508#95C2418yes—
509#95C2418yes—
510C24—22no—
511C24—24no—
512C24—41no—
513C24—42no—
514C24—41no—
515C24C2512yes—
516C24C2511yescharacteristic
517C2×Q8—28no—
518C2×Q8—28no—
519C2×Q8—28no—
520C2×Q8—44no—
521C2×Q8—44no—
522C2×Q8—44no—
523C2×Q8—44no—
524C2×Q8—44no—
525C2×Q8—44no—
526C22×C4—22no—
527C22×C4—24no—
528C22×C4—24no—
529C22×C4—22no—
530C22×C4—22no—
531C22×C4—24no—
532C22×C4—24no—
533C22×C4—24no—
534C22×C4—24no—
535C22×C4—28no—
536C22×C4—28no—
537C22×C4—42no—
538C22×C4—41no—
539C22×C4—41no—
540C22×C4—44no—
541C22×C4—42no—
542C22×C4—41no—
543C22×C4—42no—
544C22×C4—41no—
545C22×C4—42no—
546C22×C4—42no—
547C22×C4—42no—
548C22×C4—42no—
549C22×C4—42no—
550C22×C4—42no—
551C22×C4—416no—
552C22×C4C2×#4114yes—
553C22×C4C2×#4114yes—
554C22×C4C2×#4114yes—
555C22×C42−1+418yes—
556C22×C4C2512yes—
557C22×C4C2518yes—
558C22×C4C2514yes—
559C22×C4C2514yes—
560C22×C4C2514yes—
561C22×C4C2518yes—
562C22×C4C2518yes—
563C22×C4C2518yes—
564C22×C4C2512yes—
565C22×C4C2511yescharacteristic
566C22×C4C2511yescharacteristic
567C22×C4C2512yes—
568C22×C4C2512yes—
569C22×C4C2514yes—
570C22×C4C2512yes—
571C4⋊C4—24no—
572C4⋊C4—24no—
573C4⋊C4—28no—
574C4⋊C4—28no—
575C4⋊C4—28no—
576C4⋊C4—216no—
577C4⋊C4—216no—
578C4⋊C4—28no—
579C4⋊C4—28no—
580C4⋊C4—28no—
581C4⋊C4—216no—
582C4⋊C4—28no—
583C4⋊C4—216no—
584C4⋊C4—216no—
585C4⋊C4—44no—
586C4⋊C4—48no—
587C4⋊C4—48no—
588C4⋊C4—44no—
589C4⋊C4—44no—
590C4⋊C4—44no—
591C4⋊C4—48no—
592C4⋊C4—44no—
593C4⋊C4—48no—
594C4⋊C4—48no—
595C4⋊C4—44no—
596C4⋊C4—44no—
597C4⋊C4—44no—
598C4⋊C4—44no—
599C4⋊C4—48no—
600C4⋊C4—48no—
601C4⋊C4—48no—
602C4⋊C4—48no—
603C4⋊C4—48no—
604C4⋊C4—48no—
605C4⋊C4—48no—
606C22⋊C4—216no—
607C22⋊C4—216no—
608C22⋊C4—24no—
609C22⋊C4—24no—
610C22⋊C4—28no—
611C22⋊C4—216no—
612C22⋊C4—48no—
613C22⋊C4—48no—
614C22⋊C4—48no—
615C22⋊C4—48no—
616C22⋊C4—48no—
617C42—28no—
618C42—216no—
619C42—28no—
620C42—28no—
621C42—28no—
622C42—28no—
623C42—28no—
624C42—44no—
625C42—44no—
626C42—48no—
627C42—44no—
628C42—48no—
629C42—48no—
630C42—48no—
631C42C2×#4118yes—
632C42C2×#4118yes—
633C23—22no—
634C23—21no—
635C23—21no—
636C23—22no—
637C23—24no—
638C23—24no—
639C23—22no—
640C23—22no—
641C23—44no—
642C23—44no—
643C23—44no—
644C23—84no—
645C23—84no—
646C23C22×#4112yes—
647C23C2×2−1+414yes—
648C23C2611yescharacteristic
649Q8—416no—
650Q8—88no—
651Q8—88no—
652C2×C4—28no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—22no—
658C2×C4—24no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—24no—
662C2×C4—28no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—28no—
666C2×C4—28no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—24no—
670C2×C4—24no—
671C2×C4—24no—
672C2×C4—28no—
673C2×C4—28no—
674C2×C4—24no—
675C2×C4—28no—
676C2×C4—24no—
677C2×C4—28no—
678C2×C4—28no—
679C2×C4—28no—
680C2×C4—24no—
681C2×C4—28no—
682C2×C4—24no—
683C2×C4—28no—
684C2×C4—24no—
685C2×C4—24no—
686C2×C4—22no—
687C2×C4—28no—
688C2×C4—28no—
689C2×C4—22no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—48no—
693C2×C4—416no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—84no—
697C2×C4—84no—
698C2×C4—84no—
699C2×C4C22×#4112yes—
700C2×C4C22×#4114yes—
701C2×C4C22×#4112yes—
702C2×C4C22×#4114yes—
703C2×C4C2×2−1+418yes—
704C2×C4#58514yes—
705C22—22no—
706C22—22no—
707C22—24no—
708C22—24no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—44no—
713C22—44no—
714C22—88no—
715C22#350914yes—
716C22C23×#4111yescharacteristic
717C22C22×2−1+412yes—
718C22C2×#58512yes—
719C22H(7,𝔽2)11yescharacteristic
720C222−1+611yescharacteristic
721C4—22no—
722C4—22no—
723C4—24no—
724C4—24no—
725C4—24no—
726C4—28no—
727C4—42no—
728C4—48no—
729C4—42no—
730C4—48no—
731C4—48no—
732C4—48no—
733C4—44no—
734C4—44no—
735C4—44no—
736C2—24no—
737C2—82no—
738C2C2×#350911yescharacteristic
739C2#6281411yescharacteristic
740C2#6284511yescharacteristic
741C2#6290412yes—
742C2#6292912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #51627C2semidirect
2. #51616C2semidirect
3. #44853C2semidirect
4. #44851C2semidirect
5. #39677C2semidirect
8. #36977C2semidirect
10. #36199C2semidirect
12. #35120C2semidirect
13. #35120C2semidirect
15. #34642C2semidirect
16. #34399C2semidirect
25. #2806C22semidirect
26. #2802C22semidirect
31. #2774C22semidirect
33. #2766C22semidirect
34. #2762C22semidirect
36. #2755C22semidirect
39. #2726C22semidirect
42. #2719C22semidirect
45. #2660C22semidirect
46. #2660C22semidirect
56. #2612C22semidirect
57. #2612C22semidirect
58. #2612C22semidirect
60. #2606C22semidirect
61. #2606C22semidirect
62. #2605C22semidirect
63. #2605C22semidirect
67. #2590C22semidirect
68. #2584C22semidirect
71. #2570C22semidirect
72. #2565C22semidirect
78. #2528C22semidirect
80. #2524C22semidirect
81. #2524C22semidirect
84. #2514C22semidirect
85. #2511C22semidirect
86. #2508C22semidirect
87. #2506C22semidirect
96. #2462C22semidirect
97. #2453C22semidirect
98. #2399C22semidirect
99. #2382C22semidirect
100. #2371C22semidirect
101. #2370C22semidirect
103. #2356C22semidirect
104. #2356C22semidirect
105. #2351C22semidirect
109. #2331C22semidirect
114. #2327C22semidirect
117. #2320C22semidirect

Maximal subgroups

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

subgroupindex
#516272
#516162
#448532
#448512
#396772
#379582
#378852
#369772
#367282
#361992
#353252
#351202
#350122
#346422
#343992
C2×#23462