Finite Group Databasebeta

← #7630623

Subgroups — #7630623

Order 512.

5546 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35629C214yesmaximal, complemented
2#35557C214yesmaximal, complemented
3#35545C218yesmaximal, complemented
4#35544C218yesmaximal, complemented
5#35370C218yesmaximal, complemented
6#35313C214yesmaximal, complemented
7#35267C218yesmaximal, complemented
8#35267C218yesmaximal, complemented
9#35266C214yesmaximal, complemented
10#34052C211yesmaximal, characteristic, complemented
11C2×#2317C211yesmaximal, characteristic, complemented
12C2×#2316C211yesmaximal, characteristic, complemented
13#34044C211yesmaximal, characteristic
14#34044C211yesmaximal, characteristic
15C2×#2311C211yesmaximal, characteristic
16#33996C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2863C22116yescomplemented
25#2860C22116yescomplemented
26#2645C22116yescomplemented
27#2644C22116yescomplemented
28#2514C22116yescomplemented
29#2513C22116yescomplemented
30#2512C22116yescomplemented
31#2511C22116yescomplemented
32#2510C22132yescomplemented
33#2510C22132yescomplemented
34#2497C22116yescomplemented
35#2497C22116yescomplemented
36#2496C22116yescomplemented
37#2496C22116yescomplemented
38#2495C22116yescomplemented
39#2495C22116yescomplemented
40#2494C22116yescomplemented
41#2494C22116yescomplemented
42#2493C22132yes—
43#2481C22132yescomplemented
44#2481C22132yescomplemented
45#2480C22132yes—
46#2317C2214yes—
47#2317C2214yes—
48#2317C2214yes—
49#2317C2214yescomplemented
50#2317C2214yescomplemented
51#2317C2214yescomplemented
52#2316C2214yes—
53#2316C2214yes—
54#2316C2214yescomplemented
55#2316C2214yescomplemented
56#2316C2214yescomplemented
57#2315C2214yes—
58#2311C2212yes—
59#2311C2214yes—
60#2311C2214yes—
61#2311C2214yes—
62#2311C2214yes—
63#2311C2214yes—
64#2311C2214yes—
65#2311C2214yes—
66#2311C2212yes—
67#2310C2212yes—
68#2310C2212yes—
69#2309C2212yes—
70#2309C2212yes—
71#2308C2214yes—
72#2308C2214yes—
73#2297C2214yes—
74#2297C2214yes—
75#2297C2214yes—
76#2297C2214yes—
77#2296C2214yes—
78#2296C2214yescomplemented
79C2×#391C2212yes—
80C2×#390C2212yes—
81C2×#389C2214yescomplemented
82C2×#389C2214yescomplemented
83C2×#389C2214yescomplemented
84C2×#386C2214yes—
85C2×#385C2214yes—
86#2285C2214yescomplemented
87C2×#384C2212yescomplemented
88C2×#382C2212yescomplemented
89#2280C2214yes—
90#2280C2214yes—
91C2×#380C2214yes—
92C2×#380C2214yes—
93#2274C2214yes—
94#2274C2212yescomplemented
95#2274C2214yescomplemented
96#2274C2212yescomplemented
97#2273C2214yes—
98C4×#117C2214yes—
99C4×#117C2214yescomplemented
100C4×#117C2214yescomplemented
101C22×#41—28no—
102C22×#41—28no—
103C22×#41—28no—
104C22×#41C2312yes—
105C22×#41C2312yes—
106C22×#41C2312yes—
107C22×#41C2312yes—
108C23×Q8C2311yescharacteristic
109C23×D4C2312yes—
110C23×D4C2311yescharacteristic
111C24×C4C2312yes—
112C24×C4C2312yes—
113C24×C4C2311yescharacteristic
114C2×#126C23132yes—
115C2×#117—28no—
116C2×#117—28no—
117C2×#117—28no—
118C2×#117—28no—
119C2×#117—28no—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C2×#117C2312yes—
129C2×#117C2312yes—
130C2×#117C2312yes—
131C2×#117C2312yes—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C2×#117C2312yes—
137C2×#117C2312yes—
138C2×#117C2312yes—
139C2×#117C2312yes—
140C22×C4⋊C4—28no—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2318yes—
154C22×C4⋊C4C2318yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2312yes—
158C22×C22⋊C4C2312yes—
159C22×C22⋊C4C2312yes—
160C22×C22⋊C4C2318yes—
161C22×C42C2312yes—
162C22×C42C2312yes—
163C22×C42C2312yes—
164C22×C42C2312yes—
165C22×C42C2312yes—
166#401C23132yes—
167#400C23132yes—
168#400C23132yes—
169#400C23132yes—
170#397C23132yes—
171#397C23132yes—
172#397C23132yes—
173#396C23132yes—
174#391C2318yes—
175#391C2318yes—
176#391C2314yes—
177#391C2318yes—
178#391C2314yes—
179#391C2318yes—
180#391C2318yes—
181#390C2318yes—
182#390C2318yes—
183#390C2314yes—
184#390C2318yes—
185#390C2314yes—
186#390C2318yes—
187#390C2318yes—
188#389C23116yes—
189#389C23116yes—
190#389C23116yes—
191#389C23116yes—
192#389C2318yes—
193#389C2318yes—
194#389C2318yes—
195#389C2318yes—
196#389C23116yes—
197#389C23116yes—
198#388C23132yes—
199#388C23132yes—
200#388C23132yes—
201#387C23132yes—
202#387C23132yes—
203#387C23132yes—
204#386C2318yes—
205#386C23116yes—
206#386C23116yes—
207#386C23116yes—
208#386C2318yes—
209#386C23116yes—
210#386C23116yes—
211#385C23116yes—
212#385C23116yes—
213#385C23116yes—
214#385C23116yes—
215#384C2318yes—
216#384C2318yes—
217#384C2318yes—
218#383C23132yes—
219#382C2318yes—
220#382C2318yes—
221#382C2318yes—
222#381C2318yes—
223#381C2318yes—
224#380C2318yes—
225#380C2318yes—
226#380C23116yes—
227#380C23116yes—
228#380C2318yes—
229#380C2318yes—
230#380C23116yes—
231#380C23116yes—
232#380C23116yes—
233#379C2318yes—
234#379C2318yes—
235C4×C4⋊C4C2318yes—
236C4×C4⋊C4C2318yes—
237C4×C4⋊C4C2318yes—
238C4×C4⋊C4C2318yes—
239C4×C4⋊C4C2318yes—
240C4×C4⋊C4C23116yes—
241C4×C4⋊C4C2318yes—
242C4×C22⋊C4C23116yes—
243C4×C22⋊C4C2318yes—
244C4×C22⋊C4C23116yes—
245C4×C22⋊C4C2318yes—
246C4×C22⋊C4C23116yes—
247C4×C22⋊C4C2318yes—
248C4×C22⋊C4C2318yes—
249#376C2318yes—
250C2×#95C2318yes—
251C2×#95C2318yes—
252C2×#95C2318yes—
253C43C2318yes—
254C25C2412yes—
255C25C2411yescharacteristic
256C2×#41—28no—
257C2×#41—28no—
258C2×#41—28no—
259C2×#41—28no—
260C2×#41—28no—
261C2×#41—28no—
262C2×#41—28no—
263C2×#41—28no—
264C2×#41—28no—
265C2×#41—28no—
266C2×#41—28no—
267C2×#41—28no—
268C2×#41—464no—
269C22×Q8—24no—
270C22×Q8—24no—
271C22×Q8—24no—
272C22×Q8C2412yes—
273C22×Q8C2412yes—
274C22×D4—28no—
275C22×D4—28no—
276C22×D4—28no—
277C22×D4—24no—
278C22×D4—24no—
279C22×D4—24no—
280C22×D4C2414yes—
281C22×D4C2414yes—
282C22×D4C2412yes—
283C22×D4C2412yes—
284C23×C4—28no—
285C23×C4—28no—
286C23×C4—28no—
287C23×C4—28no—
288C23×C4—84no—
289C23×C4#4118yes—
290C23×C4#4118yes—
291C23×C4#4118yes—
292C23×C4C2412yes—
293C23×C4C2414yes—
294C23×C4C2414yes—
295C23×C4C2414yes—
296C23×C4C2414yes—
297C23×C4C2412yes—
298C23×C4C2414yes—
299C23×C4C2414yes—
300C23×C4C2414yes—
301C23×C4C2412yes—
302C23×C4C2414yes—
303C23×C4C2414yes—
304C23×C4C2411yescharacteristic
305C23×C4C2411yescharacteristic
306C23×C4C2411yescharacteristic
307C23×C4C2411yescharacteristic
308C23×C4C2411yescharacteristic
309C23×C4C2411yescharacteristic
310#126—832no—
311#117—28no—
312#117—28no—
313#117—28no—
314#117—28no—
315#117—28no—
316#117—28no—
317#117—28no—
318#117—28no—
319#117—28no—
320#117—28no—
321#117—28no—
322#117—28no—
323#117—28no—
324#117—28no—
325#117—28no—
326#117—28no—
327#117—28no—
328#117—28no—
329#117—28no—
330#117—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—88no—
344C2×C4⋊C4—88no—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C24132yes—
362C2×C4⋊C4C24132yes—
363C2×C4⋊C4C24116yes—
364C2×C4⋊C4C24116yes—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—88no—
371C2×C22⋊C4C2414yes—
372C2×C22⋊C4C2414yes—
373C2×C22⋊C4C2414yes—
374C2×C22⋊C4C2414yes—
375C2×C22⋊C4C2414yes—
376C2×C22⋊C4C2414yes—
377C2×C22⋊C4C2414yes—
378C2×C22⋊C4C2414yes—
379C2×C22⋊C4C2414yes—
380C2×C22⋊C4C2414yes—
381C2×C22⋊C4C24132yes—
382C2×C22⋊C4C24116yes—
383C2×C22⋊C4C24116yes—
384C2×C22⋊C4C24116yes—
385C2×C22⋊C4C24116yes—
386C2×C42—28no—
387C2×C42—28no—
388C2×C42—28no—
389C2×C42—28no—
390C2×C42—28no—
391C2×C42C2414yes—
392C2×C42C2414yes—
393C2×C42C2414yes—
394C2×C42C2414yes—
395C2×C42C2414yes—
396C2×C42C2414yes—
397C2×C42C2414yes—
398C2×C42C2414yes—
399C2×C42C2414yes—
400C2×C42C2414yes—
401C2×C42C24116yes—
402C2×C42C24116yes—
403#95C24132yes—
404#95C24132yes—
405#95C24132yes—
406#95C24116yes—
407#95C24116yes—
408#95C24116yes—
409#95C24116yes—
410#95C24116yes—
411#95C24116yes—
412#95C24116yes—
413#95C24116yes—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—42no—
418C24—42no—
419C24—44no—
420C24—44no—
421C24C2×#4114yes—
422C24C2×#4114yes—
423C24C2×#4114yes—
424C24C2514yes—
425C24C2511yescharacteristic
426C24C2511yescharacteristic
427C24C2511yescharacteristic
428#41—432no—
429#41—432no—
430#41—432no—
431#41—432no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—416no—
439C2×D4—28no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—432no—
452C2×D4—416no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—432no—
485C22×C4—816no—
486C22×C4#126164yescomplemented
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4114yes—
500C22×C4C2×#4114yes—
501C22×C4C2×#4114yes—
502C22×C4C2×#4114yes—
503C22×C4C2×#4114yes—
504C22×C4C2×#4114yes—
505C22×C4C2514yes—
506C22×C4C2514yes—
507C22×C4C2518yes—
508C22×C4C2518yes—
509C22×C4C2518yes—
510C22×C4C2518yes—
511C22×C4C2518yes—
512C22×C4C2514yes—
513C22×C4C2511yescharacteristic
514C22×C4C2511yescharacteristic
515C22×C4C2511yescharacteristic
516C22×C4C2511yescharacteristic
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—24no—
528C4⋊C4—28no—
529C4⋊C4—24no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—432no—
534C4⋊C4—432no—
535C4⋊C4—416no—
536C4⋊C4—416no—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—28no—
543C22⋊C4—28no—
544C22⋊C4—28no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—432no—
548C22⋊C4—416no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C4—416no—
552C42—28no—
553C42—28no—
554C42—28no—
555C42—28no—
556C42—28no—
557C42—28no—
558C42—28no—
559C42—28no—
560C42—28no—
561C42—28no—
562C42—416no—
563C42—416no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—28no—
574C23—28no—
575C23—44no—
576C23—48no—
577C23—48no—
578C23—44no—
579C23—416no—
580C23—48no—
581C23—84no—
582C23—88no—
583C23—84no—
584C23C2×#126116yes—
585C23C22×#4112yes—
586C23C22×#4112yes—
587C23C22×#4112yes—
588C23C22×#4112yes—
589C23C22×#4112yes—
590C23C22×#4112yes—
591C23C22×#4112yes—
592C23C22×#4112yes—
593C23C22×#4112yes—
594C23C2611yescharacteristic
595Q8—48no—
596Q8—48no—
597D4—416no—
598D4—416no—
599D4—48no—
600D4—48no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—216no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—416no—
623C2×C4—416no—
624C2×C4—48no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—84no—
628C2×C4—88no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4C2×#126116yes—
635C2×C4C2×#126116yes—
636C2×C4C2×#126116yes—
637C2×C4C2×#126116yes—
638C2×C4C2×#126116yes—
639C2×C4C2×#126116yes—
640C2×C4C22×#4112yes—
641C2×C4C22×#4112yes—
642C2×C4C22×#4112yes—
643C2×C4C22×#4112yes—
644C2×C4C22×#4112yes—
645C2×C4C22×#4112yes—
646C2×C4C22×#4112yes—
647C2×C4C22×#4112yes—
648C2×C4C22×#4112yes—
649C2×C4C22×#4112yes—
650C2×C4C22×#4112yes—
651C2×C4C22×#4112yes—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—416no—
665C22—44no—
666C22—48no—
667C22—48no—
668C22—44no—
669C22—44no—
670C22—816no—
671C22C22×#12614yes—
672C22C22×#12614yes—
673C22C22×#12614yes—
674C22C23×#4111yescharacteristic
675C22C23×#4111yescharacteristic
676C22C23×#4111yescharacteristic
677C22H(7,𝔽2)12yes—
678C222−1+612yes—
679C4—48no—
680C4—44no—
681C4—48no—
682C4—48no—
683C4—48no—
684C4—48no—
685C4—44no—
686C4—44no—
687C4C22×#12614yes—
688C4C22×#12614yes—
689C4C22×#12614yes—
690C4C22×#12614yes—
691C2—24no—
692C2—24no—
693C2—24no—
694C2—84no—
695C2C23×#12611yescharacteristic
696C2#6285511yescharacteristic
697C2#6285612yes—
698C2#6285612yes—
699C2#6286011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35629C2semidirect
2. #35557C2semidirect
3. #35545C2semidirect
4. #35544C2semidirect
5. #35370C2semidirect
6. #35313C2semidirect
7. #35267C2semidirect
8. #35267C2semidirect
9. #35266C2semidirect
10. #34052C2semidirect
11. C2×#2317C2semidirect
12. C2×#2316C2semidirect
16. #33996C2semidirect
24. #2863C22semidirect
25. #2860C22semidirect
26. #2645C22semidirect
27. #2644C22semidirect
28. #2514C22semidirect
29. #2513C22semidirect
30. #2512C22semidirect
31. #2511C22semidirect
32. #2510C22semidirect
33. #2510C22semidirect
34. #2497C22semidirect
35. #2497C22semidirect
36. #2496C22semidirect
37. #2496C22semidirect
38. #2495C22semidirect
39. #2495C22semidirect
40. #2494C22semidirect
41. #2494C22semidirect
43. #2481C22semidirect
44. #2481C22semidirect
49. #2317C22semidirect
50. #2317C22semidirect
51. #2317C22semidirect
54. #2316C22semidirect
55. #2316C22semidirect
56. #2316C22semidirect
78. #2296C22semidirect
81. C2×#389C22semidirect
82. C2×#389C22semidirect
83. C2×#389C22semidirect
86. #2285C22semidirect
87. C2×#384C22semidirect
88. C2×#382C22semidirect
94. #2274C22semidirect
95. #2274C22semidirect
96. #2274C22semidirect
99. C4×#117C22semidirect
100. C4×#117C22semidirect
486. C22×C4#126semidirect

Maximal subgroups

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

subgroupindex
#356292
#355572
#355452
#355442
#353702
#353132
#352672
#352662
#340522
C2×#23172
C2×#23162
#340442
C2×#23112
#339962