Finite Group Databasebeta

← #7647088

Subgroups — #7647088

Order 512.

4998 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#42498C218yesmaximal, complemented
2#42497C218yesmaximal, complemented
3#39497C214yesmaximal, complemented
4#39369C214yesmaximal, complemented
5#38967C214yesmaximal, complemented
6#38966C214yesmaximal, complemented
7#37477C214yesmaximal, complemented
8#36749C218yesmaximal, complemented
9#35529C214yesmaximal
10#35275C212yesmaximal, complemented
11#35253C214yesmaximal, complemented
12#35167C212yesmaximal, complemented
13#35159C212yesmaximal, complemented
14#34948C212yesmaximal, complemented
15#34181C211yesmaximal, characteristic
16C2×#2354C211yesmaximal, characteristic
17#34058C211yesmaximal, characteristic
18C2×#544C2214yes—
19C2×#544C2214yescomplemented
20C2×#518C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2868C2214yescomplemented
24#2859C2214yescomplemented
25#2857C2218yes—
26#2851C2218yes—
27#2841C22116yescomplemented
28#2790C22116yescomplemented
29#2790C22116yescomplemented
30#2788C22116yescomplemented
31#2768C22116yescomplemented
32#2742C22116yescomplemented
33#2729C22116yescomplemented
34#2720C22116yescomplemented
35#2677C22116yescomplemented
36#2660C2218yescomplemented
37#2659C2218yes—
38#2659C2218yescomplemented
39#2646C2218yes—
40#2646C2218yescomplemented
41#2629C2218yescomplemented
42#2629C2218yescomplemented
43#2628C2218yescomplemented
44#2628C2218yescomplemented
45#2571C2212yescomplemented
46#2571C2212yescomplemented
47#2570C2214yescomplemented
48#2570C2214yescomplemented
49#2569C2218yes—
50#2566C2218yes—
51#2566C2218yes—
52#2563C2218yes—
53#2552C2214yescomplemented
54#2552C2218yescomplemented
55#2552C2212yescomplemented
56#2552C2214yescomplemented
57#2552C2212yescomplemented
58#2551C2214yescomplemented
59#2551C2218yescomplemented
60#2551C2214yescomplemented
61#2550C2218yescomplemented
62#2550C2218yescomplemented
63#2542C2218yes—
64#2509C2218yescomplemented
65#2508C2218yescomplemented
66#2507C2218yescomplemented
67#2506C2218yescomplemented
68#2454C2218yes—
69#2454C2218yes—
70#2452C2218yes—
71#2452C2218yes—
72#2442C2214yes—
73#2442C2214yes—
74#2442C2218yescomplemented
75#2441C2218yescomplemented
76#2440C2218yes—
77#2440C2218yes—
78#2440C2218yescomplemented
79#2439C2218yescomplemented
80#2438C22116yescomplemented
81#2437C22116yes—
82#2437C22116yes—
83#2402C2218yescomplemented
84#2354C2218yes—
85#2354C2214yes—
86#2354C2214yes—
87#2349C2218yes—
88D4×C4⋊C4C2214yes—
89D4×C4⋊C4C2214yes—
90#2319C2214yes—
91#2319C2214yes—
92#2317C2214yes—
93#2317C2214yescomplemented
94#2317C2214yescomplemented
95#2316C2211yescharacteristic
96#2316C2211yescharacteristic
97#2316C2212yes—
98#2316C2214yescomplemented
99#2315C2214yes—
100#2312C2214yes—
101#2309C2212yes—
102C4×#125C2214yescomplemented
103C4×#125C2214yescomplemented
104#2296C2214yes—
105#2293C2214yes—
106#2293C2214yes—
107C2×#390C2212yes—
108C2×#388C2214yes—
109#2285C2214yes—
110#2285C2212yes—
111C2×#384C2211yescharacteristic
112#2280C2214yes—
113C2×#380C2212yes—
114C2×#380C2214yes—
115C4×#117C2212yes—
116C2×C4×C4⋊C4C2212yes—
117C2×C4×C4⋊C4C2211yescharacteristic
118C23×D4C2311yescharacteristic
119C24×C4C2312yes—
120#544—416no—
121#544—416no—
122C2×C4⋊Q8C2318yes—
123C2×C4⋊Q8C2318yes—
124C2×C4⋊Q8C2318yes—
125C2×C4⋊Q8C2318yes—
126C2×C4⋊Q8C2318yes—
127C2×#125C2314yes—
128C2×#125C2314yes—
129C2×#125C2318yes—
130C2×#125C2318yes—
131C2×#125C2318yes—
132C2×C22⋊Q8C23116yes—
133C2×C22⋊Q8C23116yes—
134C2×C22⋊Q8C23116yes—
135#518—48no—
136C2×C4×D4Q8116yescomplemented
137C2×C4×D4C2318yes—
138C2×C4×D4C2318yes—
139C2×C4×D4C2314yes—
140C2×C4×D4C2314yes—
141C2×#117—48no—
142C2×#117C2312yes—
143C2×#117C2312yes—
144C2×#117C2314yes—
145C2×#117C2314yes—
146C2×#117C2314yes—
147C2×#117C2314yes—
148C2×#117C2314yes—
149C2×#117C2318yes—
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2314yes—
153C22×C4⋊C4C2314yes—
154C22×C4⋊C4C2312yes—
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2311yescharacteristic
158C22×C22⋊C4C2318yes—
159C22×C22⋊C4C2314yes—
160C22×C22⋊C4C2314yes—
161C22×C22⋊C4C2312yes—
162C22×C42C2312yes—
163C22×C42C2314yes—
164C22×C42C2311yescharacteristic
165#400C2318yes—
166#400C2318yes—
167#400C2318yes—
168#400C2318yes—
169#400C23116yes—
170#400C23116yes—
171#399C23116yes—
172#399C23116yes—
173#398C2318yes—
174#398C2318yes—
175#398C2318yes—
176#398C2318yes—
177#398C23116yes—
178#398C23116yes—
179#398C23116yes—
180#398C23116yes—
181#398C23116yes—
182#398C23116yes—
183#396C23116yes—
184#396C23116yes—
185#396C23116yes—
186#396C23116yes—
187#395C23116yes—
188#395C23116yes—
189#395C23116yes—
190#395C23116yes—
191#393C23116yes—
192#393C23116yes—
193#391C23116yes—
194#390C2314yes—
195#390C2318yes—
196#390C2314yes—
197#390C2314yes—
198#390C2314yes—
199#390C2318yes—
200#389C2314yes—
201#389C2314yes—
202#389C2314yes—
203#389C2314yes—
204#389C2314yes—
205#389C2314yes—
206#389C2314yes—
207#389C2314yes—
208#389C2314yes—
209#389C2314yes—
210#389C2314yes—
211#389C2314yes—
212#389C2318yes—
213#389C2318yes—
214#389C2318yes—
215#389C2318yes—
216#389C2318yes—
217#389C2318yes—
218#389C2318yes—
219#389C2318yes—
220#389C2318yes—
221#389C2318yes—
222#389C23116yes—
223#389C2318yes—
224#389C2318yes—
225#388C2318yes—
226#388C2318yes—
227#388C2318yes—
228#388C2318yes—
229#388C23116yes—
230#388C23116yes—
231#387C2318yes—
232#387C2318yes—
233#387C2318yes—
234#387C2318yes—
235#387C23116yes—
236#387C23116yes—
237#385C23116yes—
238#385C23116yes—
239#385C23116yes—
240#385C23116yes—
241#385C2318yes—
242#385C2318yes—
243#385C23116yes—
244#385C23116yes—
245#385C23116yes—
246#384C2314yes—
247#384C2318yes—
248#384C2314yes—
249#384C2312yes—
250#384C2312yes—
251#384C2318yes—
252#384C2318yes—
253#384C2314yes—
254#384C2314yes—
255#384C2318yes—
256#382C2314yes—
257#382C2314yes—
258#382C2314yes—
259#382C2318yes—
260#382C2314yes—
261#382C2314yes—
262#382C2318yes—
263#381C2318yes—
264#380C23116yes—
265#380C23116yes—
266#380C2314yes—
267#380C2314yes—
268#380C2314yes—
269#380C2318yes—
270#380C2318yes—
271#380C2318yes—
272#380C2314yes—
273#380C2314yes—
274#380C2318yes—
275#380C2318yes—
276#380C2318yes—
277#380C2318yes—
278#380C2314yes—
279#379C2318yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2314yes—
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2318yes—
284C4×C4⋊C4C2318yes—
285C4×C4⋊C4C2318yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2312yes—
289C4×C4⋊C4C2314yes—
290C4×C4⋊C4C2312yes—
291C4×C4⋊C4C2312yes—
292C4×C4⋊C4C2314yes—
293C4×C4⋊C4C2318yes—
294C4×C4⋊C4C2318yes—
295C4×C22⋊C4C2314yes—
296C4×C22⋊C4C2318yes—
297C4×C22⋊C4C2314yes—
298#376C2314yes—
299C2×#95C2312yes—
300C2×#95C2318yes—
301C2×#95C2314yes—
302C43C2314yes—
303C25C2412yes—
304C22×Q8C24116yes—
305C22×D4—24no—
306C22×D4—24no—
307C22×D4C2×Q818yes—
308C22×D4C2414yes—
309C23×C4—44no—
310C23×C4—44no—
311C23×C4C2×Q8116yes—
312C23×C4C2412yes—
313C23×C4C2411yescharacteristic
314C23×C4C2414yes—
315C23×C4C2414yes—
316C23×C4C2412yes—
317C23×C4C2418yes—
318C23×C4C2418yes—
319C23×C4C2411yescharacteristic
320C23×C4C2412yes—
321C23×C4C2414yes—
322C23×C4C2411yescharacteristic
323C4⋊Q8—416no—
324C4⋊Q8—416no—
325C4⋊Q8—416no—
326C4⋊Q8—416no—
327C4⋊Q8—88no—
328#125—216no—
329#125—48no—
330#125—416no—
331#125—416no—
332#125—416no—
333C22⋊Q8—432no—
334C22⋊Q8—432no—
335C22⋊Q8—816no—
336C4×D4—416no—
337C4×D4—416no—
338C4×D4C2×Q8132yes—
339C4×D4C2×Q8132yes—
340#117—216no—
341#117—28no—
342#117—28no—
343#117—216no—
344#117—48no—
345#117—48no—
346#117—416no—
347#117—48no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—48no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—48no—
354C2×C4⋊C4—82no—
355C2×C4⋊C4C2×Q818yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2418yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2418yes—
370C2×C4⋊C4C2418yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2418yes—
376C2×C4⋊C4C2414yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C2414yes—
379C2×C4⋊C4C2414yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C24116yes—
383C2×C4⋊C4C24116yes—
384C2×C4⋊C4C24116yes—
385C2×C4⋊C4C24116yes—
386C2×C4⋊C4C24116yes—
387C2×C4⋊C4C2418yes—
388C2×C4⋊C4C2418yes—
389C2×C4⋊C4C2414yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C2418yes—
392C2×C4⋊C4C2414yes—
393C2×C4⋊C4C2414yes—
394C2×C4⋊C4C2411yescharacteristic
395C2×C22⋊C4—48no—
396C2×C22⋊C4—416no—
397C2×C22⋊C4—84no—
398C2×C22⋊C4C2×Q8116yes—
399C2×C22⋊C4C2418yes—
400C2×C22⋊C4C2418yes—
401C2×C22⋊C4C2414yes—
402C2×C22⋊C4C2414yes—
403C2×C22⋊C4C2418yes—
404C2×C22⋊C4C24116yes—
405C2×C22⋊C4C24116yes—
406C2×C22⋊C4C2418yes—
407C2×C22⋊C4C2418yes—
408C2×C22⋊C4C2418yes—
409C2×C22⋊C4C2418yes—
410C2×C22⋊C4C2418yes—
411C2×C42—28no—
412C2×C42—48no—
413C2×C42C2×Q818yes—
414C2×C42C2414yes—
415C2×C42C2418yes—
416C2×C42C2414yes—
417C2×C42C2418yes—
418C2×C42C2414yes—
419C2×C42C2414yes—
420C2×C42C2414yes—
421C2×C42C2414yes—
422C2×C42C2418yes—
423C2×C42C2414yes—
424C2×C42C2414yes—
425C2×C42C2412yes—
426C2×C42C2412yes—
427#95C2418yes—
428#95C24116yes—
429#95C2414yes—
430#95C2414yes—
431#95C24116yes—
432#95C2414yes—
433#95C2414yes—
434#95C2414yes—
435#95C2414yes—
436#95C2418yes—
437#95C2418yes—
438#95C2418yes—
439#95C2418yes—
440#95C2418yes—
441#95C2418yes—
442#95C2418yes—
443#95C2418yes—
444#95C2418yes—
445#95C2418yes—
446#95C2418yes—
447#95C24116yes—
448#95C24116yes—
449C24—24no—
450C24—24no—
451C24—42no—
452C24—42no—
453C24—42no—
454C24—42no—
455C24C22×Q818yes—
456C24C2514yes—
457C24C2511yescharacteristic
458C2×Q8—416no—
459C2×Q8—416no—
460C2×Q8—416no—
461C2×D4—232no—
462C2×D4—28no—
463C2×D4—28no—
464C2×D4C22×Q8116yes—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—432no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4C22×Q814yes—
492C22×C4C22×Q8116yes—
493C22×C4C22×Q814yes—
494C22×C4C22×Q818yes—
495C22×C4C22×Q8116yes—
496C22×C4C22×Q814yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C42−1+414yes—
499C22×C4C2512yes—
500C22×C4C2514yes—
501C22×C4C2514yes—
502C22×C4C2514yes—
503C22×C4C2518yes—
504C22×C4C2514yes—
505C22×C4C2514yes—
506C22×C4C2518yes—
507C22×C4C2518yes—
508C22×C4C2511yescharacteristic
509C22×C4C2512yes—
510C22×C4C2514yes—
511C22×C4C2512yes—
512C22×C4C2512yes—
513C22×C4C2511yescharacteristic
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—216no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—216no—
524C4⋊C4—216no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—28no—
528C4⋊C4—24no—
529C4⋊C4—48no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—416no—
542C4⋊C4—416no—
543C4⋊C4—416no—
544C4⋊C4—416no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—416no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—44no—
551C4⋊C4C22×Q8116yes—
552C4⋊C4H(5,𝔽2)14yes—
553C22⋊C4—216no—
554C22⋊C4—28no—
555C22⋊C4—28no—
556C22⋊C4—216no—
557C22⋊C4—48no—
558C22⋊C4—416no—
559C22⋊C4—416no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—48no—
564C22⋊C4C22×Q8132yes—
565C42—216no—
566C42—216no—
567C42—28no—
568C42—28no—
569C42—28no—
570C42—28no—
571C42—28no—
572C42—28no—
573C42—48no—
574C42—44no—
575C42—44no—
576C42C22×Q818yes—
577C42C22×Q818yes—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—24no—
583C23—24no—
584C23—24no—
585C23—24no—
586C23—24no—
587C23—24no—
588C23—216no—
589C23—48no—
590C23—48no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23C23×Q818yes—
596C23C23×Q812yes—
597C23C2×H(5,𝔽2)12yes—
598C23C2×2−1+412yes—
599C23C2611yescharacteristic
600Q8—1616no—
601D4—232no—
602C2×C4—24no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—232no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—28no—
639C2×C4—416no—
640C2×C4—48no—
641C2×C4—44no—
642C2×C4—48no—
643C2×C4—88no—
644C2×C4—84no—
645C2×C4—168no—
646C2×C4—168no—
647C2×C4#544116yescomplemented
648C2×C4C23×Q814yes—
649C2×C4C23×Q812yes—
650C2×C4C23×Q818yes—
651C2×C4C23×Q814yes—
652C2×C4C23×Q812yes—
653C2×C4C2×H(5,𝔽2)14yes—
654C2×C4C2×H(5,𝔽2)14yes—
655C2×C4C2×H(5,𝔽2)12yes—
656C2×C4C2×2−1+414yes—
657C2×C4#58512yes—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—28no—
665C22—28no—
666C22—28no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—816no—
672C22C2×#54414yes—
673C22C24×Q811yescharacteristic
674C22C22×H(5,𝔽2)11yescharacteristic
675C22C22×H(5,𝔽2)11yescharacteristic
676C22C22×2−1+411yescharacteristic
677C22C2×#58511yescharacteristic
678C22H(7,𝔽2)11yescharacteristic
679C222−1+611yescharacteristic
680C4—22no—
681C4—28no—
682C4—24no—
683C4—24no—
684C4—44no—
685C4—42no—
686C4—48no—
687C4—48no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—48no—
693C4C2×#54418yes—
694C4#350114yes—
695C2—24no—
696C2—48no—
697C2C22×#54411yescharacteristic
698C2C2×#350111yescharacteristic
699C2#6285411yescharacteristic
700C2#6291511yescharacteristic
701C2#6291611yescharacteristic
702C2#6292911yescharacteristic
703C2#6293111yescharacteristic
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. #42498C2semidirect
2. #42497C2semidirect
3. #39497C2semidirect
4. #39369C2semidirect
5. #38967C2semidirect
6. #38966C2semidirect
7. #37477C2semidirect
8. #36749C2semidirect
10. #35275C2semidirect
11. #35253C2semidirect
12. #35167C2semidirect
13. #35159C2semidirect
14. #34948C2semidirect
19. C2×#544C22semidirect
23. #2868C22semidirect
24. #2859C22semidirect
27. #2841C22semidirect
28. #2790C22semidirect
29. #2790C22semidirect
30. #2788C22semidirect
31. #2768C22semidirect
32. #2742C22semidirect
33. #2729C22semidirect
34. #2720C22semidirect
35. #2677C22semidirect
36. #2660C22semidirect
38. #2659C22semidirect
40. #2646C22semidirect
41. #2629C22semidirect
42. #2629C22semidirect
43. #2628C22semidirect
44. #2628C22semidirect
45. #2571C22semidirect
46. #2571C22semidirect
47. #2570C22semidirect
48. #2570C22semidirect
53. #2552C22semidirect
54. #2552C22semidirect
55. #2552C22semidirect
56. #2552C22semidirect
57. #2552C22semidirect
58. #2551C22semidirect
59. #2551C22semidirect
60. #2551C22semidirect
61. #2550C22semidirect
62. #2550C22semidirect
64. #2509C22semidirect
65. #2508C22semidirect
66. #2507C22semidirect
67. #2506C22semidirect
74. #2442C22semidirect
75. #2441C22semidirect
78. #2440C22semidirect
79. #2439C22semidirect
80. #2438C22semidirect
83. #2402C22semidirect
93. #2317C22semidirect
94. #2317C22semidirect
98. #2316C22semidirect
102. C4×#125C22semidirect
103. C4×#125C22semidirect
136. C2×C4×D4Q8semidirect
647. C2×C4#544semidirect

Maximal subgroups

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

subgroupindex
#424982
#424972
#394972
#393692
#389672
#389662
#374772
#367492
#355292
#352752
#352532
#351672
#351592
#349482
#341812
C2×#23542
#340582