Finite Group Databasebeta

← #7630983

Subgroups — #7630983

Order 512.

5598 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C212yesmaximal, complemented
2#36625C212yesmaximal, complemented
3C2×#2851C212yesmaximal, complemented
4#36363C218yesmaximal, complemented
5#36281C2116yesmaximal, complemented
6#36191C214yesmaximal, complemented
7#36131C214yesmaximal, complemented
8#35104C212yesmaximal, complemented
9#35033C212yesmaximal, complemented
10#35013C214yesmaximal, complemented
11#35012C214yesmaximal, complemented
12#35002C212yesmaximal, complemented
13#34985C214yesmaximal
14#34270C214yesmaximal, complemented
15#34185C212yesmaximal
16C2×C42×D4C211yesmaximal, characteristic
17C2×#543C2214yes—
18C2×#533C2212yescomplemented
19C22×#125C2212yescomplemented
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2864C2214yescomplemented
25#2863C2218yescomplemented
26#2862C2212yescomplemented
27#2859C2212yescomplemented
28#2857C2214yes—
29#2857C2214yes—
30#2857C2214yescomplemented
31#2855C2218yes—
32#2855C2218yescomplemented
33#2851C2214yes—
34#2851C2214yes—
35#2851C2218yes—
36#2851C2214yescomplemented
37#2851C2214yescomplemented
38#2851C2218yescomplemented
39#2851C2214yescomplemented
40#2848C2218yes—
41#2659C2214yes—
42#2659C2214yes—
43#2627C2218yes—
44#2626C2218yes—
45#2608C2218yescomplemented
46#2607C2218yescomplemented
47#2606C2218yescomplemented
48#2605C2218yescomplemented
49#2586C22116yescomplemented
50#2585C22116yescomplemented
51#2574C22116yescomplemented
52#2570C2218yescomplemented
53#2556C22116yescomplemented
54#2550C2218yescomplemented
55#2549C22116yescomplemented
56#2522C22116yescomplemented
57#2520C22116yescomplemented
58#2465C22116yescomplemented
59#2462C22116yescomplemented
60#2450C22116yes—
61#2450C22116yes—
62#2449C22116yes—
63#2449C22116yes—
64#2441C2218yescomplemented
65#2440C2218yescomplemented
66#2440C2218yescomplemented
67#2439C2218yescomplemented
68#2368C22116yescomplemented
69#2368C22116yescomplemented
70#2360C22116yescomplemented
71#2360C22116yescomplemented
72#2353C22116yes—
73#2352C2218yes—
74#2352C2218yes—
75#2329C2218yes—
76#2329C2218yes—
77#2315C2214yes—
78#2313C2214yes—
79#2312C2214yes—
80#2308C2214yes—
81C4×C4⋊Q8C2214yescomplemented
82C4×#125C2214yescomplemented
83C4×C22⋊Q8C2218yes—
84C4×C22⋊Q8C2218yes—
85C4×C22⋊Q8C2218yescomplemented
86#2296C2214yescomplemented
87#2295C2218yescomplemented
88C2×#389C2214yescomplemented
89C2×#387C2218yescomplemented
90C2×#385C2218yes—
91#2285C2212yescomplemented
92#2283C2214yes—
93C2×#382C2212yescomplemented
94#2280C2214yes—
95C2×#380C2214yes—
96#2273C2214yes—
97C42×D4C2214yes—
98C42×D4C2214yes—
99C42×D4C2218yes—
100C2×C4×C4⋊C4C2211yescharacteristic
101C2×C4×C4⋊C4C2212yes—
102C2×C4×C22⋊C4C2211yescharacteristic
103C2×C4×C22⋊C4C2212yes—
104C2×C4×C22⋊C4C2211yescharacteristic
105C2×C4×C22⋊C4C2212yes—
106C2×C43C2211yescharacteristic
107C23×D4C2311yescharacteristic
108C24×C4C2312yes—
109C24×C4C2312yes—
110C24×C4C2311yescharacteristic
111C24×C4C2311yescharacteristic
112#543—232no—
113#533—48no—
114C2×C4⋊Q8C2314yes—
115C2×C4⋊Q8C2314yes—
116C2×#125—48no—
117C2×#125C2314yes—
118C2×#125C2314yes—
119C2×#125C2318yes—
120C2×C22⋊Q8C2318yes—
121C2×C22⋊Q8C2318yes—
122C2×C22⋊Q8C23116yes—
123C2×C22⋊Q8C23116yes—
124C2×C22⋊Q8C2318yes—
125C2×C22⋊Q8C2318yes—
126C2×C4×Q8C2318yes—
127C2×C4×Q8C2318yes—
128C2×C4×D4—216no—
129C2×C4×D4Q8116yescomplemented
130C2×C4×D4C2318yes—
131C2×C4×D4C2318yes—
132C2×C4×D4C2314yes—
133C2×C4×D4C2314yes—
134C2×#117C2314yes—
135C2×#117C2314yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2314yes—
139C22×C4⋊C4C2318yes—
140C22×C4⋊C4C2318yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4C2318yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2311yescharacteristic
145C22×C22⋊C4C2311yescharacteristic
146C22×C22⋊C4C2312yes—
147C22×C42—44no—
148C22×C42—44no—
149C22×C42C2311yescharacteristic
150C22×C42C2312yes—
151C22×C42C2312yes—
152C22×C42C2312yes—
153C22×C42C2312yes—
154C22×C42C2311yescharacteristic
155C22×C42C2314yes—
156C22×C42C2312yes—
157C22×C42C2314yes—
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160C22×C42C2312yes—
161C22×C42C2312yes—
162C22×C42C2312yes—
163C22×C42C2311yescharacteristic
164#400C23116yes—
165#399C23116yes—
166#398C23116yes—
167#398C23116yes—
168#398C23116yes—
169#398C23116yes—
170#396C23116yes—
171#396C23116yes—
172#395C23116yes—
173#395C23116yes—
174#395C23116yes—
175#393C23116yes—
176#391C2318yes—
177#391C2318yes—
178#391C23116yes—
179#391C23116yes—
180#391C2318yes—
181#391C2318yes—
182#391C2318yes—
183#391C2318yes—
184#389C2318yes—
185#389C2318yes—
186#389C2318yes—
187#389C2318yes—
188#389C2318yes—
189#389C23116yes—
190#389C23116yes—
191#389C2318yes—
192#389C2318yes—
193#389C2318yes—
194#389C2318yes—
195#389C2318yes—
196#389C2318yes—
197#389C2318yes—
198#388C23116yes—
199#388C23116yes—
200#387C23116yes—
201#387C23116yes—
202#387C23116yes—
203#387C23116yes—
204#387C23116yes—
205#387C23116yes—
206#387C23116yes—
207#387C23116yes—
208#387C23116yes—
209#387C23116yes—
210#387C23116yes—
211#387C23116yes—
212#387C23116yes—
213#387C23116yes—
214#386C23116yes—
215#386C23116yes—
216#385C23116yes—
217#385C23116yes—
218#385C23116yes—
219#385C23116yes—
220#385C23116yes—
221#385C23116yes—
222#385C23116yes—
223#385C23116yes—
224#384C2314yes—
225#384C2314yes—
226#383C2318yes—
227#382C2314yes—
228#382C2314yes—
229#382C2314yes—
230#382C2318yes—
231#382C2314yes—
232#381C2318yes—
233#381C2318yes—
234#380C2318yes—
235#380C2318yes—
236#380C2318yes—
237#380C2318yes—
238#380C2318yes—
239#380C2318yes—
240#380C2318yes—
241#380C2318yes—
242#380C2318yes—
243#379C2318yes—
244C4×C4⋊C4C2318yes—
245C4×C4⋊C4C23116yes—
246C4×C4⋊C4C2318yes—
247C4×C4⋊C4C2314yes—
248C4×C4⋊C4C2318yes—
249C4×C4⋊C4C23116yes—
250C4×C4⋊C4C2314yes—
251C4×C4⋊C4C2318yes—
252C4×C4⋊C4C2318yes—
253C4×C22⋊C4C2318yes—
254C4×C22⋊C4C2318yes—
255C4×C22⋊C4C2318yes—
256C4×C22⋊C4C2314yes—
257C4×C22⋊C4C2318yes—
258C4×C22⋊C4C23116yes—
259C4×C22⋊C4C23116yes—
260C4×C22⋊C4C2318yes—
261C4×C22⋊C4C2318yes—
262C4×C22⋊C4C2314yes—
263C4×C22⋊C4C2314yes—
264C4×C22⋊C4C2318yes—
265C4×C22⋊C4C2314yes—
266#376C2318yes—
267C2×#95C2312yes—
268C2×#95C2312yes—
269C2×#95C2318yes—
270C2×#95C2318yes—
271C2×#95C2318yes—
272C2×#95C2312yes—
273C43C2312yes—
274C43C2312yes—
275C43C2314yes—
276C25C2411yescharacteristic
277C25C2411yescharacteristic
278C22×Q8C2418yes—
279C22×Q8C2418yes—
280C22×D4—28no—
281C22×D4C2×Q818yes—
282C22×D4C2414yes—
283C23×C4—28no—
284C23×C4—28no—
285C23×C4—42no—
286C23×C4—42no—
287C23×C4—44no—
288C23×C4—44no—
289C23×C4—42no—
290C23×C4—42no—
291C23×C4—44no—
292C23×C4—44no—
293C23×C4C2×Q818yes—
294C23×C4C2×Q818yes—
295C23×C4C2412yes—
296C23×C4C2414yes—
297C23×C4C2414yes—
298C23×C4C2414yes—
299C23×C4C2414yes—
300C23×C4C2412yes—
301C23×C4C2411yescharacteristic
302C23×C4C2412yes—
303C23×C4C2411yescharacteristic
304C23×C4C2418yes—
305C23×C4C2418yes—
306C23×C4C2412yes—
307C23×C4C2411yescharacteristic
308C23×C4C2411yescharacteristic
309C23×C4C2411yescharacteristic
310C23×C4C2412yes—
311C23×C4C2412yes—
312C23×C4C2412yes—
313C23×C4C2412yes—
314C4⋊Q8—48no—
315C4⋊Q8—48no—
316#125—216no—
317#125—48no—
318#125—416no—
319C22⋊Q8—232no—
320C22⋊Q8—232no—
321C22⋊Q8—264no—
322C22⋊Q8—264no—
323C22⋊Q8—416no—
324C22⋊Q8—416no—
325C4×Q8—416no—
326C4×Q8—416no—
327C4×D4—232no—
328C4×D4—232no—
329C4×D4C2×Q8132yes—
330C4×D4C2×Q8132yes—
331#117—48no—
332#117—48no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4—416no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4—416no—
338C2×C4⋊C4C2×Q818yes—
339C2×C4⋊C4C2418yes—
340C2×C4⋊C4C2418yes—
341C2×C4⋊C4C2418yes—
342C2×C4⋊C4C24116yes—
343C2×C4⋊C4C24116yes—
344C2×C4⋊C4C2418yes—
345C2×C4⋊C4C2418yes—
346C2×C4⋊C4C24116yes—
347C2×C4⋊C4C24116yes—
348C2×C4⋊C4C2418yes—
349C2×C4⋊C4C24116yes—
350C2×C4⋊C4C2418yes—
351C2×C4⋊C4C24116yes—
352C2×C4⋊C4C2418yes—
353C2×C4⋊C4C2418yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2418yes—
356C2×C4⋊C4C2414yes—
357C2×C22⋊C4—232no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4C2×Q818yes—
360C2×C22⋊C4C2×Q818yes—
361C2×C22⋊C4#41116yes—
362C2×C22⋊C4C2418yes—
363C2×C22⋊C4C24116yes—
364C2×C22⋊C4C24116yes—
365C2×C22⋊C4C2418yes—
366C2×C22⋊C4C2418yes—
367C2×C22⋊C4C24116yes—
368C2×C22⋊C4C2414yes—
369C2×C22⋊C4C2414yes—
370C2×C22⋊C4C2418yes—
371C2×C42—28no—
372C2×C42—28no—
373C2×C42—28no—
374C2×C42—24no—
375C2×C42—48no—
376C2×C42—44no—
377C2×C42—48no—
378C2×C42—44no—
379C2×C42—44no—
380C2×C42—44no—
381C2×C42—42no—
382C2×C42—42no—
383C2×C42—44no—
384C2×C42—42no—
385C2×C42C2×Q818yes—
386C2×C42C2414yes—
387C2×C42C2414yes—
388C2×C42C2414yes—
389C2×C42C2414yes—
390C2×C42C2414yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2412yes—
394C2×C42C2412yes—
395C2×C42C2411yescharacteristic
396C2×C42C2418yes—
397C2×C42C2418yes—
398C2×C42C24116yes—
399C2×C42C2414yes—
400C2×C42C2412yes—
401C2×C42C2414yes—
402C2×C42C2412yes—
403C2×C42C2414yes—
404C2×C42C2412yes—
405C2×C42C2412yes—
406C2×C42C2412yes—
407C2×C42C2411yescharacteristic
408#95C24116yes—
409#95C2418yes—
410#95C2418yes—
411#95C2418yes—
412#95C24116yes—
413#95C24116yes—
414#95C24116yes—
415#95C24116yes—
416#95C24116yes—
417#95C2418yes—
418#95C2418yes—
419#95C24116yes—
420#95C2418yes—
421#95C2418yes—
422#95C24116yes—
423#95C24116yes—
424#95C24116yes—
425C24—24no—
426C24—41no—
427C24—41no—
428C24—41no—
429C24—42no—
430C24—41no—
431C24—42no—
432C24C22×Q814yes—
433C24C22×Q814yes—
434C24C2×#4118yes—
435C24C2512yes—
436C24C2511yescharacteristic
437C24C2512yes—
438C2×Q8—216no—
439C2×Q8—216no—
440C2×Q8—48no—
441C2×Q8—48no—
442C2×Q8—48no—
443C2×Q8—48no—
444C2×D4—232no—
445C2×D4—216no—
446C2×D4C22×Q8116yes—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—216no—
461C22×C4—216no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—41no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—432no—
494C22×C4—432no—
495C22×C4C22×Q814yes—
496C22×C4C22×Q814yes—
497C22×C4C22×Q818yes—
498C22×C4C22×Q818yes—
499C22×C4C22×Q814yes—
500C22×C4C22×Q814yes—
501C22×C4C22×Q814yes—
502C22×C4C22×Q818yes—
503C22×C4C22×Q818yes—
504C22×C4C2×#4118yes—
505C22×C4C2×#4118yes—
506C22×C4C25116yes—
507C22×C4C2514yes—
508C22×C4C25116yes—
509C22×C4C2511yescharacteristic
510C22×C4C2512yes—
511C22×C4C2512yes—
512C22×C4C2512yes—
513C22×C4C2512yes—
514C22×C4C2514yes—
515C22×C4C2514yes—
516C22×C4C2512yes—
517C22×C4C2512yes—
518C22×C4C2511yescharacteristic
519C4⋊C4—216no—
520C4⋊C4—216no—
521C4⋊C4—232no—
522C4⋊C4—216no—
523C4⋊C4—232no—
524C4⋊C4—216no—
525C4⋊C4—232no—
526C4⋊C4—216no—
527C4⋊C4—232no—
528C4⋊C4—48no—
529C4⋊C4—416no—
530C4⋊C4—48no—
531C4⋊C4—416no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4C22×Q8116yes—
537C22⋊C4—232no—
538C22⋊C4—232no—
539C22⋊C4—216no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—416no—
543C22⋊C4C22×Q8116yes—
544C22⋊C4C22×Q8116yes—
545C22⋊C4C2×#41132yes—
546C42—28no—
547C42—28no—
548C42—24no—
549C42—22no—
550C42—28no—
551C42—28no—
552C42—24no—
553C42—24no—
554C42—28no—
555C42—24no—
556C42—44no—
557C42—44no—
558C42—42no—
559C42—42no—
560C42—44no—
561C42—42no—
562C42—48no—
563C42—48no—
564C42—416no—
565C42C22×Q818yes—
566C42C22×Q818yes—
567C42H(5,𝔽2)14yes—
568C23—22no—
569C23—21no—
570C23—22no—
571C23—24no—
572C23—21no—
573C23—22no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—28no—
579C23—44no—
580C23—44no—
581C23—48no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—48no—
586C23—48no—
587C23#533116yescomplemented
588C23C23×Q814yes—
589C23C23×Q814yes—
590C23C23×Q812yes—
591C23C22×#4114yes—
592C23C22×#4118yes—
593C23C2611yescharacteristic
594Q8—816no—
595Q8—816no—
596D4—232no—
597C2×C4—216no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—216no—
617C2×C4—216no—
618C2×C4—216no—
619C2×C4—216no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—216no—
623C2×C4—216no—
624C2×C4—216no—
625C2×C4—216no—
626C2×C4—24no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—416no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—416no—
638C2×C4—48no—
639C2×C4—816no—
640C2×C4—816no—
641C2×C4C23×Q814yes—
642C2×C4C23×Q814yes—
643C2×C4C23×Q812yes—
644C2×C4C23×Q814yes—
645C2×C4C23×Q814yes—
646C2×C4C23×Q812yes—
647C2×C4C22×#4118yes—
648C2×C4C22×#4118yes—
649C2×C4C2×H(5,𝔽2)14yes—
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58514yes—
652C2×C4#58512yes—
653C22—22no—
654C22—24no—
655C22—24no—
656C22—28no—
657C22—24no—
658C22—28no—
659C22—22no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—416no—
665C22—416no—
666C22—44no—
667C22—44no—
668C22C2×#53314yes—
669C22C2×#53318yes—
670C22C24×Q811yescharacteristic
671C22C23×#4112yes—
672C22C22×H(5,𝔽2)11yescharacteristic
673C22C2×#58511yescharacteristic
674C22C2×#58512yes—
675C4—22no—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—24no—
680C4—28no—
681C4—28no—
682C4—24no—
683C4—24no—
684C4—44no—
685C4—416no—
686C4—416no—
687C4#350114yes—
688C2—24no—
689C2—28no—
690C2—44no—
691C2C22×#53311yescharacteristic
692C2C2×#350111yescharacteristic
693C2#6277312yes—
694C2#6278112yes—
695C2#6281311yescharacteristic
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. #36635C2semidirect
2. #36625C2semidirect
3. C2×#2851C2semidirect
4. #36363C2semidirect
5. #36281C2semidirect
6. #36191C2semidirect
7. #36131C2semidirect
8. #35104C2semidirect
9. #35033C2semidirect
10. #35013C2semidirect
11. #35012C2semidirect
12. #35002C2semidirect
14. #34270C2semidirect
18. C2×#533C22semidirect
19. C22×#125C22semidirect
24. #2864C22semidirect
25. #2863C22semidirect
26. #2862C22semidirect
27. #2859C22semidirect
30. #2857C22semidirect
32. #2855C22semidirect
36. #2851C22semidirect
37. #2851C22semidirect
38. #2851C22semidirect
39. #2851C22semidirect
45. #2608C22semidirect
46. #2607C22semidirect
47. #2606C22semidirect
48. #2605C22semidirect
49. #2586C22semidirect
50. #2585C22semidirect
51. #2574C22semidirect
52. #2570C22semidirect
53. #2556C22semidirect
54. #2550C22semidirect
55. #2549C22semidirect
56. #2522C22semidirect
57. #2520C22semidirect
58. #2465C22semidirect
59. #2462C22semidirect
64. #2441C22semidirect
65. #2440C22semidirect
66. #2440C22semidirect
67. #2439C22semidirect
68. #2368C22semidirect
69. #2368C22semidirect
70. #2360C22semidirect
71. #2360C22semidirect
81. C4×C4⋊Q8C22semidirect
82. C4×#125C22semidirect
85. C4×C22⋊Q8C22semidirect
86. #2296C22semidirect
87. #2295C22semidirect
88. C2×#389C22semidirect
89. C2×#387C22semidirect
91. #2285C22semidirect
93. C2×#382C22semidirect
129. C2×C4×D4Q8semidirect
587. C23#533semidirect

Maximal subgroups

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

subgroupindex
#366352
#366252
C2×#28512
#363632
#362812
#361912
#361312
#351042
#350332
#350132
#350122
#350022
#349852
#342702
#341852
C2×C42×D42