Finite Group Databasebeta

← #7687868

Subgroups — #7687868

Order 512.

5211 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#59673C212yesmaximal, complemented
3#59257C214yesmaximal, complemented
4#59247C218yesmaximal, complemented
5#45115C214yesmaximal, complemented
6#43804C218yesmaximal, complemented
7#40149C218yesmaximal, complemented
8#39470C214yesmaximal, complemented
9#36594C211yesmaximal, characteristic, complemented
10#36463C214yesmaximal, complemented
11#36194C212yesmaximal, complemented
12#36176C214yesmaximal, complemented
13#35759C214yesmaximal, complemented
14#34943C212yesmaximal, complemented
15#34925C211yesmaximal, characteristic
16#34206C214yesmaximal, complemented
17C4×#554C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#554C2212yes—
20C2×#532C2214yes—
21C2×#520C2214yes—
22#2868C2211yescharacteristic
23#2858C2212yes—
24#2857C2214yes—
25#2854C2218yes—
26#2804C2214yes—
27#2799C2218yes—
28#2750C22116yes—
29#2740C22116yes—
30#2700C22116yes—
31#2693C2218yes—
32#2681C22116yes—
33#2636C2218yes—
34#2616C2214yes—
35#2604C2218yes—
36#2581C2218yes—
37#2580C2218yes—
38#2573C22116yes—
39#2572C2218yes—
40#2571C2212yes—
41#2571C2214yes—
42#2563C2214yes—
43#2563C2214yes—
44#2557C22116yes—
45#2555C2214yes—
46#2551C2211yescharacteristic
47#2550C2218yes—
48#2547C2212yes—
49#2545C2214yes—
50#2541C2212yes—
51#2538C2214yes—
52#2538C2214yes—
53#2537C2211yescharacteristic
54#2532C22116yes—
55#2523C2218yes—
56#2513C2218yes—
57#2512C2218yes—
58#2509C2218yes—
59#2508C2218yes—
60#2503C2214yes—
61#2497C2218yes—
62#2495C2218yes—
63#2495C2218yes—
64#2492C2218yes—
65#2491C2214yes—
66#2490C2218yes—
67#2488C22116yes—
68#2487C2214yes—
69#2486C2218yes—
70#2466C22116yes—
71#2459C22116yes—
72#2445C2218yes—
73#2439C2218yes—
74#2428C2214yes—
75#2420C2218yes—
76#2420C2218yes—
77#2415C2218yes—
78#2398C2218yes—
79#2388C2214yes—
80#2370C2214yes—
81#2369C2214yes—
82#2369C2212yes—
83#2368C2218yes—
84#2367C22116yes—
85#2366C2212yes—
86#2366C2218yes—
87#2365C2218yes—
88#2360C2214yes—
89#2357C2218yes—
90#2355C2214yes—
91#2355C2212yes—
92#2354C2214yes—
93#2354C2214yes—
94#2352C2218yes—
95#2352C2218yes—
96#2349C2214yes—
97D4×C4⋊C4C2212yes—
98D4×C4⋊C4C2214yes—
99#2346C2214yes—
100#2345C2214yes—
101#2325C22116yes—
102#2311C2218yes—
103C4×C4⋊Q8C2212yes—
104C4×C4⋊Q8C2212yes—
105C4×#127C2211yescharacteristic
106C4×#125C2214yes—
107C4×C22⋊Q8C2214yes—
108C4×C22⋊Q8C2218yes—
109C4×#121C2214yes—
110C4×#121C2218yes—
111C4×#121C2218yes—
112#2285C2214yes—
113#2283C2218yes—
114C42×Q8C2211yescharacteristic
115C42×D4C2212yes—
116C42×D4C2214yes—
117C2×C4×C4⋊C4C2214yes—
118C22×#41C2314yes—
119#556—48no—
120#554—216no—
121#532—232no—
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2314yes—
125C2×C4⋊Q8C2312yes—
126C2×C4⋊Q8C2312yes—
127C2×#127C2311yescharacteristic
128C2×#127C2314yes—
129C2×#126C2318yes—
130C2×#125C2314yes—
131C2×#124C2318yes—
132C2×#124C2314yes—
133C2×#124C2314yes—
134C2×#123C23116yes—
135C2×C22⋊Q8C2314yes—
136C2×C22⋊Q8C2318yes—
137C2×C22⋊Q8C2314yes—
138C2×C22⋊Q8C2318yes—
139C2×#121C2318yes—
140C2×#121C2314yes—
141C2×#121C2314yes—
142C2×#121C2314yes—
143#520—416no—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2314yes—
147C2×C4×Q8C2314yes—
148C2×C4×Q8C2318yes—
149C2×C4×D4C2314yes—
150C2×C4×D4C2312yes—
151C2×C4×D4C2314yes—
152C2×C4×D4C2318yes—
153C2×C4×D4C2312yes—
154C2×C4×D4C2318yes—
155C2×C4×D4C2318yes—
156C2×C4×D4C23116yes—
157C2×C4×D4C2314yes—
158C2×C4×D4C2312yes—
159C2×C4×D4C2312yes—
160C2×#117C2314yes—
161C2×#117C2318yes—
162C22×C4⋊C4C2314yes—
163C22×C4⋊C4C2314yes—
164C22×C4⋊C4C2314yes—
165C22×C42C2314yes—
166C22×C42C2318yes—
167#400C2318yes—
168#400C23116yes—
169#400C2318yes—
170#400C2318yes—
171#400C23116yes—
172#399C2318yes—
173#398C23116yes—
174#397C23116yes—
175#397C23116yes—
176#397C2318yes—
177#397C23116yes—
178#396C23116yes—
179#396C23116yes—
180#395C2314yes—
181#395C2318yes—
182#394C2314yes—
183#394C2318yes—
184#394C23116yes—
185#394C23116yes—
186#394C2318yes—
187#393C23116yes—
188#393C23116yes—
189#393C2314yes—
190#392C2314yes—
191#392C2318yes—
192#391C2314yes—
193#391C2314yes—
194#391C23116yes—
195#391C23116yes—
196#391C2318yes—
197#391C2314yes—
198#391C2318yes—
199#390C2314yes—
200#390C2312yes—
201#390C2318yes—
202#390C2318yes—
203#390C23116yes—
204#390C2318yes—
205#390C2318yes—
206#390C2314yes—
207#389C2314yes—
208#389C2314yes—
209#389C2314yes—
210#389C2314yes—
211#389C2318yes—
212#389C2318yes—
213#389C2318yes—
214#389C2318yes—
215#389C23116yes—
216#389C2314yes—
217#389C2318yes—
218#389C2318yes—
219#389C2314yes—
220#388C23116yes—
221#388C2318yes—
222#388C23116yes—
223#388C23116yes—
224#388C23116yes—
225#388C23116yes—
226#388C23116yes—
227#388C23116yes—
228#387C2318yes—
229#387C2318yes—
230#387C2318yes—
231#387C2318yes—
232#387C23116yes—
233#387C23116yes—
234#387C2318yes—
235#387C23116yes—
236#386C23116yes—
237#386C23116yes—
238#386C23116yes—
239#386C23116yes—
240#386C2318yes—
241#386C2318yes—
242#385C23116yes—
243#385C2318yes—
244#385C2318yes—
245#385C2318yes—
246#385C2318yes—
247#384C2311yescharacteristic
248#384C2312yes—
249#384C2314yes—
250#384C2314yes—
251#383C23116yes—
252#382C2314yes—
253#382C2318yes—
254#382C2318yes—
255#382C2318yes—
256#382C2318yes—
257#381C23116yes—
258#381C23116yes—
259#380C2314yes—
260#380C2318yes—
261#380C2318yes—
262#380C2318yes—
263#380C2314yes—
264#380C2318yes—
265#380C2318yes—
266#380C2318yes—
267#380C2318yes—
268#380C2314yes—
269C4×C4⋊C4C2312yes—
270C4×C4⋊C4C2314yes—
271C4×C4⋊C4C2318yes—
272C4×C4⋊C4C2318yes—
273C4×C4⋊C4C2314yes—
274C4×C4⋊C4C2318yes—
275C4×C4⋊C4C2312yes—
276C4×C4⋊C4C2318yes—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2314yes—
279C4×C4⋊C4C2314yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2314yes—
282C4×C4⋊C4C2318yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2314yes—
285C4×C4⋊C4C2318yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2314yes—
289C4×C22⋊C4C2318yes—
290C4×C22⋊C4C23116yes—
291C4×C22⋊C4C2314yes—
292C4×C22⋊C4C2314yes—
293C4×C22⋊C4C23116yes—
294C4×C22⋊C4C2318yes—
295C4×C22⋊C4C2314yes—
296C4×C22⋊C4C2318yes—
297C4×C22⋊C4C2314yes—
298#376C2318yes—
299C2×#95C2318yes—
300C43C2312yes—
301C43C2311yescharacteristic
302C2×#41—232no—
303C2×#41—48no—
304C22×Q8C2411yescharacteristic
305C22×Q8C2414yes—
306C22×Q8C2414yes—
307C22×Q8C2414yes—
308C22×D4C2414yes—
309C22×D4C2414yes—
310C22×D4C2418yes—
311C22×D4C2412yes—
312C22×D4C2418yes—
313C22×D4C2412yes—
314C23×C4C2414yes—
315C23×C4C2414yes—
316C23×C4C2414yes—
317C23×C4C2418yes—
318C23×C4C2418yes—
319C23×C4C2414yes—
320C23×C4C2418yes—
321C4⋊Q8—216no—
322C4⋊Q8—28no—
323C4⋊Q8—28no—
324C4⋊Q8—44no—
325C4⋊Q8—82no—
326#127—216no—
327#127—24no—
328#126—416no—
329#125—216no—
330#124—232no—
331#124—48no—
332#124—84no—
333#123—816no—
334C22⋊Q8—216no—
335C22⋊Q8—232no—
336C22⋊Q8—48no—
337C22⋊Q8—88no—
338#121—216no—
339#121—216no—
340#121—232no—
341#121—48no—
342C4×Q8—24no—
343C4×Q8—28no—
344C4×Q8—48no—
345C4×Q8—48no—
346C4×Q8—416no—
347C4×D4—216no—
348C4×D4—232no—
349C4×D4—216no—
350C4×D4—28no—
351C4×D4—48no—
352C4×D4—416no—
353C4×D4—44no—
354C4×D4—416no—
355C4×D4—44no—
356C4×D4—432no—
357C4×D4#41116yes—
358#117—216no—
359#117—416no—
360C2×C4⋊C4—216no—
361C2×C4⋊C4—216no—
362C2×C4⋊C4—48no—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C2418yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C2414yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2418yes—
378C2×C4⋊C4C2418yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C2414yes—
383C2×C4⋊C4C2418yes—
384C2×C4⋊C4C24116yes—
385C2×C4⋊C4C2418yes—
386C2×C4⋊C4C2418yes—
387C2×C4⋊C4C2414yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2411yescharacteristic
390C2×C22⋊C4C2418yes—
391C2×C22⋊C4C2418yes—
392C2×C22⋊C4C2418yes—
393C2×C22⋊C4C2418yes—
394C2×C22⋊C4C2418yes—
395C2×C22⋊C4C2414yes—
396C2×C22⋊C4C24116yes—
397C2×C22⋊C4C2414yes—
398C2×C22⋊C4C2418yes—
399C2×C22⋊C4C2414yes—
400C2×C22⋊C4C24116yes—
401C2×C22⋊C4C2418yes—
402C2×C22⋊C4C24116yes—
403C2×C22⋊C4C2418yes—
404C2×C22⋊C4C2418yes—
405C2×C22⋊C4C24116yes—
406C2×C22⋊C4C2414yes—
407C2×C42—216no—
408C2×C42—416no—
409C2×C42C2418yes—
410C2×C42C2411yescharacteristic
411C2×C42C2414yes—
412C2×C42C2412yes—
413C2×C42C2414yes—
414C2×C42C2418yes—
415C2×C42C2414yes—
416C2×C42C2412yes—
417C2×C42C2414yes—
418C2×C42C2414yes—
419C2×C42C24116yes—
420C2×C42C2412yes—
421C2×C42C2411yescharacteristic
422C2×C42C2418yes—
423C2×C42C2418yes—
424C2×C42C2412yes—
425C2×C42C2411yescharacteristic
426#95C2414yes—
427#95C2418yes—
428#95C2418yes—
429#95C2418yes—
430#95C2414yes—
431#95C2418yes—
432#95C24116yes—
433#95C24116yes—
434#95C2418yes—
435#95C2418yes—
436#95C2418yes—
437#95C2414yes—
438#95C2418yes—
439#95C2414yes—
440#95C2414yes—
441#95C24116yes—
442#95C2414yes—
443#95C2412yes—
444#95C24116yes—
445#95C24116yes—
446#95C24116yes—
447#95C2418yes—
448C24C2514yes—
449C24C2514yes—
450#41—464no—
451C2×Q8—24no—
452C2×Q8—216no—
453C2×Q8—28no—
454C2×Q8—22no—
455C2×Q8—28no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×D4—216no—
462C2×D4—216no—
463C2×D4—216no—
464C2×D4—28no—
465C2×D4—216no—
466C2×D4—24no—
467C2×D4—28no—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—28no—
471C2×D4—48no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—48no—
475C2×D4—48no—
476C2×D4—42no—
477C2×D4C2×#4118yes—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—216no—
481C22×C4—28no—
482C22×C4—216no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—28no—
486C22×C4—28no—
487C22×C4—216no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—416no—
496C22×C4C2×#41116yes—
497C22×C4C2514yes—
498C22×C4C2514yes—
499C22×C4C2514yes—
500C22×C4C2518yes—
501C22×C4C2518yes—
502C22×C4C2514yes—
503C22×C4C2518yes—
504C22×C4C2512yes—
505C22×C4C2511yescharacteristic
506C22×C4C2514yes—
507C22×C4C2512yes—
508C22×C4C2512yes—
509C22×C4C2511yescharacteristic
510C22×C4C2511yescharacteristic
511C22×C4C2512yes—
512C4⋊C4—24no—
513C4⋊C4—216no—
514C4⋊C4—28no—
515C4⋊C4—216no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—216no—
520C4⋊C4—28no—
521C4⋊C4—216no—
522C4⋊C4—28no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—216no—
526C4⋊C4—28no—
527C4⋊C4—24no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—416no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4C2×#4118yes—
538C4⋊C4H(5,𝔽2)14yes—
539C22⋊C4—216no—
540C22⋊C4—216no—
541C22⋊C4—216no—
542C22⋊C4—216no—
543C22⋊C4—216no—
544C22⋊C4—28no—
545C22⋊C4—416no—
546C22⋊C4—48no—
547C22⋊C4—416no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—416no—
553C22⋊C4—48no—
554C22⋊C4—416no—
555C22⋊C4C2×#41116yes—
556C42—28no—
557C42—28no—
558C42—28no—
559C42—28no—
560C42—216no—
561C42—22no—
562C42—24no—
563C42—28no—
564C42—216no—
565C42—24no—
566C42—48no—
567C42—48no—
568C42—416no—
569C42—42no—
570C42—41no—
571C42C2×#4118yes—
572C422−1+414yes—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—28no—
580C23—24no—
581C23—24no—
582C23—28no—
583C23—24no—
584C23C22×#4118yes—
585C23C2611yescharacteristic
586Q8—416no—
587Q8—44no—
588Q8—88no—
589Q8—88no—
590D4—216no—
591D4—432no—
592D4—416no—
593D4—416no—
594D4—84no—
595D4—816no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—232no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—416no—
629C2×C4—416no—
630C2×C4—88no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4—168no—
634C2×C4C22×#4114yes—
635C2×C4C22×#4118yes—
636C2×C4C22×#4114yes—
637C2×C4C22×#4114yes—
638C2×C4C2×H(5,𝔽2)14yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×2−1+412yes—
641C2×C4C2×2−1+414yes—
642C2×C4#58514yes—
643C22—28no—
644C22—28no—
645C22—28no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—48no—
651C22—44no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22C23×#4112yes—
656C22C22×H(5,𝔽2)11yescharacteristic
657C22C22×2−1+411yescharacteristic
658C22C2×#58512yes—
659C22H(7,𝔽2)11yescharacteristic
660C4—24no—
661C4—24no—
662C4—22no—
663C4—28no—
664C4—44no—
665C4—44no—
666C4—48no—
667C4—41no—
668C4—48no—
669C4—44no—
670C4—48no—
671C4—44no—
672C4—42no—
673C4#348114yes—
674C4#350718yes—
675C2—48no—
676C2—84no—
677C2C2×#348111yescharacteristic
678C2C2×#350712yes—
679C2#6291711yescharacteristic
680C2#6299412yes—
681C2#6302411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radical#34925C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59864C2semidirect
2. #59673C2semidirect
3. #59257C2semidirect
4. #59247C2semidirect
5. #45115C2semidirect
6. #43804C2semidirect
7. #40149C2semidirect
8. #39470C2semidirect
9. #36594C2semidirect
10. #36463C2semidirect
11. #36194C2semidirect
12. #36176C2semidirect
13. #35759C2semidirect
14. #34943C2semidirect
16. #34206C2semidirect
17. C4×#554C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#596732
#592572
#592472
#451152
#438042
#401492
#394702
#365942
#364632
#361942
#361762
#357592
#349432
#349252
#342062
C4×#5542