Finite Group Databasebeta

← #7739326

Subgroups — #7739326

Order 512.

5260 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59924C212yesmaximal, complemented
2#59672C212yesmaximal, complemented
3#59458C214yesmaximal, complemented
4#51601C218yesmaximal, complemented
5#44946C218yesmaximal, complemented
6#44941C218yesmaximal, complemented
7#39369C214yesmaximal, complemented
8#38964C214yesmaximal, complemented
9#37306C214yesmaximal, complemented
10#36407C214yesmaximal, complemented
11#36180C214yesmaximal, complemented
12#35532C214yesmaximal, complemented
13#35275C212yesmaximal, complemented
14#34978C212yesmaximal, complemented
15#34368C211yesmaximal, characteristic
16#34207C211yesmaximal, characteristic
17#34155C211yesmaximal, characteristic
18C2×#554C2218yes—
19C2×#544C2214yes—
20C2×#544C2214yes—
21C2×#532C2211yescharacteristic
22C2×#520C2212yes—
23#2868C2214yes—
24#2859C2214yes—
25#2857C2218yes—
26#2811C2218yes—
27#2809C2218yes—
28#2804C2218yes—
29#2763C22116yes—
30#2720C22116yes—
31#2720C22116yes—
32#2692C22116yes—
33#2692C22116yes—
34#2683C2218yes—
35#2677C22116yes—
36#2660C2218yes—
37#2628C2218yes—
38#2612C2218yes—
39#2580C2218yes—
40#2574C22116yes—
41#2571C2212yes—
42#2567C2218yes—
43#2567C2218yes—
44#2567C2218yes—
45#2566C2218yes—
46#2565C2218yes—
47#2556C22116yes—
48#2552C2214yes—
49#2552C2218yes—
50#2552C2214yes—
51#2552C2212yes—
52#2552C2212yes—
53#2551C2214yes—
54#2551C2218yes—
55#2551C2214yes—
56#2551C2212yes—
57#2542C2218yes—
58#2542C2218yes—
59#2508C2218yes—
60#2506C2218yes—
61#2502C22116yes—
62#2454C2218yes—
63#2452C2218yes—
64#2452C2218yes—
65#2452C2218yes—
66#2449C22116yes—
67#2449C22116yes—
68#2441C2218yes—
69#2441C2214yes—
70#2441C2214yes—
71#2440C2218yes—
72#2440C22116yes—
73#2440C2218yes—
74#2440C2218yes—
75#2440C2218yes—
76#2439C2218yes—
77#2439C2214yes—
78#2439C2214yes—
79#2396C2218yes—
80#2379C2214yes—
81#2372C2214yes—
82#2370C2218yes—
83#2369C2211yescharacteristic
84#2369C2212yes—
85#2369C2218yes—
86#2369C2211yescharacteristic
87#2366C2212yes—
88#2366C2212yes—
89#2363C2214yes—
90#2363C2214yes—
91#2361C2218yes—
92#2354C2214yes—
93#2354C2214yes—
94D4×C4⋊C4C2214yes—
95D4×C4⋊C4C2214yes—
96D4×C4⋊C4C2218yes—
97#2326C2214yes—
98#2324C2218yes—
99#2317C2214yes—
100#2317C2214yes—
101#2311C2214yes—
102#2311C2212yes—
103C4×C4⋊Q8C2212yes—
104C4×C4⋊Q8C2214yes—
105C4×C4⋊Q8C2214yes—
106C4×#127C2212yes—
107C4×#125C2214yes—
108C4×#125C2214yes—
109#2297C2214yes—
110#2296C2214yes—
111#2293C2214yes—
112#2285C2212yes—
113C2×#384C2214yes—
114C2×#384C2211yescharacteristic
115C42×D4C2214yes—
116C4×#117C2212yes—
117C4×#117C2211yescharacteristic
118C22×#41C2312yes—
119#554—432no—
120#544—416no—
121#544—416no—
122#532Q8116yescomplemented
123C2×C4⋊Q8C2314yes—
124C2×C4⋊Q8C2314yes—
125C2×C4⋊Q8C2318yes—
126C2×C4⋊Q8C2318yes—
127C2×C4⋊Q8C2318yes—
128C2×C4⋊Q8C2318yes—
129C2×C4⋊Q8C2318yes—
130C2×C4⋊Q8C2311yescharacteristic
131C2×#127C2311yescharacteristic
132C2×#125C2314yes—
133C2×#125C2314yes—
134C2×#125C2318yes—
135C2×#125C2318yes—
136C2×#125C2318yes—
137C2×#124C2312yes—
138C2×C22⋊Q8C23116yes—
139C2×C22⋊Q8C23116yes—
140C2×C22⋊Q8C23116yes—
141C2×C22⋊Q8C23116yes—
142C2×C22⋊Q8C23116yes—
143C2×#121C2314yes—
144#520—48no—
145C2×C4×Q8C2318yes—
146C2×C4×Q8C2314yes—
147C2×C4×D4C2314yes—
148C2×C4×D4C2314yes—
149C2×C4×D4C2318yes—
150C2×C4×D4C2314yes—
151C2×C4×D4C2318yes—
152C2×C4×D4C2318yes—
153C2×C4×D4C2314yes—
154C2×#117C2312yes—
155C2×#117C2314yes—
156C2×#117C2312yes—
157C2×#117C2311yescharacteristic
158C2×#117C2311yescharacteristic
159C2×#117C2314yes—
160C2×#117C2314yes—
161C2×#117C2314yes—
162C22×C4⋊C4C2314yes—
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2314yes—
165C22×C4⋊C4C2314yes—
166C22×C4⋊C4C23116yes—
167C22×C42C2314yes—
168C22×C42C2311yescharacteristic
169#400C2318yes—
170#400C23116yes—
171#399C23116yes—
172#399C2318yes—
173#399C23116yes—
174#398C23116yes—
175#398C23116yes—
176#398C23116yes—
177#398C2318yes—
178#398C2318yes—
179#398C23116yes—
180#396C2318yes—
181#396C23116yes—
182#396C2318yes—
183#396C23116yes—
184#395C23116yes—
185#395C23116yes—
186#395C23116yes—
187#395C23116yes—
188#395C2318yes—
189#395C23116yes—
190#393C23116yes—
191#393C23116yes—
192#393C2318yes—
193#391C2314yes—
194#391C2318yes—
195#391C2318yes—
196#391C2314yes—
197#391C2318yes—
198#391C2314yes—
199#391C23116yes—
200#391C23116yes—
201#390C2314yes—
202#390C2318yes—
203#390C2314yes—
204#390C2318yes—
205#390C2318yes—
206#390C2314yes—
207#390C2314yes—
208#390C2314yes—
209#389C2314yes—
210#389C2314yes—
211#389C2314yes—
212#389C2314yes—
213#389C2314yes—
214#389C2314yes—
215#389C2318yes—
216#389C2318yes—
217#389C2314yes—
218#389C2318yes—
219#389C2314yes—
220#389C2314yes—
221#389C2318yes—
222#389C2318yes—
223#389C2318yes—
224#389C23116yes—
225#389C2318yes—
226#389C2318yes—
227#389C2318yes—
228#389C23116yes—
229#389C2318yes—
230#389C2314yes—
231#388C23116yes—
232#388C23116yes—
233#388C2318yes—
234#388C2318yes—
235#387C2318yes—
236#387C2318yes—
237#387C2318yes—
238#387C2318yes—
239#387C23116yes—
240#387C23116yes—
241#386C2318yes—
242#386C2318yes—
243#386C2318yes—
244#386C2318yes—
245#386C2318yes—
246#385C23116yes—
247#385C23116yes—
248#385C23116yes—
249#385C23116yes—
250#385C23116yes—
251#385C23116yes—
252#385C23116yes—
253#384C2312yes—
254#384C2314yes—
255#384C2314yes—
256#384C2318yes—
257#384C2318yes—
258#384C2318yes—
259#384C2312yes—
260#384C2318yes—
261#384C2312yes—
262#384C2318yes—
263#384C2314yes—
264#384C2312yes—
265#384C2318yes—
266#382C2314yes—
267#382C2314yes—
268#382C2318yes—
269#382C2318yes—
270#382C2314yes—
271#380C2314yes—
272#380C2318yes—
273#380C2318yes—
274#380C23116yes—
275#380C2314yes—
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2318yes—
279C4×C4⋊C4C2314yes—
280C4×C4⋊C4C2318yes—
281C4×C4⋊C4C2314yes—
282C4×C4⋊C4C2318yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2318yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C2318yes—
288C4×C4⋊C4C2314yes—
289C4×C4⋊C4C2318yes—
290C4×C4⋊C4C2312yes—
291C4×C4⋊C4C2314yes—
292C4×C4⋊C4C2312yes—
293C4×C22⋊C4C2318yes—
294C4×C22⋊C4C2314yes—
295C4×C22⋊C4C2314yes—
296C4×C22⋊C4C2318yes—
297C4×C22⋊C4C2314yes—
298C4×C22⋊C4C2318yes—
299#376C2312yes—
300#376C2314yes—
301C43C2312yes—
302C43C2314yes—
303C2×#41—44no—
304C2×#41—44no—
305C2×#41C2×Q8116yes—
306C22×Q8C2418yes—
307C22×Q8C24116yes—
308C22×Q8C2412yes—
309C22×D4C2414yes—
310C22×D4C2414yes—
311C22×D4C2412yes—
312C23×C4C2418yes—
313C23×C4C2412yes—
314C23×C4C2414yes—
315C23×C4C2411yescharacteristic
316C23×C4C2418yes—
317C23×C4C2414yes—
318C23×C4C2418yes—
319C4⋊Q8—216no—
320C4⋊Q8—48no—
321C4⋊Q8—416no—
322C4⋊Q8—416no—
323C4⋊Q8—416no—
324C4⋊Q8—416no—
325C4⋊Q8—416no—
326C4⋊Q8C2×Q818yes—
327#127C2×Q818yes—
328#125—216no—
329#125—232no—
330#125—216no—
331#125—416no—
332#125—416no—
333#124C2×Q8116yes—
334C22⋊Q8—432no—
335C22⋊Q8—432no—
336C22⋊Q8—432no—
337C22⋊Q8—432no—
338C22⋊Q8—432no—
339#121C2×Q8132yes—
340C4×Q8—48no—
341C4×Q8—416no—
342C4×D4—216no—
343C4×D4—216no—
344C4×D4—48no—
345C4×D4—416no—
346C4×D4—416no—
347C4×D4—416no—
348C4×D4C2×Q8132yes—
349#117—24no—
350#117—216no—
351#117—28no—
352#117—28no—
353#117—48no—
354#117—48no—
355#117—48no—
356#117—42no—
357C2×C4⋊C4—48no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—48no—
360C2×C4⋊C4—48no—
361C2×C4⋊C4—432no—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2418yes—
367C2×C4⋊C4C24116yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2414yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C24116yes—
375C2×C4⋊C4C2418yes—
376C2×C4⋊C4C2418yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C2418yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C2414yes—
382C2×C4⋊C4C24116yes—
383C2×C4⋊C4C2418yes—
384C2×C4⋊C4C2418yes—
385C2×C4⋊C4C2414yes—
386C2×C4⋊C4C2414yes—
387C2×C4⋊C4C2414yes—
388C2×C4⋊C4C2414yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C24116yes—
392C2×C4⋊C4C24116yes—
393C2×C4⋊C4C2418yes—
394C2×C4⋊C4C2414yes—
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2412yes—
398C2×C22⋊C4C2412yes—
399C2×C22⋊C4C2418yes—
400C2×C22⋊C4C2414yes—
401C2×C22⋊C4C2414yes—
402C2×C22⋊C4C2418yes—
403C2×C22⋊C4C2418yes—
404C2×C22⋊C4C2418yes—
405C2×C22⋊C4C2418yes—
406C2×C22⋊C4C2412yes—
407C2×C22⋊C4C2418yes—
408C2×C22⋊C4C24116yes—
409C2×C22⋊C4C24116yes—
410C2×C22⋊C4C2418yes—
411C2×C42—48no—
412C2×C42C2×Q818yes—
413C2×C42C2414yes—
414C2×C42C2412yes—
415C2×C42C2418yes—
416C2×C42C2418yes—
417C2×C42C2414yes—
418C2×C42C2414yes—
419C2×C42C2414yes—
420C2×C42C2414yes—
421C2×C42C2414yes—
422C2×C42C2414yes—
423C2×C42C2414yes—
424C2×C42C2412yes—
425C2×C42C2418yes—
426C2×C42C2414yes—
427C2×C42C2414yes—
428C2×C42C2412yes—
429C2×C42C2411yescharacteristic
430C2×C42C2411yescharacteristic
431#95C2414yes—
432#95C2414yes—
433#95C2414yes—
434#95C2414yes—
435#95C2414yes—
436#95C2414yes—
437#95C2418yes—
438#95C2418yes—
439#95C2418yes—
440#95C2418yes—
441#95C24116yes—
442#95C2418yes—
443#95C2418yes—
444#95C24116yes—
445#95C24116yes—
446#95C2418yes—
447#95C2418yes—
448#95C2418yes—
449#95C2418yes—
450#95C24116yes—
451C24C2511yescharacteristic
452C24C2514yes—
453#41—432no—
454C2×Q8—216no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—48no—
458C2×Q8—48no—
459C2×Q8—416no—
460C2×Q8—416no—
461C2×Q8—416no—
462C2×Q8C22×Q818yes—
463C2×D4—28no—
464C2×D4—28no—
465C2×D4—44no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4C22×Q8116yes—
470C2×D4C22×Q8116yes—
471C2×D4C22×Q818yes—
472C22×C4—24no—
473C22×C4—28no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—28no—
477C22×C4—22no—
478C22×C4—216no—
479C22×C4—216no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—416no—
484C22×C4—416no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4C22×Q818yes—
488C22×C4C22×Q814yes—
489C22×C4C22×Q8116yes—
490C22×C4C2514yes—
491C22×C4C2514yes—
492C22×C4C2518yes—
493C22×C4C2514yes—
494C22×C4C2512yes—
495C22×C4C2518yes—
496C22×C4C2518yes—
497C22×C4C2511yescharacteristic
498C22×C4C2512yes—
499C22×C4C2514yes—
500C22×C4C2512yes—
501C22×C4C2514yes—
502C22×C4C2514yes—
503C22×C4C2512yes—
504C22×C4C2511yescharacteristic
505C4⋊C4—22no—
506C4⋊C4—28no—
507C4⋊C4—216no—
508C4⋊C4—28no—
509C4⋊C4—216no—
510C4⋊C4—232no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—232no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—24no—
518C4⋊C4—216no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—416no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—416no—
535C4⋊C4—416no—
536C4⋊C4—48no—
537C4⋊C4C22×Q8116yes—
538C4⋊C4H(5,𝔽2)14yes—
539C4⋊C4H(5,𝔽2)18yes—
540C4⋊C42−1+418yes—
541C22⋊C4—24no—
542C22⋊C4—216no—
543C22⋊C4—28no—
544C22⋊C4—216no—
545C22⋊C4—28no—
546C22⋊C4—216no—
547C22⋊C4—48no—
548C22⋊C4—42no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—416no—
552C22⋊C4—416no—
553C22⋊C4C22×Q8132yes—
554C42—24no—
555C42—216no—
556C42—28no—
557C42—28no—
558C42—216no—
559C42—28no—
560C42—28no—
561C42—28no—
562C42—28no—
563C42—28no—
564C42—28no—
565C42—44no—
566C42—44no—
567C42—48no—
568C42—42no—
569C42C22×Q818yes—
570C42C22×Q814yes—
571C42C22×Q814yes—
572C23—22no—
573C23—24no—
574C23—21no—
575C23—28no—
576C23—22no—
577C23—28no—
578C23—24no—
579C23—21no—
580C23C23×Q818yes—
581C23C23×Q812yes—
582C23C2611yescharacteristic
583Q8—48no—
584Q8—816no—
585Q8—1616no—
586D4—232no—
587D4—48no—
588D4—416no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—21no—
600C2×C4—232no—
601C2×C4—28no—
602C2×C4—216no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—216no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—432no—
623C2×C4—168no—
624C2×C4—164no—
625C2×C4—168no—
626C2×C4C23×Q814yes—
627C2×C4C23×Q812yes—
628C2×C4C23×Q818yes—
629C2×C4C23×Q812yes—
630C2×C4C23×Q814yes—
631C2×C4C2×H(5,𝔽2)14yes—
632C2×C4C2×H(5,𝔽2)18yes—
633C2×C4C2×H(5,𝔽2)12yes—
634C2×C4C2×H(5,𝔽2)14yes—
635C2×C4C2×2−1+418yes—
636C2×C4C2×2−1+414yes—
637C2×C4#58512yes—
638C22—22no—
639C22—28no—
640C22—24no—
641C22—216no—
642C22—48no—
643C22—42no—
644C22—44no—
645C22—41no—
646C22—41no—
647C22—44no—
648C22C24×Q811yescharacteristic
649C22C22×H(5,𝔽2)12yes—
650C22C22×H(5,𝔽2)11yescharacteristic
651C22C22×2−1+412yes—
652C22C2×#58511yescharacteristic
653C4—28no—
654C4—28no—
655C4—22no—
656C4—28no—
657C4—24no—
658C4—24no—
659C4—48no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—48no—
664C4—48no—
665C4—44no—
666C4#350114yes—
667C4#350318yes—
668C2—42no—
669C2—48no—
670C2C2×#350111yescharacteristic
671C2C2×#350312yes—
672C2#6301312yes—
673C2#6303711yescharacteristic
674C2#6304011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#532C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59924C2semidirect
2. #59672C2semidirect
3. #59458C2semidirect
4. #51601C2semidirect
5. #44946C2semidirect
6. #44941C2semidirect
7. #39369C2semidirect
8. #38964C2semidirect
9. #37306C2semidirect
10. #36407C2semidirect
11. #36180C2semidirect
12. #35532C2semidirect
13. #35275C2semidirect
14. #34978C2semidirect
122. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#599242
#596722
#594582
#516012
#449462
#449412
#393692
#389642
#373062
#364072
#361802
#355322
#352752
#349782
#343682
#342072
#341552