Finite Group Databasebeta

← #35577305

Subgroups — #35577305

Order 21248.

2322 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128296C212yesmaximal, complemented
2#24915789C212yesmaximal, complemented
3#24915762C212yesmaximal, complemented
4#24913577C211yesmaximal, characteristic, complemented
5#24913207C212yesmaximal, complemented
6#24912548C214yesmaximal, complemented
7#24912524C214yesmaximal, complemented
8#24912131C214yesmaximal, complemented
9#24911425C214yesmaximal, complemented
10#24910168C212yesmaximal, complemented
11#121×Dic83C212yesmaximal, complemented
12C2×#19127552C211yesmaximal, characteristic, complemented
13C83×#2656C211yesmaximal, characteristic
14#19128296—216no—
15C2×#14581785C2214yes—
16C2×#14581782C2214yes—
17C2×#14581779C2212yes—
18C2×#14581773C2214yes—
19C2×#14581773C2214yes—
20C2×#14581771C2212yes—
21C2×#14581769C2214yes—
22C2×D4×Dic83C2214yes—
23C2×D4×Dic83C2212yes—
24C22×Dic83⋊C4C2211yescharacteristic
25C22×Dic83⋊C4C2211yescharacteristic
26C22×C4×Dic83C2211yescharacteristic
27C22×C4×Dic83C2211yescharacteristic
28C22×C4×Dic83C2211yescharacteristic
29#19127633C2214yes—
30#19127632C2214yes—
31#19127624C2214yes—
32#19127624C2214yes—
33#19127622C2214yes—
34#19127615C2214yes—
35#19127606C2212yes—
36#19127604C2212yes—
37#19127604C2212yes—
38#19127604C2212yes—
39#19127604C2212yes—
40#19127600C2212yes—
41#19127597C2212yes—
42#19127597C2212yes—
43#19127595C2214yes—
44#19127595C2214yes—
45#19127594C2214yes—
46#19127594C2214yes—
47#19127589C2214yes—
48#19127589C2214yes—
49#19127587C2214yes—
50#19127581C2214yes—
51#19127581C2214yes—
52#19127581C2214yes—
53#19127576C2214yes—
54#19127566C2212yes—
55#19127561C2212yes—
56#19127561C2212yes—
57C4⋊C4×Dic83C2212yes—
58#19127552C2214yes—
59#19127552C2214yes—
60C22⋊C4×Dic83C2214yes—
61C2×C83⋊#95C2211yescharacteristic
62C2×#121×C83C2212yes—
63C2×#121×C83C2212yes—
64C2×#121×C83C2212yes—
65#399×C83C2214yes—
66#392×C83C2214yes—
67C2×#95×C83C2211yescharacteristic
68C22×Dic166C2312yes—
69C23×Dic83C2311yescharacteristic
70C23×Dic83C2311yescharacteristic
71C23×Dic83C2314yes—
72C23×Dic83C2311yescharacteristic
73C23×Dic83C2311yescharacteristic
74#14581785—216no—
75#14581782—216no—
76#14581779—44no—
77#14581773—216no—
78#14581773—216no—
79#14581771—28no—
80#14581769—216no—
81D4×Dic83—216no—
82D4×Dic83—44no—
83C2×C4⋊Dic83C2312yes—
84C2×C4⋊Dic83C2314yes—
85C2×C4⋊Dic83C2312yes—
86C2×C4⋊Dic83C2312yes—
87C2×C4⋊Dic83C2312yes—
88C2×C4⋊Dic83C2312yes—
89C2×Dic83⋊C4—24no—
90C2×Dic83⋊C4D418yescomplemented
91C2×Dic83⋊C4C2314yes—
92C2×Dic83⋊C4C2314yes—
93C2×Dic83⋊C4C2314yes—
94C2×Dic83⋊C4C2314yes—
95C2×Dic83⋊C4C2314yes—
96C2×C22⋊Dic83C2314yes—
97C2×C22⋊Dic83C2314yes—
98C2×C22⋊Dic83C2314yes—
99C2×C22⋊Dic83C2314yes—
100C2×C22⋊Dic83C2314yes—
101C2×C22⋊Dic83C2314yes—
102C2×C22⋊Dic83C2314yes—
103C2×C22⋊Dic83C2314yes—
104C2×C4×Dic83—24no—
105C2×C4×Dic83—42no—
106C2×C4×Dic83—42no—
107C2×C4×Dic83C2312yes—
108C2×C4×Dic83C2312yes—
109C2×C4×Dic83C2312yes—
110C2×C4×Dic83C2312yes—
111C2×C4×Dic83C2314yes—
112C2×C4×Dic83C2312yes—
113C2×C4×Dic83C2312yes—
114C83⋊#95C2314yes—
115C83⋊#95C2314yes—
116C83⋊#95C2314yes—
117C83⋊#95C2314yes—
118C83⋊#95C2314yes—
119C83⋊#95C2314yes—
120C83⋊#95C2314yes—
121C22×D4×C83C2314yes—
122C22×D4×C83C2312yes—
123C22×D4×C83C2312yes—
124C22×D4×C83C2312yes—
125C23×C4×C83C2311yescharacteristic
126C23×C4×C83C2311yescharacteristic
127C23×C4×C83C2311yescharacteristic
128#121×C83—28no—
129#121×C83—28no—
130#121×C83—28no—
131C2×C4⋊C4×C83C2312yes—
132C2×C4⋊C4×C83C2312yes—
133C2×C4⋊C4×C83C2312yes—
134C2×C22⋊C4×C83C2314yes—
135C2×C22⋊C4×C83C2314yes—
136C2×C22⋊C4×C83C2314yes—
137#95×C83C2314yes—
138C2×Dic166—24no—
139C2×Dic166—42no—
140C2×Dic166—42no—
141C22×Dic83—22no—
142C22×Dic83—22no—
143C22×Dic83—22no—
144C22×Dic83—22no—
145C22×Dic83—22no—
146C22×Dic83—22no—
147C22×Dic83—28no—
148C22×Dic83—41no—
149C22×Dic83—41no—
150C22×Dic83—41no—
151C22×Dic83—41no—
152C22×Dic83—44no—
153C22×Dic83—44no—
154C22×Dic83C2×D414yes—
155C22×Dic83C2×D414yes—
156C22×Dic83C2414yes—
157C22×Dic83C2412yes—
158C22×Dic83C2414yes—
159C22×Dic83C2412yes—
160C22×Dic83C2412yes—
161C22×Dic83C2411yescharacteristic
162C22×Dic83C2411yescharacteristic
163C4⋊Dic83—24no—
164C4⋊Dic83—28no—
165C4⋊Dic83—42no—
166C4⋊Dic83—42no—
167C4⋊Dic83—42no—
168C4⋊Dic83—42no—
169Dic83⋊C4—28no—
170Dic83⋊C4—28no—
171Dic83⋊C4—28no—
172Dic83⋊C4—28no—
173Dic83⋊C4C2×D4116yes—
174C22⋊Dic83—28no—
175C22⋊Dic83—28no—
176C22⋊Dic83—28no—
177C22⋊Dic83—28no—
178C22⋊Dic83—28no—
179C22⋊Dic83—28no—
180C22⋊Dic83—44no—
181C22⋊Dic83—44no—
182C4×Dic83—24no—
183C4×Dic83—24no—
184C4×Dic83—28no—
185C4×Dic83—24no—
186C4×Dic83—24no—
187C4×Dic83—42no—
188C4×Dic83—42no—
189C24×C83C2414yes—
190C24×C83C2411yescharacteristic
191C2×D4×C83—28no—
192C2×D4×C83—28no—
193C2×D4×C83—28no—
194C2×D4×C83—24no—
195C2×D4×C83—24no—
196C2×D4×C83—24no—
197C2×D4×C83—42no—
198C2×D4×C83—42no—
199C2×D4×C83—42no—
200C2×D4×C83—42no—
201C2×D4×C83—42no—
202C2×D4×C83—42no—
203C22×C4×C83—41no—
204C22×C4×C83—41no—
205C22×C4×C83—41no—
206C22×C4×C83—41no—
207C22×C4×C83—41no—
208C22×C4×C83—41no—
209C22×C4×C83C2×D414yes—
210C22×C4×C83#4114yes—
211C22×C4×C83#4114yes—
212C22×C4×C83C2414yes—
213C22×C4×C83C2412yes—
214C22×C4×C83C2412yes—
215C22×C4×C83C2412yes—
216C4⋊C4×C83—24no—
217C4⋊C4×C83—24no—
218C4⋊C4×C83—24no—
219C22⋊C4×C83—28no—
220C22⋊C4×C83—28no—
221C22⋊C4×C83—28no—
222C23×C83—21no—
223C23×C83—21no—
224C23×C83—21no—
225C23×C83—21no—
226C23×C83—24no—
227C23×C83—24no—
228C23×C83—24no—
229C23×C83—24no—
230C23×C83—24no—
231C23×C83—24no—
232C23×C83—24no—
233C23×C83C22×D412yes—
234C23×C83C2×#4112yes—
235C23×C83C2×#4112yes—
236C23×C83C2511yescharacteristic
237D4×C83—48no—
238D4×C83—416no—
239D4×C83—84no—
240D4×C83—84no—
241C2×C4×C83—22no—
242C2×C4×C83—22no—
243C2×C4×C83—22no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—24no—
248C2×C4×C83—24no—
249C2×C4×C83—24no—
250C2×C4×C83—44no—
251C2×C4×C83—44no—
252C2×C4×C83—44no—
253C2×C4×C83C22×D414yes—
254C2×C4×C83C2×#4114yes—
255C2×C4×C83C2×#4114yes—
256C2×Dic83—28no—
257C2×Dic83—28no—
258C2×Dic83—22no—
259C2×Dic83—21no—
260C2×Dic83—21no—
261C2×Dic83—22no—
262C2×Dic83—22no—
263C2×Dic83—21no—
264C2×Dic83—21no—
265C2×Dic83—22no—
266C2×Dic83—24no—
267C2×Dic83—24no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—24no—
271C2×Dic83—24no—
272C2×Dic83—22no—
273C2×Dic83—24no—
274C2×Dic83—24no—
275C2×Dic83—44no—
276C2×Dic83—44no—
277C2×Dic83—88no—
278C2×Dic83C22×D414yes—
279C2×Dic83C22×D414yes—
280C2×Dic83H(5,𝔽2)12yes—
281C2×Dic832−1+412yes—
282Dic166—84no—
283C22×C83—22no—
284C22×C83—22no—
285C22×C83—22no—
286C22×C83—22no—
287C22×C83—22no—
288C22×C83—22no—
289C22×C83—44no—
290C22×C83—44no—
291C22×C83—44no—
292C22×C83—44no—
293C22×C83—44no—
294C22×C83—44no—
295C22×C83—44no—
296C22×C83#53214yes—
297C22×C83C23×D411yescharacteristic
298C22×C83C22×#4111yescharacteristic
299C22×C83C22×#4111yescharacteristic
300C22×C83C2×H(5,𝔽2)11yescharacteristic
301C22×C83C2×2−1+411yescharacteristic
302C22×C83#58511yescharacteristic
303C22×C83#58511yescharacteristic
304C4×C83—24no—
305C4×C83—24no—
306C4×C83—24no—
307C4×C83—44no—
308Dic83—24no—
309Dic83—22no—
310Dic83—22no—
311Dic83—24no—
312Dic83—42no—
313Dic83—44no—
314Dic83—44no—
315#35004—831nomaximal, Sylow
316C2×C83—24no—
317C2×C83—84no—
318C2×C83C2×#53211yescharacteristic
319C2×C83#345911yescharacteristic
320C2×C83#346111yescharacteristic
321C2×C83#349111yescharacteristic
322C2×C83#349211yescharacteristic
323C2×C83#349611yescharacteristic
324C2×C83#349711yescharacteristic
325C2×#539—832no—
326#2854—832no—
327#2850—832no—
328#2659—831no—
329#2656D8311yescharacteristic
330#2618—832no—
331#2556—834no—
332#2554—834no—
333#2519—834no—
334#2461—834no—
335#2325—832no—
336C4×#121—832no—
337C2×C4×C4⋊C4—831no—
338C83#3500411yescharacteristic, complemented, Sylow
339#539—16616no—
340C2×#126—834no—
341C2×#125—834no—
342C2×#124—832no—
343C2×#123—834no—
344C2×#123—834no—
345C2×C22⋊Q8—832no—
346C2×C22⋊Q8—834no—
347C2×#121C2×D8312yes—
348C2×#121C2×D8312yes—
349C2×#121C2×D8312yes—
350C2×C4×D4—832no—
351C2×C4×D4—834no—
352C22×C4⋊C4—831no—
353C22×C4⋊C4—831no—
354C22×C42—831no—
355C22×C42—831no—
356C22×C42—831no—
357#399C2×D8314yes—
358#398—834no—
359#398—834no—
360#396—834no—
361#396—834no—
362#395—834no—
363#394—834no—
364#392C2×D8314yes—
365#391—832no—
366#390—832no—
367#390—832no—
368#390—832no—
369#390—832no—
370#389—832no—
371#389—832no—
372#389—832no—
373#388—834no—
374#388—834no—
375#388—834no—
376#388—834no—
377#388—834no—
378#388—834no—
379#387—834no—
380#386—834no—
381#386—834no—
382#386—834no—
383#385—834no—
384#382—832no—
385#380—832no—
386#380—832no—
387C4×C4⋊C4—832no—
388C4×C4⋊C4—834no—
389C4×C4⋊C4—834no—
390C4×C22⋊C4—834no—
391C2×#95—831no—
392C2×#95C2×D8311yescharacteristic
393C22×Q8—832no—
394C22×D4C22×D8314yes—
395C22×D4C22×D8312yes—
396C22×D4C22×D8312yes—
397C22×D4C22×D8312yes—
398C23×C4—831no—
399C23×C4—831no—
400C23×C4—834no—
401C23×C4—831no—
402C23×C4—831no—
403C23×C4C22×D8311yescharacteristic
404C23×C4C22×D8311yescharacteristic
405C23×C4C22×D8311yescharacteristic
406#126—16616no—
407#125—16616no—
408#124—3324no—
409#123—16616no—
410#123—16616no—
411C22⋊Q8—16616no—
412C22⋊Q8—1668no—
413#121—28no—
414#121—28no—
415#121—28no—
416C4×D4—3324no—
417C4×D4—16616no—
418C2×C4⋊C4—832no—
419C2×C4⋊C4—832no—
420C2×C4⋊C4—832no—
421C2×C4⋊C4—832no—
422C2×C4⋊C4—838no—
423C2×C4⋊C4—834no—
424C2×C4⋊C4—834no—
425C2×C4⋊C4—834no—
426C2×C4⋊C4—834no—
427C2×C4⋊C4—834no—
428C2×C4⋊C4—832no—
429C2×C4⋊C4—834no—
430C2×C4⋊C4—1664no—
431C2×C4⋊C4C22×D8312yes—
432C2×C4⋊C4C22×D8312yes—
433C2×C4⋊C4C22×D8312yes—
434C2×C22⋊C4—834no—
435C2×C22⋊C4—834no—
436C2×C22⋊C4—834no—
437C2×C22⋊C4—834no—
438C2×C22⋊C4—834no—
439C2×C22⋊C4—834no—
440C2×C22⋊C4—834no—
441C2×C22⋊C4—834no—
442C2×C22⋊C4C22×D8314yes—
443C2×C22⋊C4C22×D8314yes—
444C2×C22⋊C4C22×D8314yes—
445C2×C42—3322no—
446C2×C42—3322no—
447C2×C42—832no—
448C2×C42—832no—
449C2×C42—834no—
450C2×C42—832no—
451C2×C42—832no—
452C2×C42—832no—
453C2×C42—832no—
454C2×C42—1664no—
455#95—834no—
456#95—834no—
457#95—834no—
458#95—834no—
459#95—834no—
460#95—834no—
461#95—834no—
462#95C22×D8314yes—
463C24C23×D8314yes—
464C24C23×D8311yescharacteristic
465C2×Q8—3322no—
466C2×Q8—3322no—
467C2×Q8—1664no—
468C2×D4—28no—
469C2×D4—28no—
470C2×D4—28no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—3321no—
487C22×C4—3321no—
488C22×C4—3321no—
489C22×C4—3321no—
490C22×C4—3324no—
491C22×C4—3324no—
492C22×C4—834no—
493C22×C4—834no—
494C22×C4—831no—
495C22×C4—831no—
496C22×C4—832no—
497C22×C4—832no—
498C22×C4—832no—
499C22×C4—834no—
500C22×C4—834no—
501C22×C4—1662no—
502C22×C4—1668no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1662no—
506C22×C4—1662no—
507C22×C4—1662no—
508C22×C4D4×D8314yes—
509C22×C4#1305058914yes—
510C22×C4#1305058914yes—
511C22×C4C23×D8314yes—
512C22×C4C23×D8312yes—
513C22×C4C23×D8312yes—
514C22×C4C23×D8312yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—3322no—
519C4⋊C4—3322no—
520C4⋊C4—3322no—
521C4⋊C4—3322no—
522C4⋊C4—8316no—
523C4⋊C4—1668no—
524C4⋊C4—1668no—
525C4⋊C4—1668no—
526C4⋊C4—1668no—
527C4⋊C4—1664no—
528C4⋊C4—1668no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—3324no—
533C22⋊C4—3324no—
534C22⋊C4—1668no—
535C22⋊C4—1668no—
536C22⋊C4—1668no—
537C22⋊C4—1668no—
538C22⋊C4—1668no—
539C22⋊C4—1668no—
540C42—3322no—
541C42—3322no—
542C42—1664no—
543C42—1664no—
544C42—1668no—
545C42—1664no—
546C42—1664no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23C2×D4×D8312yes—
559C23C2×#1305058912yes—
560C23C2×#1305058912yes—
561C23C24×D8311yescharacteristic
562Q8—6644no—
563D4—48no—
564D4—416no—
565D4—84no—
566D4—84no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—3324no—
580C2×C4—3324no—
581C2×C4—834no—
582C2×C4—834no—
583C2×C4—832no—
584C2×C4—832no—
585C2×C4—6648no—
586C2×C4—1664no—
587C2×C4—1664no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4—1664no—
591C2×C4—1664no—
592C2×C4—1662no—
593C2×C4—1664no—
594C2×C4—1664no—
595C2×C4—1668no—
596C2×C4—1668no—
597C2×C4—1661no—
598C2×C4—1661no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4—1661no—
602C2×C4—1662no—
603C2×C4—1661no—
604C2×C4—1662no—
605C2×C4C2×D4×D8314yes—
606C2×C4C2×#1305058914yes—
607C2×C4C2×#1305058914yes—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22#1912824914yes—
622C22C22×D4×D8311yescharacteristic
623C22C22×#1305058911yescharacteristic
624C22C22×#1305058911yescharacteristic
625C22H(5,𝔽2)×D8311yescharacteristic
626C222−1+4×D8311yescharacteristic
627C22#1912853611yescharacteristic
628C22#1912853611yescharacteristic
629C4—24no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—3322no—
634C4—3324no—
635C4—3324no—
636C4—1662no—
637C4—1662no—
638C4—1664no—
639C4—1664no—
640C2—24no—
641C2—84no—
642C2C2×#1912824911yescharacteristic
643C2#2492086911yescharacteristic
644C2#2492089111yescharacteristic
645C2#2492116011yescharacteristic
646C2#2492117011yescharacteristic
647C2#2492122411yescharacteristic
648C2#2492123211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
Fitting, prime radicalC83×#2656C2
hypercenter#2656D83
hypocenterC83#35004
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C2×#19128296C2semidirect
2. #24915789C2semidirect
3. #24915762C2semidirect
4. #24913577C2semidirect
5. #24913207C2semidirect
6. #24912548C2semidirect
7. #24912524C2semidirect
8. #24912131C2semidirect
9. #24911425C2semidirect
10. #24910168C2semidirect
11. #121×Dic83C2semidirect
12. C2×#19127552C2semidirect
90. C2×Dic83⋊C4D4semidirect
338. C83#35004semidirect

Maximal subgroups

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

subgroupindex
C2×#191282962
#249157892
#249157622
#249135772
#249132072
#249125482
#249125242
#249121312
#249114252
#249101682
#121×Dic832
C2×#191275522
C83×#26562
#3500483