Finite Group Databasebeta

← #27400796

Subgroups — #27400796

Order 13568.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic106C211yesmaximal, characteristic
2M16×Dic106C211yesmaximal, characteristic, complemented
3#20277192C211yesmaximal, characteristic, complemented
4#20275694C212yesmaximal, complemented
5#20275692C211yesmaximal, characteristic, complemented
6#20275659C211yesmaximal, characteristic, complemented
7#20275657C212yesmaximal, complemented
8#20274735C211yesmaximal, characteristic
9#20274733C212yesmaximal
10#20274527C212yesmaximal
11C4×#104×C53C211yesmaximal, characteristic
12Q8×Dic106C411yescharacteristic, complemented
13#16811352C411yescharacteristic, complemented
14#16811349C412yescomplemented
15#16811332—22no—
16#16811332C412yescomplemented
17#16811330—21no—
18#16811330C411yescharacteristic, complemented
19D4×Dic106—21no—
20D4×Dic106C411yescharacteristic, complemented
21#16811180—21no—
22#16811180C2211yescharacteristic
23#16811176—22no—
24#16811176C2212yes—
25#41×Dic53C2212yes—
26C2×C4×Dic106—21no—
27C2×C4×Dic106C2211yescharacteristic
28#16810753—21no—
29#16810751—22no—
30C8×Dic106—21no—
31#16810647C2211yescharacteristic
32#16810646C2212yes—
33C2×#13133144C2212yes—
34M16×Dic53C2212yes—
35#16810529—21no—
36#16810521—21no—
37#16810507—22no—
38C4×#11719919C2212yes—
39C4×C4⋊Dic53C2211yescharacteristic
40#16810469C2212yes—
41#16810384C2212yes—
42#16810384C2212yes—
43#16810383C2212yes—
44#16810383C2212yes—
45#16810382C2212yes—
46C4×#41×C53C2211yescharacteristic
47C2×#104×C53C2211yescharacteristic
48C2×#104×C53C2211yescharacteristic
49C4×M16×C53C2211yescharacteristic
50C43×C53C2211yescharacteristic
51C53×#339C2211yescharacteristic
52C53×#339C2211yescharacteristic
53C22×Dic106—21no—
54C22×Dic106D411yescharacteristic
55C2×#11719917C2312yes—
56#13133217—21no—
57#13133217—21no—
58#13133217C2×C411yescharacteristic
59#13133217C2×C411yescharacteristic
60#13133216—22no—
61#13133216—22no—
62#13133216C2×C412yes—
63#13133216C2×C412yes—
64#13133209—22no—
65#13133209C2×C412yes—
66#13133208—22no—
67#13133208—22no—
68#13133208C2×C412yes—
69#13133208C2×C412yes—
70#13133196—21no—
71#13133196—22no—
72#13133196—21no—
73#13133196—21no—
74#13133196—21no—
75#13133194—22no—
76#13133194—22no—
77#13133194—22no—
78#13133194—22no—
79#13133194—22no—
80#13133194—22no—
81Q8×Dic53C2×C412yes—
82Q8×Dic53C2×C412yes—
83C4×Dic106—21no—
84C4×Dic106—21no—
85C4×Dic106—21no—
86C4×Dic106—22no—
87C4×Dic106—22no—
88C4×Dic106—22no—
89C4×Dic106—21no—
90C4×Dic106C2×C411yescharacteristic
91C4×Dic106C2×C411yescharacteristic
92C4×Dic106C2×C412yes—
93C4×Dic106D411yescharacteristic
94C4×Dic106D412yes—
95C4×Dic106D411yescharacteristic
96D4×Dic53—22no—
97D4×Dic53—22no—
98D4×Dic53C2×C412yes—
99D4×Dic53C2×C412yes—
100#13133176—22no—
101#13133176C2312yes—
102C2×C4⋊Dic53—21no—
103C2×C4⋊Dic53—21no—
104C2×C4⋊Dic53D411yescharacteristic
105C2×#11719919—22no—
106C2×#11719919—22no—
107C2×#11719919D412yes—
108C2×C4×Dic53—22no—
109C2×C4×Dic53C2312yes—
110C2×C4×Dic53C2312yes—
111C4⋊#88322—21no—
112C4⋊#88322—21no—
113#13133146—22no—
114#13133146—22no—
115#13133144—24no—
116#13133127—22no—
117C8×Dic53—22no—
118#13133124—21no—
119#13133124—22no—
120#13133124—22no—
121#13133124—22no—
122C2×#41×C53C2311yescharacteristic
123C2×M16×C53C2311yescharacteristic
124C2×M16×C53C2311yescharacteristic
125C4×Q8×C53C2×C411yescharacteristic
126C4×Q8×C53C2×C411yescharacteristic
127C4×D4×C53—21no—
128C4×D4×C53—21no—
129C4×D4×C53C2×C411yescharacteristic
130C4×D4×C53C2×C411yescharacteristic
131#117×C53—21no—
132#117×C53C2311yescharacteristic
133C2×C42×C53—21no—
134C2×C42×C53—21no—
135C2×C42×C53—21no—
136C2×C42×C53C2311yescharacteristic
137C2×C42×C53C2311yescharacteristic
138C2×C42×C53C2311yescharacteristic
139#104×C53—22no—
140#104×C53Q814yescomplemented
141#97×C53—21no—
142C4×C8×C53—21no—
143#41×C53—21no—
144#41×C53—21no—
145#41×C53C2×Q812yes—
146#41×C53#4112yes—
147C2×Q8×C53C22×C411yescharacteristic
148C2×D4×C53—21no—
149C2×D4×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C2×Dic106—21no—
164C2×Dic106—22no—
165C2×Dic106—21no—
166C2×Dic106—22no—
167C2×Dic106—42no—
168C2×Dic106—44no—
169C2×Dic106C22⋊C411yescharacteristic
170C2×Dic106C22⋊C411yescharacteristic
171C2×Dic106C22⋊C412yes—
172M16×C53—21no—
173M16×C53—21no—
174M16×C53—21no—
175M16×C53C2×Q812yescomplemented
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53C22×C411yescharacteristic
181C4⋊C4×C53C22×C411yescharacteristic
182C22⋊C4×C53—21no—
183C22⋊C4×C53—21no—
184C22⋊C4×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—42no—
192C42×C53—42no—
193C42×C53C22×C411yescharacteristic
194C42×C53C22×C411yescharacteristic
195C42×C53C2×D411yescharacteristic
196C42×C53C2×D411yescharacteristic
197C42×C53C2×Q812yescomplemented
198C22⋊Dic53—22no—
199C22⋊Dic53—22no—
200C22⋊Dic53—22no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—22no—
204C4⋊Dic53—22no—
205C4⋊Dic53—22no—
206C4⋊Dic53—22no—
207C4⋊Dic53—42no—
208C4⋊Dic53C22⋊C411yescharacteristic
209C4⋊Dic53C22⋊C412yes—
210C4⋊Dic53C22⋊C411yescharacteristic
211C4⋊Dic53C22×C412yes—
212C4⋊Dic53C22×C412yes—
213#11719919—22no—
214#11719919—22no—
215#11719919—22no—
216#11719919—22no—
217#11719919—22no—
218#11719919—22no—
219#11719919—44no—
220#11719919C22⋊C412yes—
221#11719919C22⋊C412yes—
222#11719919C22⋊C412yes—
223#11719919C22⋊C412yes—
224C4×Dic53—22no—
225C4×Dic53—22no—
226C4×Dic53—22no—
227C4×Dic53—22no—
228C4×Dic53—22no—
229C4×Dic53C22×C412yes—
230C4×Dic53C22×C412yes—
231C4×Dic53C2×D412yes—
232C4×Dic53C2×D412yes—
233#11719917—22no—
234#11719917—22no—
235C2×#88322—22no—
236C23×C53—21no—
237C23×C53C22×D411yescharacteristic
238Q8×C53—21no—
239Q8×C53C4×Q812yes—
240D4×C53—22no—
241D4×C53—22no—
242D4×C53—21no—
243D4×C53C4×Q812yes—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—22no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—42no—
258C2×C4×C53—81no—
259C2×C4×C53C2×C22⋊C411yescharacteristic
260C2×C4×C53C2×C22⋊C411yescharacteristic
261C2×C4×C53C2×C22⋊C411yescharacteristic
262C2×C4×C53C2×C22⋊C411yescharacteristic
263C2×C4×C53C23×C411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×Q811yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C532−1+411yescharacteristic
268C2×C4×C532−1+411yescharacteristic
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—44no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—82no—
278C2×Dic53C2×C22⋊C412yes—
279C2×Dic53C2×C22⋊C412yes—
280C2×Dic53C2×C22⋊C412yes—
281C2×Dic53C2×C22⋊C412yes—
282Dic106—22no—
283Dic106—22no—
284Dic106—42no—
285Dic106—44no—
286Dic106—42no—
287Dic106—82no—
288Dic106—81no—
289C8×C53—22no—
290C8×C53—41no—
291#88322—42no—
292#22529—531nomaximal, Sylow
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—22no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53C22×C22⋊C411yescharacteristic
299C22×C53#54811yescharacteristic
300C22×C53D4×Q811yescharacteristic
301C4×C53—22no—
302C4×C53—21no—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53—41no—
308C4×C53—41no—
309C4×C53#42112yes—
310C4×C53C2×C4×Q811yescharacteristic
311C4×C53#51911yescharacteristic
312C4×C53#54811yescharacteristic
313C4×C53D4×Q811yescharacteristic
314Dic53—22no—
315Dic53—22no—
316Dic53—22no—
317Dic53—22no—
318Dic53—42no—
319Dic53—42no—
320Dic53—82no—
321Q8×#41—531no—
322Q8×M16—531no—
323#2295—531no—
324#1995—532no—
325#1994—531no—
326#1986—531no—
327#1986—532no—
328#1806—532no—
329#1806—531no—
330#1759—532no—
331C4×#104D5311yescharacteristic
332C2×C53—21no—
333C2×C53—21no—
334C2×C53—42no—
335C2×C53Q8×C22⋊C411yescharacteristic
336C2×C53C2×#42111yescharacteristic
337C2×C53#290111yescharacteristic
338Q82—531no—
339#557—532no—
340#557—531no—
341#554—531no—
342#554—532no—
343#554—1061no—
344#554—1062no—
345D4×Q8—531no—
346D4×Q8—1061no—
347#533—532no—
348#533—531no—
349#533—1061no—
350#533—1062no—
351C4×#41—532no—
352C4×#41C2×D5311yescharacteristic
353C2×C4×Q8—531no—
354C2×C4×Q8—1061no—
355#446—1061no—
356#446—1062no—
357C8×Q8—1061no—
358#423—532no—
359#423—531no—
360C2×#104—532no—
361C2×#104C2×D5311yescharacteristic
362C2×#104C2×D5311yescharacteristic
363C4×M16—532no—
364C4×M16C2×D5311yescharacteristic
365#391—1061no—
366#389—1061no—
367#387—1062no—
368C4×C4⋊C4—532no—
369C4×C4⋊C4—531no—
370#376—532no—
371C43C2×D5311yescharacteristic
372#339—532no—
373#339—532no—
374#339—532no—
375#339—532no—
376#339—532no—
377#339C2×D5311yescharacteristic
378#339C2×D5311yescharacteristic
379C53#2252911yescharacteristic, complemented, Sylow
380C2×#41C22×D5311yescharacteristic
381C22×Q8—531no—
382C22×Q8—1061no—
383C2×M16—532no—
384C2×M16C22×D5311yescharacteristic
385C2×M16C22×D5311yescharacteristic
386C4⋊Q8—532no—
387C4⋊Q8—532no—
388C4⋊Q8—531no—
389C4⋊Q8—531no—
390C4⋊Q8—1061no—
391C4⋊Q8—1061no—
392C4⋊Q8—1062no—
393C4⋊Q8—1062no—
394#125—532no—
395#125—532no—
396#125—532no—
397#125—1062no—
398#125—1062no—
399#125—1062no—
400C22⋊Q8—1062no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C22⋊Q8—1062no—
404C22⋊Q8—1062no—
405C22⋊Q8—1062no—
406C22⋊Q8—1061no—
407C22⋊Q8—1062no—
408C22⋊Q8—1061no—
409C22⋊Q8—1061no—
410C22⋊Q8—1061no—
411C4×Q8—531no—
412C4×Q8—531no—
413C4×Q8—532no—
414C4×Q8—531no—
415C4×Q8—531no—
416C4×Q8—532no—
417C4×Q8—532no—
418C4×Q8—532no—
419C4×Q8—1062no—
420C4×Q8—1062no—
421C4×Q8—1062no—
422C4×Q8—1061no—
423C4×Q8—1061no—
424C4×Q8—1061no—
425C4×Q8—1061no—
426C4×Q8C4×D5311yescharacteristic
427C4×Q8C4×D5311yescharacteristic
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—532no—
431C4×D4—532no—
432C4×D4—1062no—
433C4×D4—1062no—
434C4×D4C4×D5311yescharacteristic
435C4×D4C4×D5311yescharacteristic
436#117—21no—
437#117—532no—
438#117—1062no—
439#117C22×D5311yescharacteristic
440C2×C4⋊C4—532no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—1062no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1062no—
445C2×C4⋊C4—1061no—
446C2×C42—21no—
447C2×C42—21no—
448C2×C42—21no—
449C2×C42—532no—
450C2×C42—532no—
451C2×C42—1062no—
452C2×C42C22×D5311yescharacteristic
453C2×C42C22×D5311yescharacteristic
454C2×C42C22×D5311yescharacteristic
455C4⋊C8—1062no—
456C4⋊C8—1062no—
457C4⋊C8—1061no—
458C4⋊C8—1061no—
459#104—22no—
460#104—1064no—
461#104Dic10614yescomplemented
462#97—21no—
463#97—1062no—
464C4×C8—21no—
465C4×C8—1062no—
466#95—1062no—
467#95—1062no—
468#95—1062no—
469#95—1061no—
470#41—21no—
471#41—21no—
472#41C2×Dic10612yes—
473#41#1171994512yes—
474C2×Q8—531no—
475C2×Q8—531no—
476C2×Q8—532no—
477C2×Q8—1062no—
478C2×Q8—1062no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×Q8—2122no—
482C2×Q8—2124no—
483C2×Q8C2×C4×D5311yescharacteristic
484C2×D4—21no—
485C2×D4C2×C4×D5311yescharacteristic
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4D4×D5311yescharacteristic
497C22×C4D4×D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499M16—21no—
500M16—21no—
501M16—21no—
502M16—1062no—
503M16—1062no—
504M16C2×Dic10612yescomplemented
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—1062no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—532no—
511C4⋊C4—531no—
512C4⋊C4—532no—
513C4⋊C4—532no—
514C4⋊C4—532no—
515C4⋊C4—532no—
516C4⋊C4—531no—
517C4⋊C4—532no—
518C4⋊C4—532no—
519C4⋊C4—1062no—
520C4⋊C4—1062no—
521C4⋊C4—1062no—
522C4⋊C4—1062no—
523C4⋊C4—1062no—
524C4⋊C4—1062no—
525C4⋊C4—1062no—
526C4⋊C4—1062no—
527C4⋊C4—1062no—
528C4⋊C4—1061no—
529C4⋊C4—1062no—
530C4⋊C4—1061no—
531C4⋊C4—2124no—
532C4⋊C4—2122no—
533C4⋊C4C2×C4×D5311yescharacteristic
534C4⋊C4C2×C4×D5311yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—1062no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—42no—
549C42—532no—
550C42—532no—
551C42—532no—
552C42—532no—
553C42—1062no—
554C42—1062no—
555C42—1062no—
556C42—1062no—
557C42—1062no—
558C42C2×Dic10612yescomplemented
559C42C2×C4×D5311yescharacteristic
560C42C2×C4×D5311yescharacteristic
561C42D4×D5311yescharacteristic
562C42D4×D5311yescharacteristic
563C23—21no—
564C23C2×D4×D5311yescharacteristic
565Q8—21no—
566Q8—1062no—
567Q8—1062no—
568Q8—4241no—
569Q8—4242no—
570Q8—2122no—
571Q8—2124no—
572Q8—2122no—
573Q8C4×Dic10612yes—
574D4—22no—
575D4—22no—
576D4—21no—
577D4C4×Dic10612yes—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—81no—
593C2×C4—532no—
594C2×C4—532no—
595C2×C4—532no—
596C2×C4—532no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—4242no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2124no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4C22⋊C4×D5311yescharacteristic
607C2×C4C22⋊C4×D5311yescharacteristic
608C2×C4C22⋊C4×D5311yescharacteristic
609C2×C4C22⋊C4×D5311yescharacteristic
610C2×C4C22×C4×D5311yescharacteristic
611C2×C4C2×D4×D5311yescharacteristic
612C2×C4C22×Dic10611yescharacteristic
613C2×C4C2×#1171994511yescharacteristic
614C2×C4#1313326411yescharacteristic
615C2×C4#1313326411yescharacteristic
616C8—22no—
617C8—41no—
618C8—2122no—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22C2×C22⋊C4×D5311yescharacteristic
625C22#1681128211yescharacteristic
626C22D4×Dic10611yescharacteristic
627C4—22no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—1062no—
636C4—1062no—
637C4—1062no—
638C4—1062no—
639C4—4242no—
640C4—2122no—
641C4—2122no—
642C4#1681063812yes—
643C4C2×C4×Dic10611yescharacteristic
644C4#1681110011yescharacteristic
645C4#1681128211yescharacteristic
646C4D4×Dic10611yescharacteristic
647C2—21no—
648C2—21no—
649C2—42no—
650C2C22⋊C4×Dic10611yescharacteristic
651C2C2×#1681063811yescharacteristic
652C2#2028325911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#104×C53C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#41×C53C23
cosocleC22×C4×C53C24
hypercenterC4×#104D53
hypocenterC53#22529
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. M16×Dic106C2semidirect
3. #20277192C2semidirect
4. #20275694C2semidirect
5. #20275692C2semidirect
6. #20275659C2semidirect
7. #20275657C2semidirect
12. Q8×Dic106C4semidirect
13. #16811352C4semidirect
14. #16811349C4semidirect
16. #16811332C4semidirect
18. #16811330C4semidirect
20. D4×Dic106C4semidirect
140. #104×C53Q8semidirect
175. M16×C53C2×Q8semidirect
197. C42×C53C2×Q8semidirect
379. C53#22529semidirect
461. #104Dic106semidirect
504. M16C2×Dic106semidirect
558. C42C2×Dic106semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1062
M16×Dic1062
#202771922
#202756942
#202756922
#202756592
#202756572
#202747352
#202747332
#202745272
C4×#104×C532
#2252953