Finite Group Databasebeta

← #27362994

Subgroups — #27362994

Order 13568.

1032 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic
3C2×#16810823C211yesmaximal, characteristic, complemented
4#20273818C212yesmaximal, complemented
5#20273811C212yesmaximal, complemented
6#20273793C212yesmaximal, complemented
7#20273787C212yesmaximal, complemented
8#20273405C211yesmaximal, characteristic, complemented
9#20273144C211yesmaximal, characteristic, complemented
10#20273102C211yesmaximal, characteristic, complemented
11Q8×#41×C53C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C22×#11719924—21no—
14C2×#13133214C2211yescharacteristic
15#16810997C2212yes—
16#16810971C2212yes—
17#16810823—21no—
18#16810823—22no—
19#16810823—21no—
20#16810823—21no—
21#16810823C2212yes—
22#16810823C2212yes—
23#16810823C2212yes—
24#16810818—21no—
25#16810818C2212yes—
26#16810773—22no—
27#16810676C2211yescharacteristic
28#16810647C2211yescharacteristic
29C2×C4⋊#88322C2211yescharacteristic
30#16810627—21no—
31C2×#13133137—21no—
32#16810372C2212yescomplemented
33#16810369C2212yescomplemented
34Q8⋊#88322C2212yescomplemented
35Q8⋊#88322C2212yescomplemented
36Q8⋊#88322C2212yescomplemented
37D4⋊#88322C2212yescomplemented
38Q82×C53C2211yescharacteristic, complemented
39C53×#557—21no—
40C53×#557C2211yescharacteristic, complemented
41C53×#554—22no—
42C53×#554—21no—
43C53×#554—21no—
44C53×#554C2211yescharacteristic, complemented
45D4×Q8×C53—21no—
46D4×Q8×C53C2211yescharacteristic, complemented
47C53×#533—22no—
48C53×#533—21no—
49C53×#533—21no—
50C53×#533C2211yescharacteristic
51C4×#41×C53—21no—
52C4×#41×C53C2211yescharacteristic
53C2×C4×Q8×C53—21no—
54C2×C4×Q8×C53C2211yescharacteristic
55C22×D212—21no—
56C22×D212—21no—
57C22×D212—21no—
58#13133242—44no—
59C2×#11719924—22no—
60C2×#11719924—22no—
61C2×#11719924—22no—
62C2×#11719924—24no—
63C2×#11719924—44no—
64C2×#11719922—22no—
65C2×#11719917—21no—
66C22×#88322—21no—
67#13133214—21no—
68#13133214—21no—
69#13133214—21no—
70#13133214C2312yes—
71C4⋊#88322—21no—
72C4⋊#88322—21no—
73C4⋊#88322—21no—
74C4⋊#88322C2312yes—
75C4⋊#88322C2312yes—
76#13133137—24no—
77#13133137—22no—
78#13133137—22no—
79#13133137—22no—
80#13133137—22no—
81C22⋊#88322—21no—
82C4×#88322C2312yes—
83C2×#41×C53D411yescharacteristic
84C22×Q8×C53—21no—
85C22×Q8×C53D411yescharacteristic
86C4⋊Q8×C53—22no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—22no—
92C4⋊Q8×C53C2311yescharacteristic
93C4⋊Q8×C53C2311yescharacteristic
94#125×C53—22no—
95#125×C53—22no—
96#125×C53—22no—
97#125×C53—21no—
98#125×C53—21no—
99#125×C53—21no—
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53—22no—
102C22⋊Q8×C53—22no—
103C22⋊Q8×C53—22no—
104C22⋊Q8×C53—22no—
105C22⋊Q8×C53—22no—
106C22⋊Q8×C53—21no—
107C22⋊Q8×C53—21no—
108C22⋊Q8×C53—22no—
109C22⋊Q8×C53—21no—
110C22⋊Q8×C53D412yes—
111C4×Q8×C53—22no—
112C4×Q8×C53—22no—
113C4×Q8×C53—22no—
114C4×Q8×C53—21no—
115C4×Q8×C53—21no—
116C4×Q8×C53—21no—
117C4×Q8×C53—21no—
118C4×Q8×C53—21no—
119C4×Q8×C53—21no—
120C4×Q8×C53—21no—
121C4×Q8×C53D411yescharacteristic
122C4×Q8×C53D411yescharacteristic
123C4×Q8×C53D411yescharacteristic
124C4×Q8×C53D412yes—
125C4×Q8×C53C2311yescharacteristic
126C4×Q8×C53C2311yescharacteristic
127C4×Q8×C53C2311yescharacteristic
128C4×D4×C53—22no—
129C4×D4×C53—22no—
130C4×D4×C53—21no—
131C4×D4×C53—21no—
132C4×D4×C53—21no—
133C4×D4×C53—21no—
134C4×D4×C53D411yescharacteristic
135C4×D4×C53C2311yescharacteristic
136#117×C53—22no—
137#117×C53—21no—
138#117×C53—21no—
139#117×C53D411yescharacteristic
140C2×C4⋊C4×C53—22no—
141C2×C4⋊C4×C53—21no—
142C2×C4⋊C4×C53—21no—
143C2×C4⋊C4×C53D411yescharacteristic
144C2×C42×C53—22no—
145C2×C42×C53—21no—
146C2×C42×C53—21no—
147C2×C42×C53C2311yescharacteristic
148C23×D53—41no—
149#41×C53—22no—
150#41×C53SD1614yescomplemented
151C2×Q8×C53—22no—
152C2×Q8×C53—22no—
153C2×Q8×C53—21no—
154C2×Q8×C53—21no—
155C2×Q8×C53—22no—
156C2×Q8×C53—41no—
157C2×Q8×C53—42no—
158C2×Q8×C53—42no—
159C2×Q8×C53C2×D411yescharacteristic
160C2×Q8×C53C2×D411yescharacteristic
161C2×Q8×C53C2×D411yescharacteristic
162C2×Q8×C53C2×D412yes—
163C2×D4×C53—21no—
164C2×D4×C53C2×D411yescharacteristic
165C22×C4×C53—21no—
166C22×C4×C53—22no—
167C22×C4×C53—21no—
168C22×C4×C53—22no—
169C22×C4×C53—21no—
170C22×C4×C53—21no—
171C22×C4×C53—21no—
172C22×C4×C53C2×D411yescharacteristic
173C22×C4×C53C2×D411yescharacteristic
174C22×C4×C53C2×D411yescharacteristic
175C2×D212—22no—
176C2×D212—22no—
177C2×D212—22no—
178C2×D212—42no—
179C2×D212—42no—
180C2×D212—42no—
181C2×D212—42no—
182C2×D212—41no—
183C2×D212—41no—
184C2×D212—41no—
185C2×D212—82no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53—22no—
189C4⋊C4×C53—22no—
190C4⋊C4×C53—22no—
191C4⋊C4×C53—22no—
192C4⋊C4×C53—22no—
193C4⋊C4×C53—22no—
194C4⋊C4×C53—21no—
195C4⋊C4×C53—22no—
196C4⋊C4×C53—21no—
197C4⋊C4×C53—21no—
198C4⋊C4×C53—21no—
199C4⋊C4×C53—22no—
200C4⋊C4×C53—21no—
201C4⋊C4×C53—21no—
202C4⋊C4×C53—21no—
203C4⋊C4×C53—21no—
204C4⋊C4×C53—21no—
205C4⋊C4×C53—21no—
206C4⋊C4×C53C2×D411yescharacteristic
207C4⋊C4×C53C2×D411yescharacteristic
208C4⋊C4×C53C2×D411yescharacteristic
209C4⋊C4×C53C2×D411yescharacteristic
210C4⋊C4×C53C2×D412yes—
211C22⋊C4×C53—22no—
212C22⋊C4×C53—22no—
213C22⋊C4×C53—22no—
214C22⋊C4×C53—21no—
215C22⋊C4×C53—21no—
216C22⋊C4×C53—21no—
217C42×C53—22no—
218C42×C53—22no—
219C42×C53—21no—
220C42×C53—22no—
221C42×C53—21no—
222C42×C53—21no—
223C42×C53—21no—
224C42×C53—21no—
225C42×C53—21no—
226C42×C53—21no—
227C42×C53C2×D411yescharacteristic
228C42×C53C2×D411yescharacteristic
229C42×C53C2×D411yescharacteristic
230C42×C53C2411yescharacteristic
231#11719924—44no—
232#11719924—44no—
233#11719924—44no—
234#11719924—84no—
235#11719922—44no—
236#11719917—42no—
237C2×#88322—22no—
238C2×#88322—22no—
239C2×#88322—22no—
240C2×#88322—42no—
241C23×C53—21no—
242C23×C53#12011yescharacteristic
243C22×D53—42no—
244C22×D53—82no—
245C22×D53—82no—
246C22×D53—82no—
247Q8×C53—21no—
248Q8×C53—21no—
249Q8×C53—21no—
250Q8×C53—21no—
251Q8×C53—42no—
252Q8×C53—44no—
253Q8×C53—42no—
254Q8×C53—41no—
255Q8×C53—41no—
256Q8×C53—82no—
257Q8×C53C2×SD1612yescomplemented
258Q8×C53AGL(1,ℤ/8)12yescomplemented
259D4×C53—24no—
260D4×C53—21no—
261D4×C53C2×SD1612yescomplemented
262C2×C4×C53—21no—
263C2×C4×C53—21no—
264C2×C4×C53—21no—
265C2×C4×C53—21no—
266C2×C4×C53—24no—
267C2×C4×C53—21no—
268C2×C4×C53—22no—
269C2×C4×C53—21no—
270C2×C4×C53—21no—
271C2×C4×C53—21no—
272C2×C4×C53—21no—
273C2×C4×C53—22no—
274C2×C4×C53—41no—
275C2×C4×C53—42no—
276C2×C4×C53—44no—
277C2×C4×C53—41no—
278C2×C4×C53—41no—
279C2×C4×C53—81no—
280C2×C4×C53—81no—
281C2×C4×C53—82no—
282C2×C4×C53#12011yescharacteristic
283C2×C4×C53#12011yescharacteristic
284C2×C4×C53#12011yescharacteristic
285C2×C4×C53C2×SD1612yes—
286C2×C4×C53#13712yes—
287C2×C4×C53C22×D411yescharacteristic
288C2×C4×C53C22×D411yescharacteristic
289C2×C4×C53C22×D411yescharacteristic
290D212—44no—
291D212—42no—
292D212—42no—
293D212—42no—
294D212—42no—
295D212—82no—
296D212—82no—
297D212—82no—
298#88322—42no—
299#88322—42no—
300#88322—42no—
301#18788—531nomaximal, Sylow
302C22×C53—21no—
303C22×C53—21no—
304C22×C53—22no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53#44812yes—
308C22×C53C2×#12011yescharacteristic
309C2×D53—82no—
310C2×D53—82no—
311C2×D53—82no—
312C2×D53—164no—
313C4×C53—22no—
314C4×C53—21no—
315C4×C53—22no—
316C4×C53—21no—
317C4×C53—21no—
318C4×C53—41no—
319C4×C53—42no—
320C4×C53—41no—
321C4×C53—41no—
322C4×C53—42no—
323C4×C53—41no—
324C4×C53—41no—
325C4×C53#44812yes—
326C4×C53C22×SD1611yescharacteristic
327C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
328C4×C53C2×#13711yescharacteristic
329C4×C53#57511yescharacteristic
330Q8×#41D5311yescharacteristic, complemented
331#3208—531no—
332#3038—531no—
333C2×#460—531no—
334#1627—532no—
335#1626—532no—
336#1623—532no—
337#1622—532no—
338#1550—531no—
339#1487—531no—
340#1477—531no—
341C2×C53—22no—
342C2×C53—42no—
343C2×C53C2×#44811yescharacteristic
344C2×C53#300611yescharacteristic
345C2×C53#302111yescharacteristic
346D53—162no—
347C2×AGL(1,ℤ/8)—1061no—
348C22×SD16—1061no—
349Q82C2×D5311yescharacteristic, complemented
350#557—21no—
351#557C2×D5311yescharacteristic, complemented
352#554—22no—
353#554—21no—
354#554—21no—
355#554C2×D5311yescharacteristic, complemented
356D4×Q8—21no—
357D4×Q8C2×D5311yescharacteristic, complemented
358#533—22no—
359#533—21no—
360#533—21no—
361#533C2×D5311yescharacteristic
362C2×#127—531no—
363C4×#41—21no—
364C4×#41C2×D5311yescharacteristic
365C2×C4×Q8—21no—
366C2×C4×Q8C2×D5311yescharacteristic
367#492—532no—
368#486—532no—
369#460—532no—
370#460—532no—
371#460—532no—
372#460—1062no—
373#460—1061no—
374#460—1061no—
375#460—1061no—
376#459—532no—
377#459—1061no—
378#450—1062no—
379#431—531no—
380#423—531no—
381C2×C4⋊C8—531no—
382#418—1061no—
383C2×#102—1061no—
384#333—532no—
385#332—532no—
386Q8⋊C8—532no—
387Q8⋊C8—532no—
388Q8⋊C8—532no—
389D4⋊C8—532no—
390C53#1878811yescharacteristic, complemented, Sylow
391C2×#41C22⋊D5311yescharacteristic
392C22×Q8—21no—
393C22×Q8C22⋊D5311yescharacteristic
394C22×D4—1061no—
395C22×D4—1061no—
396C22×D4—1061no—
397AGL(1,ℤ/8)—2124no—
398C2×SD16—1064no—
399C2×SD16—1062no—
400C2×SD16—1062no—
401C2×SD16—1062no—
402C2×SD16—2124no—
403C2×D8—1062no—
404C2×M16—1061no—
405C22×C8—1061no—
406C4⋊Q8—22no—
407C4⋊Q8—21no—
408C4⋊Q8—22no—
409C4⋊Q8—21no—
410C4⋊Q8—21no—
411C4⋊Q8—21no—
412C4⋊Q8C22×D5311yescharacteristic
413C4⋊Q8C22×D5311yescharacteristic
414#127—532no—
415#127—1061no—
416#127—1061no—
417#127—1061no—
418#125—22no—
419#125—22no—
420#125—22no—
421#125—21no—
422#125—21no—
423#125—21no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—22no—
427C22⋊Q8—22no—
428C22⋊Q8—22no—
429C22⋊Q8—21no—
430C22⋊Q8—22no—
431C22⋊Q8—21no—
432C22⋊Q8—21no—
433C22⋊Q8—22no—
434C22⋊Q8C22⋊D5312yes—
435C4×Q8—22no—
436C4×Q8—22no—
437C4×Q8—22no—
438C4×Q8—21no—
439C4×Q8—21no—
440C4×Q8—21no—
441C4×Q8—21no—
442C4×Q8—21no—
443C4×Q8—21no—
444C4×Q8—21no—
445C4×Q8C22⋊D5311yescharacteristic
446C4×Q8C22⋊D5311yescharacteristic
447C4×Q8C22⋊D5311yescharacteristic
448C4×Q8C22⋊D5312yes—
449C4×Q8C22×D5311yescharacteristic
450C4×Q8C22×D5311yescharacteristic
451C4×Q8C22×D5311yescharacteristic
452C4×D4—22no—
453C4×D4—22no—
454C4×D4—21no—
455C4×D4—21no—
456C4×D4—21no—
457C4×D4—21no—
458C4×D4C22⋊D5311yescharacteristic
459C4×D4C22×D5311yescharacteristic
460#117—22no—
461#117—21no—
462#117—21no—
463#117C22⋊D5311yescharacteristic
464C2×C4⋊C4—22no—
465C2×C4⋊C4—21no—
466C2×C4⋊C4—21no—
467C2×C4⋊C4C22⋊D5311yescharacteristic
468C2×C42—22no—
469C2×C42—21no—
470C2×C42—21no—
471C2×C42C22×D5311yescharacteristic
472C4⋊C8—532no—
473C4⋊C8—532no—
474C4⋊C8—1061no—
475C4⋊C8—1061no—
476C4⋊C8—1061no—
477#102—1062no—
478#102—1062no—
479#102—1062no—
480#102—1062no—
481#102—1064no—
482C22⋊C8—1061no—
483C4×C8—532no—
484C24—2121no—
485#41—22no—
486#41#1171992414yescomplemented
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—22no—
492C2×Q8—41no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8C2×C22⋊D5311yescharacteristic
496C2×Q8C2×C22⋊D5311yescharacteristic
497C2×Q8C2×C22⋊D5311yescharacteristic
498C2×Q8C2×C22⋊D5312yes—
499C2×D4—21no—
500C2×D4—1062no—
501C2×D4—1062no—
502C2×D4—1062no—
503C2×D4—4242no—
504C2×D4—2122no—
505C2×D4—2122no—
506C2×D4—2121no—
507C2×D4—2121no—
508C2×D4—2121no—
509C2×D4—2122no—
510C2×D4—2122no—
511C2×D4C2×C22⋊D5311yescharacteristic
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4C2×C22⋊D5311yescharacteristic
520C22×C4C2×C22⋊D5311yescharacteristic
521C22×C4C2×C22⋊D5311yescharacteristic
522SD16—4244no—
523SD16—2124no—
524SD16—2124no—
525SD16—2124no—
526D8—2124no—
527M16—2122no—
528C2×C8—1062no—
529C2×C8—1062no—
530C2×C8—1062no—
531C2×C8—2122no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4C2×C22⋊D5311yescharacteristic
553C4⋊C4C2×C22⋊D5311yescharacteristic
554C4⋊C4C2×C22⋊D5311yescharacteristic
555C4⋊C4C2×C22⋊D5311yescharacteristic
556C4⋊C4C2×C22⋊D5312yes—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—22no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42C2×C22⋊D5311yescharacteristic
574C42C2×C22⋊D5311yescharacteristic
575C42C2×C22⋊D5311yescharacteristic
576C42C23×D5311yescharacteristic
577C23—21no—
578C23—4242no—
579C23—4242no—
580C23—4242no—
581C23—2122no—
582C23#1313318611yescharacteristic
583Q8—21no—
584Q8—21no—
585Q8—21no—
586Q8—21no—
587Q8—42no—
588Q8—42no—
589Q8—44no—
590Q8—41no—
591Q8—41no—
592Q8—82no—
593Q8C2×#1171992412yescomplemented
594Q8#1313324212yescomplemented
595D4—24no—
596D4—21no—
597D4—4242no—
598D4—4242no—
599D4—4242no—
600D4—2124no—
601D4—2122no—
602D4—2122no—
603D4—2122no—
604D4—2122no—
605D4C2×#1171992412yescomplemented
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4#1313318611yescharacteristic
627C2×C4#1313318611yescharacteristic
628C2×C4#1313318611yescharacteristic
629C2×C4C2×#1171992412yes—
630C2×C4#1313324812yes—
631C2×C4C22×C22⋊D5311yescharacteristic
632C2×C4C22×C22⋊D5311yescharacteristic
633C2×C4C22×C22⋊D5311yescharacteristic
634C8—2122no—
635C8—2122no—
636C8—2122no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—8484no—
643C22—4242no—
644C22—4242no—
645C22—4242no—
646C22#1681076212yes—
647C22C2×#1313318611yescharacteristic
648C4—21no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—22no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—41no—
660C4#1681076212yes—
661C4C22×#1171992411yescharacteristic
662C4C2×#1313324211yescharacteristic
663C4C2×#1313324811yescharacteristic
664C4#1681142111yescharacteristic
665C2—22no—
666C2—42no—
667C2—8482no—
668C2C2×#1681076211yescharacteristic
669C2#2028396711yescharacteristic
670C2#2028408711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×#41×C53C2
socleC22×C53C2×#120
prime radical#20284195C2
hypercenterQ8×#41D53
hypocenterC53#18788
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. #20285653C2semidirect
3. C2×#16810823C2semidirect
4. #20273818C2semidirect
5. #20273811C2semidirect
6. #20273793C2semidirect
7. #20273787C2semidirect
8. #20273405C2semidirect
9. #20273144C2semidirect
10. #20273102C2semidirect
11. Q8×#41×C53C2semidirect
32. #16810372C22semidirect
33. #16810369C22semidirect
34. Q8⋊#88322C22semidirect
35. Q8⋊#88322C22semidirect
36. Q8⋊#88322C22semidirect
37. D4⋊#88322C22semidirect
38. Q82×C53C22semidirect
40. C53×#557C22semidirect
44. C53×#554C22semidirect
46. D4×Q8×C53C22semidirect
150. #41×C53SD16semidirect
257. Q8×C53C2×SD16semidirect
258. Q8×C53AGL(1,ℤ/8)semidirect
261. D4×C53C2×SD16semidirect
330. Q8×#41D53semidirect
349. Q82C2×D53semidirect
351. #557C2×D53semidirect
355. #554C2×D53semidirect
357. D4×Q8C2×D53semidirect
390. C53#18788semidirect
486. #41#11719924semidirect
593. Q8C2×#11719924semidirect
594. Q8#13133242semidirect
605. D4C2×#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202841952
C2×#168108232
#202738182
#202738112
#202737932
#202737872
#202734052
#202731442
#202731022
Q8×#41×C532
#1878853