Finite Group Databasebeta

← #28320573

Subgroups — #28320573

Order 13568.

1338 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291194C211yesmaximal, characteristic
2#20291151C211yesmaximal, characteristic
3#20290906C212yesmaximal, complemented
4#20290906C212yesmaximal, complemented
5#20290905C212yesmaximal, complemented
6#20290904C212yesmaximal, complemented
7#20290412C212yesmaximal
8C2×#16811501C212yesmaximal
9C2×#16811248C211yesmaximal, characteristic
10C22×C22⋊#6157C2212yes—
11C22×C22⋊#6157C2212yes—
12#16811570—24no—
13#16811565C2212yes—
14#16811565C2212yes—
15C22⋊C4×#6157—24no—
16C4×C22⋊#6157C2212yes—
17C4×C22⋊#6157C2212yes—
18#16811501C2214yes—
19#16811501C2214yes—
20#16811501C2214yes—
21#16811501C2214yes—
22#16811248—24no—
23#16811248—24no—
24C2×#13133197C411yescharacteristic, complemented
25C2×#13133197C2211yescharacteristic
26C2×#13133196C411yescharacteristic, complemented
27C2×#13133195C411yescharacteristic, complemented
28C2×#13133195C411yescharacteristic, complemented
29C2×#13133195C2211yescharacteristic
30C2×#13133195C2211yescharacteristic
31C2×#13133194C411yescharacteristic, complemented
32C2×#13133194C411yescharacteristic, complemented
33C2×#13133193C411yescharacteristic, complemented
34C2×#13133193C2211yescharacteristic
35C22×#11719921C2211yescharacteristic
36C2×C22⋊C4×D53C2211yescharacteristic
37C2×C22⋊C4×D53C2211yescharacteristic
38C2×C22⋊Q8×C53C411yescharacteristic, complemented
39C23×#6157—22no—
40C23×#6157—22no—
41C2×C4⋊#6157—22no—
42C2×C4⋊#6157—22no—
43C2×C22⋊#6157—22no—
44C2×C22⋊#6157—24no—
45C2×C22⋊#6157—28no—
46C2×C22⋊#6157—22no—
47C2×C22⋊#6157Q818yescomplemented
48C2×C22⋊#6157C2314yes—
49C2×C22⋊#6157C2314yes—
50C2×C4×#6157—22no—
51C2×C4×#6157—22no—
52(C2×#6157)⋊C4—22no—
53(C2×#6157)⋊C4—22no—
54(C2×#6157)⋊C4—22no—
55(C2×#6157)⋊C4—22no—
56C24×D53C2311yescharacteristic
57C22×Dic106C2×C411yescharacteristic, complemented
58C22×Dic106C2×C411yescharacteristic, complemented
59C22×Dic106C2×C411yescharacteristic, complemented
60C23×Dic53C2×C411yescharacteristic
61C22×C4×D53C2×C411yescharacteristic
62C22×C4×D53C2×C411yescharacteristic
63C22×C4×D53C2311yescharacteristic
64C22×C4×D53C2311yescharacteristic
65#13133197—24no—
66#13133197—22no—
67#13133197—22no—
68#13133196—24no—
69#13133195—24no—
70#13133195—24no—
71#13133195—22no—
72#13133195—22no—
73#13133195—22no—
74#13133195—22no—
75#13133194—24no—
76#13133194—24no—
77#13133193—24no—
78#13133193—22no—
79#13133193—22no—
80C2×C4⋊Dic53C2×C411yescharacteristic, complemented
81C2×C4⋊Dic53C2×C411yescharacteristic, complemented
82C2×C4⋊Dic53C2×C411yescharacteristic, complemented
83C2×#11719919C2×C411yescharacteristic, complemented
84C2×#11719919C2×C411yescharacteristic, complemented
85C2×#11719919C2×C411yescharacteristic, complemented
86C2×#11719919C2×C411yescharacteristic, complemented
87C2×#11719919C2×C411yescharacteristic, complemented
88C2×#11719919C2×C411yescharacteristic, complemented
89C2×C22⋊Dic53C2×C411yescharacteristic
90C2×C22⋊Dic53C2×C411yescharacteristic
91C2×#11719921—22no—
92C2×#11719921—22no—
93C2×#11719921C2×C411yescharacteristic
94C2×#11719921C2×C411yescharacteristic
95C2×#11719921C2×C411yescharacteristic
96C2×#11719921C2×C411yescharacteristic
97C2×#11719921C2311yescharacteristic
98C2×#11719921C2311yescharacteristic
99C2×#11719921C2311yescharacteristic
100C2×#11719921C2311yescharacteristic
101C22⋊C4×D53—22no—
102C22⋊C4×D53—22no—
103C22⋊C4×D53D414yes—
104C22⋊C4×D53D414yes—
105C22×Q8×C53C2×C411yescharacteristic, complemented
106C23×C4×C53C2×C411yescharacteristic
107C22⋊Q8×C53—24no—
108C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
109C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
110C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
111C2×C22⋊C4×C53C2×C411yescharacteristic
112C2×C22⋊C4×C53C2×C411yescharacteristic
113C24×C53C22×C411yescharacteristic
114C23×D53—81no—
115C23×D53—81no—
116C23×D53C22×C411yescharacteristic
117C23×D53C2×D412yes—
118C23×D53C2×D412yes—
119C23×D53C2×Q814yes—
120C23×D53#4114yes—
121C23×D53C2411yescharacteristic
122C22×#6157—24no—
123C22×#6157—28no—
124C22×#6157—24no—
125C22×#6157—44no—
126C22×#6157—82no—
127C22×#6157—82no—
128C22×#6157—82no—
129C22×#6157—82no—
130C2×Q8×C53—22no—
131C2×Q8×C53—81no—
132C22×C4×C53—22no—
133C22×C4×C53—81no—
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C22×C411yescharacteristic
138C22×C4×C53C22×C411yescharacteristic
139C22×C4×C53C22×C411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—81no—
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C411yescharacteristic
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C22×C411yescharacteristic
146C22×Dic53C22×C411yescharacteristic
147C22×Dic53C22×C411yescharacteristic
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—81no—
151C2×C4×D53—81no—
152C2×C4×D53—81no—
153C2×C4×D53—81no—
154C2×C4×D53C2×D412yes—
155C2×C4×D53C2×D412yes—
156C2×C4×D53C2×D412yes—
157C2×C4×D53C2×D412yes—
158C2×Dic106—22no—
159C2×Dic106—22no—
160C2×Dic106—22no—
161C2×Dic106—81no—
162C2×Dic106—81no—
163C2×Dic106—81no—
164C22⋊#6157—28no—
165C22⋊#6157—24no—
166C22⋊#6157—28no—
167C22⋊#6157—44no—
168C22⋊#6157C2×Q818yes—
169C4⋊#6157—82no—
170C4⋊#6157—82no—
171C4×#6157—82no—
172C4×#6157—82no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53C22⋊C414yes—
178C22⋊Dic53—22no—
179C22⋊Dic53C22⋊C414yes—
180#11719921—22no—
181#11719921—22no—
182#11719921—21no—
183#11719921—21no—
184#11719921—21no—
185#11719921—21no—
186#11719921—21no—
187#11719921—21no—
188#11719921—21no—
189#11719921—21no—
190#11719921C22⋊C414yes—
191#11719921C22⋊C414yes—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195#11719919—22no—
196#11719919—22no—
197#11719919—22no—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53C2×C22⋊C412yes—
204C23×C53C4×Q814yes—
205C23×C53C23×C411yescharacteristic
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—84no—
215C22×D53—84no—
216C22×D53C2×C22⋊C412yes—
217C22×D53C4×Q814yes—
218C22×D53C22⋊Q814yes—
219C22×D53C22⋊Q814yes—
220C22×D53#12414yes—
221C22×D53C4⋊Q814yes—
222C22×D53C22×D411yescharacteristic
223C22×D53C22×D411yescharacteristic
224C22×D53C22×Q812yes—
225C22×D53C2×#4112yes—
226C2×#6157—28no—
227C2×#6157—44no—
228C2×#6157—44no—
229C2×#6157—44no—
230C2×#6157—44no—
231C2×#6157—44no—
232C2×#6157—88no—
233C2×#6157—88no—
234Q8×C53—82no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—82no—
246C2×C4×C53C2×C22⋊C412yes—
247C2×C4×C53C2×C22⋊C412yes—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—82no—
259C2×Dic53C2×C22⋊C412yes—
260C2×Dic53C2×C22⋊C412yes—
261C4×D53—42no—
262C4×D53—42no—
263C4×D53—42no—
264C4×D53—42no—
265C4×D53—82no—
266C4×D53—82no—
267Dic106—82no—
268Dic106—82no—
269Dic106—82no—
270#21333—531nomaximal, Sylow
271C22×C53—21no—
272C22×C53—21no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—44no—
276C22×C53#39114yes—
277C22×C53C22×C22⋊C411yescharacteristic
278C22×C53C2×C4×Q812yes—
279C2×D53—21no—
280C2×D53—21no—
281C2×D53—21no—
282C2×D53—21no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—44no—
288C2×D53—44no—
289C2×D53—44no—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53#39114yes—
293C2×D53C2×C22⋊Q812yes—
294C2×D53C2×#12411yescharacteristic
295C2×D53C2×C4⋊Q811yescharacteristic
296#6157—48no—
297#6157—84no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—41no—
304Dic53—22no—
305Dic53—22no—
306Dic53—41no—
307Dic53—41no—
308Dic53—41no—
309Dic53—41no—
310C2×#543—531no—
311#2315—531no—
312#2273—531no—
313#1996—532no—
314#1996—532no—
315#1996—532no—
316#1996—532no—
317C2×#342—532no—
318C2×#342—532no—
319C2×C53—22no—
320C2×C53—24no—
321C2×C53C2×#39111yescharacteristic
322C2×C53#288311yescharacteristic
323C2×C53#288311yescharacteristic
324D53—22no—
325D53—21no—
326D53—21no—
327D53—24no—
328#543—1064no—
329#543—1064no—
330C2×C22⋊Q8—531no—
331C2×C22⋊Q8—531no—
332C2×C22⋊Q8—531no—
333C2×C22⋊Q8—531no—
334C2×C22⋊Q8—531no—
335C2×C22⋊Q8—531no—
336C2×C22⋊Q8—531no—
337C2×C22⋊Q8—531no—
338C2×C22⋊Q8—531no—
339C2×C22⋊Q8—531no—
340C2×C22⋊Q8—531no—
341C2×C22⋊Q8#615711yescharacteristic, complemented
342C22×C22⋊C4—531no—
343C22×C22⋊C4—532no—
344C22×C22⋊C4—531no—
345C22×C22⋊C4—531no—
346C22×C22⋊C4—532no—
347#385—1064no—
348#380—532no—
349#380—532no—
350C4×C22⋊C4—532no—
351C4×C22⋊C4—532no—
352C4×C22⋊C4—1064no—
353#342—534no—
354#342—534no—
355#342—534no—
356#342—534no—
357C53#2133311yescharacteristic, complemented, Sylow
358C25—531no—
359C22×Q8—531no—
360C22×Q8—531no—
361C22×Q8—531no—
362C22×Q8C2×#615711yescharacteristic, complemented
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—1062no—
369C23×C4—1062no—
370C23×C4C2×#615711yescharacteristic
371C22⋊Q8—24no—
372C22⋊Q8—1064no—
373C22⋊Q8—1064no—
374C22⋊Q8—1064no—
375C22⋊Q8—1064no—
376C22⋊Q8—1064no—
377C22⋊Q8—1064no—
378C22⋊Q8—1062no—
379C22⋊Q8—1062no—
380C22⋊Q8—1062no—
381C22⋊Q8—1062no—
382C22⋊Q8—1062no—
383C22⋊Q8—1062no—
384C22⋊Q8—1062no—
385C22⋊Q8—1062no—
386C22⋊Q8—1064no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4C2×#615711yescharacteristic, complemented
399C2×C4⋊C4C2×#615711yescharacteristic, complemented
400C2×C4⋊C4C2×#615711yescharacteristic, complemented
401C2×C22⋊C4—531no—
402C2×C22⋊C4—531no—
403C2×C22⋊C4—531no—
404C2×C22⋊C4—534no—
405C2×C22⋊C4—534no—
406C2×C22⋊C4—534no—
407C2×C22⋊C4—538no—
408C2×C22⋊C4—531no—
409C2×C22⋊C4—531no—
410C2×C22⋊C4—531no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—534no—
416C2×C22⋊C4—1062no—
417C2×C22⋊C4—1062no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1062no—
420C2×C22⋊C4—1062no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4—1068no—
423C2×C22⋊C4—1062no—
424C2×C22⋊C4C2×#615711yescharacteristic
425C2×C22⋊C4C2×#615711yescharacteristic
426C2×C42—1062no—
427C2×C42—1062no—
428#95—1062no—
429#95—1062no—
430#95—1062no—
431#95—1062no—
432C24—534no—
433C24—534no—
434C24—532no—
435C24—532no—
436C24—531no—
437C24—531no—
438C24—4241no—
439C24—4241no—
440C24C22×#615711yescharacteristic
441C2×Q8—22no—
442C2×Q8—81no—
443C2×Q8—1062no—
444C2×Q8—1062no—
445C2×Q8—1062no—
446C2×Q8—4241no—
447C2×Q8—4241no—
448C2×Q8—4241no—
449C22×C4—22no—
450C22×C4—81no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1068no—
464C22×C4—1062no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4—4241no—
470C22×C4—4241no—
471C22×C4—4242no—
472C22×C4—4242no—
473C22×C4—4242no—
474C22×C4—4242no—
475C22×C4—4241no—
476C22×C4—2124no—
477C22×C4C22×#615711yescharacteristic
478C22×C4C22×#615711yescharacteristic
479C22×C4C22×#615711yescharacteristic
480C22×C4C22×#615711yescharacteristic
481C22×C4C22×#615711yescharacteristic
482C22×C4C22×#615711yescharacteristic
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—1062no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—4242no—
496C4⋊C4—4242no—
497C22⋊C4—22no—
498C22⋊C4—534no—
499C22⋊C4—534no—
500C22⋊C4—534no—
501C22⋊C4—538no—
502C22⋊C4—1061no—
503C22⋊C4—1061no—
504C22⋊C4—1061no—
505C22⋊C4—1061no—
506C22⋊C4—1062no—
507C22⋊C4—1061no—
508C22⋊C4—1061no—
509C22⋊C4—1062no—
510C22⋊C4—1061no—
511C22⋊C4—1061no—
512C22⋊C4—1068no—
513C22⋊C4—1064no—
514C22⋊C4—1062no—
515C22⋊C4—1068no—
516C22⋊C4—2124no—
517C22⋊C4C22⋊#615714yes—
518C42—4242no—
519C42—4242no—
520C23—41no—
521C23—41no—
522C23—532no—
523C23—531no—
524C23—531no—
525C23—534no—
526C23—534no—
527C23—534no—
528C23—534no—
529C23—534no—
530C23—532no—
531C23—532no—
532C23—4244no—
533C23—4244no—
534C23—2121no—
535C23—2121no—
536C23—2122no—
537C23—2122no—
538C23—2121no—
539C23—2121no—
540C23—2122no—
541C23—2122no—
542C23C2×C22⋊#615712yes—
543C23Q8×#615714yes—
544C23C23×#615711yescharacteristic
545Q8—82no—
546Q8—4242no—
547Q8—4242no—
548Q8—4242no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—532no—
561C2×C4—532no—
562C2×C4—1068no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—4242no—
568C2×C4—4242no—
569C2×C4—4248no—
570C2×C4—4242no—
571C2×C4—4248no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2124no—
579C2×C4—2124no—
580C2×C4—2124no—
581C2×C4—2124no—
582C2×C4—2124no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4C2×C22⋊#615712yes—
588C2×C4C2×C22⋊#615712yes—
589C22—21no—
590C22—22no—
591C22—21no—
592C22—22no—
593C22—44no—
594C22—532no—
595C22—531no—
596C22—531no—
597C22—534no—
598C22—1062no—
599C22—1061no—
600C22—1061no—
601C22—1062no—
602C22—1061no—
603C22—1061no—
604C22—2124no—
605C22—2124no—
606C22—2124no—
607C22—2124no—
608C22—2122no—
609C22—2122no—
610C22—2124no—
611C22(C2×#6157)⋊Q814yes—
612C22C22×C22⋊#615711yescharacteristic
613C22C2×Q8×#615712yes—
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—1062no—
621C4—1062no—
622C4—4244no—
623C4—2128no—
624C4—2121no—
625C4—2121no—
626C4—2121no—
627C4—2121no—
628C2—24no—
629C2—22no—
630C2—1061no—
631C2—1062no—
632C2—1061no—
633C2—1064no—
634C2C2×(C2×#6157)⋊Q811yescharacteristic
635C2#2029131411yescharacteristic
636C2#2029131511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C22⋊Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC2×C22⋊Q8#6157
hypocenterC53#21333
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20290906C2semidirect
4. #20290906C2semidirect
5. #20290905C2semidirect
6. #20290904C2semidirect
24. C2×#13133197C4semidirect
26. C2×#13133196C4semidirect
27. C2×#13133195C4semidirect
28. C2×#13133195C4semidirect
31. C2×#13133194C4semidirect
32. C2×#13133194C4semidirect
33. C2×#13133193C4semidirect
38. C2×C22⋊Q8×C53C4semidirect
47. C2×C22⋊#6157Q8semidirect
57. C22×Dic106C2×C4semidirect
58. C22×Dic106C2×C4semidirect
59. C22×Dic106C2×C4semidirect
80. C2×C4⋊Dic53C2×C4semidirect
81. C2×C4⋊Dic53C2×C4semidirect
82. C2×C4⋊Dic53C2×C4semidirect
83. C2×#11719919C2×C4semidirect
84. C2×#11719919C2×C4semidirect
85. C2×#11719919C2×C4semidirect
86. C2×#11719919C2×C4semidirect
87. C2×#11719919C2×C4semidirect
88. C2×#11719919C2×C4semidirect
105. C22×Q8×C53C2×C4semidirect
108. C2×C4⋊C4×C53C2×C4semidirect
109. C2×C4⋊C4×C53C2×C4semidirect
110. C2×C4⋊C4×C53C2×C4semidirect
341. C2×C22⋊Q8#6157semidirect
357. C53#21333semidirect
362. C22×Q8C2×#6157semidirect
398. C2×C4⋊C4C2×#6157semidirect
399. C2×C4⋊C4C2×#6157semidirect
400. C2×C4⋊C4C2×#6157semidirect

Maximal subgroups

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

subgroupindex
#202911942
#202911512
#202909062
#202909052
#202909042
#202904122
C2×#168115012
C2×#168112482
#2133353