Finite Group Databasebeta

← #16800211

Subgroups — #16800211

Order 3328.

834 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719158C211yesmaximal, characteristic, complemented
2C2×#11719062C211yesmaximal, characteristic
3#13131005C211yesmaximal, characteristic, complemented
4#13130935C211yesmaximal, characteristic, complemented
5#13130895C211yesmaximal, characteristic, complemented
6#13130894C211yesmaximal, characteristic, complemented
7#13130780C211yesmaximal, characteristic
8#13130779C211yesmaximal, characteristic
9#13130694C211yesmaximal, characteristic
10#13130635C211yesmaximal, characteristic, complemented
11#13130635C211yesmaximal, characteristic, complemented
12#13130628C211yesmaximal, characteristic, complemented
13#13130578C211yesmaximal, characteristic
14#13130578C211yesmaximal, characteristic
15#13118103C211yesmaximal, characteristic
16C22×#6122C2211yescharacteristic
17C23×#1057C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×#88131C2211yescharacteristic
20C2×C22⋊#1057C2211yescharacteristic
21C2×C22⋊#1057C2211yescharacteristic
22#11719158—24no—
23C2×#88112C2211yescharacteristic
24C2×#88112C2211yescharacteristic
25C2×#88112C2211yescharacteristic
26C2×#88110C2211yescharacteristic
27C2×#88110C2211yescharacteristic
28C2×#88110C2211yescharacteristic
29C2×#88110C2211yescharacteristic
30C2×C4⋊#1057C2211yescharacteristic
31C2×C4⋊#1057C2211yescharacteristic
32C2×C4⋊#1057C2211yescharacteristic
33C2×C4×#1057C2211yescharacteristic
34C2×C4×#1057C2211yescharacteristic
35C2×C4×#1057C2211yescharacteristic
36C2×C4×#1057C2211yescharacteristic
37#11719062—24no—
38C13×#393C411yescharacteristic
39#11718890C411yescharacteristic
40#11718610C411yescharacteristic
41#11718609C411yescharacteristic
42#11718609C411yescharacteristic
43C22⋊C4×Dic13C2211yescharacteristic
44C22⋊C4×Dic13C2211yescharacteristic
45#11718320C411yescharacteristic
46#11718318C411yescharacteristic
47#11718318C411yescharacteristic
48#11718310C2211yescharacteristic
49#11718310C2211yescharacteristic
50#11718310C2211yescharacteristic
51#11718290C2211yescharacteristic
52#11718290C2211yescharacteristic
53#11718290C2211yescharacteristic
54#11718290C2211yescharacteristic
55#11718290C2211yescharacteristic
56#11718290C2211yescharacteristic
57#11718290C2211yescharacteristic
58#11718290C2211yescharacteristic
59#11718290C2211yescharacteristic
60#11718290C2211yescharacteristic
61C22×Q8×C13C2×C411yescharacteristic
62C23×Dic13C2311yescharacteristic
63C2×#6122—22no—
64C2×#6122—22no—
65C2×#6122Q814yescomplemented
66C22×#1057—22no—
67C22×#1057—22no—
68C22×#1057—22no—
69C22×#1057C2311yescharacteristic
70C22×#1057C2311yescharacteristic
71C22×#1057C2311yescharacteristic
72C22×#1057C2311yescharacteristic
73C22×#1057C2311yescharacteristic
74C22×#1057C2311yescharacteristic
75C22×#1057C2311yescharacteristic
76C22×#1057C2311yescharacteristic
77C2×C13×C22⋊C4C2×C411yescharacteristic
78C2×C13×C22⋊C4C2×C411yescharacteristic
79C2×C13×C22⋊C4C2×C411yescharacteristic
80C2×C22⋊Dic13C2×C411yescharacteristic
81C2×C22⋊Dic13C2×C411yescharacteristic
82C2×C22⋊Dic13C2×C411yescharacteristic
83Q8×Dic13—22no—
84Q8×Dic13—22no—
85C2×C4⋊Dic13C2×C411yescharacteristic
86C2×C4⋊Dic13C2×C411yescharacteristic
87C2×C4⋊Dic13C2×C411yescharacteristic
88C2×#6101C2×C411yescharacteristic
89C2×#6101C2×C411yescharacteristic
90C2×#6101C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×#6101C2×C411yescharacteristic
93C2×#6101C2×C411yescharacteristic
94C2×#6101C2×C411yescharacteristic
95C2×#6101C2×C411yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C2×C4×Dic13C2311yescharacteristic
100C2×C4×Dic13C2311yescharacteristic
101C2×C4×Dic13C2311yescharacteristic
102#88131—22no—
103#88131—22no—
104C22⋊#1057—22no—
105C22⋊#1057—22no—
106#88112—22no—
107#88112—22no—
108#88112—22no—
109#88110—22no—
110#88110—22no—
111#88110—22no—
112#88110D414yes—
113C4⋊#1057—22no—
114C4⋊#1057—22no—
115C4⋊#1057—22no—
116C4×#1057—22no—
117C4×#1057—22no—
118C4×#1057—22no—
119C4×#1057—22no—
120C13×#95C2×C411yescharacteristic
121C13×#95C2×C411yescharacteristic
122C13×#95C2×C411yescharacteristic
123#88071C2×C411yescharacteristic
124#88071C2×C411yescharacteristic
125#88071C2×C411yescharacteristic
126#88071C2×C411yescharacteristic
127#88071C2×C411yescharacteristic
128#88071C2×C411yescharacteristic
129#88071C2×C411yescharacteristic
130#21606—131nomaximal, Sylow
131C24×C13C22×C411yescharacteristic
132C2×Q8×C13—22no—
133C2×Q8×C13—22no—
134C2×Q8×C13—41no—
135C22×C4×C13C22×C411yescharacteristic
136C22×C4×C13C22×C411yescharacteristic
137C22×C4×C13C22×C411yescharacteristic
138C22×C4×C13C22×C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×C4×C13C22×C411yescharacteristic
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13C22×C411yescharacteristic
146C22×Dic13C22×C411yescharacteristic
147C22×Dic13C22×C411yescharacteristic
148C22×Dic13C22×C411yescharacteristic
149C22×Dic13C22×C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C22×Dic13C2×Q812yes—
153C22×Dic13#4112yes—
154C22×Dic13C2411yescharacteristic
155#6122—28no—
156C2×#1057—28no—
157C2×#1057—21no—
158C2×#1057—21no—
159C2×#1057—21no—
160C2×#1057—21no—
161C2×#1057—21no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—21no—
166C2×#1057—21no—
167C2×#1057—21no—
168C2×#1057—21no—
169C2×#1057—21no—
170C2×#1057—21no—
171C2×#1057—21no—
172C2×#1057—21no—
173C2×#1057—21no—
174C2×#1057—21no—
175C2×#1057C2×D412yes—
176C2×#1057C2×D412yes—
177C2×#1057C2×Q812yes—
178C2×#1057C2×Q812yes—
179C2×#1057#4112yes—
180C2×#1057#4112yes—
181C13×C22⋊C4—22no—
182C13×C22⋊C4—22no—
183C13×C22⋊C4—41no—
184C22⋊Dic13—22no—
185C22⋊Dic13—22no—
186C22⋊Dic13—41no—
187C4⋊Dic13—22no—
188C4⋊Dic13—22no—
189C4⋊Dic13—22no—
190#6101—22no—
191#6101—22no—
192#6101—22no—
193#6101—22no—
194#6101—22no—
195#6101—41no—
196#6101—41no—
197#6101—41no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C4×Dic13—21no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—21no—
207C4×Dic13—21no—
208C4×Dic13C2×D412yes—
209C4×Dic13#4112yes—
210C2×#446—131no—
211C2×#405—131no—
212#2396—131no—
213#1984—131no—
214#1939—131no—
215#1917—131no—
216#1917—131no—
217#1850—131no—
218#1850—131no—
219#1798—131no—
220#1771—131no—
221#1771—131no—
222C8×C4⋊C4—131no—
223#1753—131no—
224#1753—131no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13C4×Q812yes—
232C23×C13C23×C411yescharacteristic
233Q8×C13—44no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13C4×D412yes—
251C2×C4×C13#13112yes—
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—22no—
259C2×Dic13—22no—
260C2×Dic13—22no—
261C2×Dic13—22no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13C4×D412yes—
275C2×Dic13C4×Q812yes—
276C2×Dic13#13112yes—
277C2×Dic13C22×D411yescharacteristic
278C2×Dic13C22×Q811yescharacteristic
279C2×Dic13C2×#4111yescharacteristic
280C2×Dic13C2×#4111yescharacteristic
281C2×Dic13C2×#4111yescharacteristic
282C2×Dic13C2×#4111yescharacteristic
283#1057—22no—
284#1057—22no—
285#1057—22no—
286#1057—41no—
287#1057—41no—
288#1057—41no—
289#1057—41no—
290#1057C22⋊Q814yes—
291C22×M16—131no—
292C23×C8—131no—
293C2×C4×Q8—131no—
294C2×#117—131no—
295#446—264no—
296C2×C4⋊C8—131no—
297C2×C4⋊C8—131no—
298C2×C4⋊C8—131no—
299C2×C4⋊C8—131no—
300C2×C4⋊C8—131no—
301C2×C4⋊C8—131no—
302C2×C22⋊C8—131no—
303C2×C22⋊C8—131no—
304#405—264no—
305C2×#97—131no—
306C2×#97—131no—
307C2×#97—131no—
308C2×#97—131no—
309C2×C4×C8—131no—
310C2×C4×C8—131no—
311C2×C4×C8—131no—
312C2×C4×C8—131no—
313#400—131no—
314#399—131no—
315#398—131no—
316#398—131no—
317#396—131no—
318#396—131no—
319#395—131no—
320#393#26111yescharacteristic
321C4×C4⋊C4—131no—
322C4×C4⋊C4—131no—
323C4×C4⋊C4—131no—
324C4×C22⋊C4—131no—
325C4×C22⋊C4—131no—
326#336—131no—
327#336—131no—
328#336—131no—
329#336—131no—
330#336—131no—
331#336—131no—
332#336—131no—
333#336—131no—
334#336—131no—
335#336—131no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—41no—
340C22×C13—41no—
341C22×C13—41no—
342C22×C13—41no—
343C22×C13C2×C4×D411yescharacteristic
344C22×C13C2×C4×Q811yescharacteristic
345C22×C13C4×#4111yescharacteristic
346C22×C13C2×#13111yescharacteristic
347C22×C13C2×#13111yescharacteristic
348C22×C13#56811yescharacteristic
349C22×C13#56811yescharacteristic
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—41no—
353C4×C13—41no—
354C4×C13—41no—
355C4×C13—41no—
356Dic13—22no—
357Dic13—22no—
358Dic13—22no—
359Dic13—41no—
360Dic13—41no—
361Dic13—41no—
362Dic13—41no—
363Dic13C2×C22⋊Q811yescharacteristic
364Dic13#52511yescharacteristic
365Dic13#52911yescharacteristic
366Dic13#53311yescharacteristic
367C22×Q8C2×#26111yescharacteristic
368C23×C4—131no—
369C2×M16—134no—
370C2×M16—262no—
371C2×M16—262no—
372C22×C8—131no—
373C22×C8—131no—
374C22×C8—131no—
375C22×C8—131no—
376C22×C8—131no—
377C22×C8—131no—
378C22×C8—131no—
379C22×C8—131no—
380C22×C8—262no—
381C22×C8—262no—
382C22×C8—262no—
383C4×Q8—262no—
384C4×Q8—262no—
385#117—262no—
386#117—262no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—131no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4—131no—
397C2×C4⋊C4—131no—
398C2×C22⋊C4—131no—
399C2×C22⋊C4—131no—
400C2×C22⋊C4—131no—
401C2×C22⋊C4C2×#26111yescharacteristic
402C2×C22⋊C4C2×#26111yescharacteristic
403C2×C22⋊C4C2×#26111yescharacteristic
404C2×C42—131no—
405C2×C42—131no—
406C2×C42—131no—
407C2×C42—131no—
408C2×C42—131no—
409C2×C42—131no—
410C4⋊C8—262no—
411C4⋊C8—262no—
412C4⋊C8—262no—
413C4⋊C8—262no—
414C4⋊C8—262no—
415C4⋊C8—262no—
416C22⋊C8—262no—
417C22⋊C8—262no—
418#97—134no—
419#97—262no—
420#97—262no—
421#97—262no—
422C4×C8—262no—
423C4×C8—262no—
424C4×C8—262no—
425C4×C8—262no—
426#95—131no—
427#95—131no—
428#95—131no—
429#95—131no—
430#95—131no—
431#95—131no—
432#95—131no—
433#95C2×#26111yescharacteristic
434#95C2×#26111yescharacteristic
435#95C2×#26111yescharacteristic
436C2×C13—42no—
437C2×C13C4×C22⋊Q811yescharacteristic
438C2×C13#293311yescharacteristic
439C2×C13#293511yescharacteristic
440C2×C13#296211yescharacteristic
441C2×C13#296211yescharacteristic
442C2×C13#296611yescharacteristic
443C2×C13#296711yescharacteristic
444C24C22×#26111yescharacteristic
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—41no—
448C22×C4—131no—
449C22×C4—132no—
450C22×C4—132no—
451C22×C4—131no—
452C22×C4—131no—
453C22×C4—131no—
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—261no—
461C22×C4—261no—
462C22×C4C22×#26111yescharacteristic
463C22×C4C22×#26111yescharacteristic
464C22×C4C22×#26111yescharacteristic
465C22×C4C22×#26111yescharacteristic
466C22×C4C22×#26111yescharacteristic
467C22×C4C22×#26111yescharacteristic
468M16—268no—
469C2×C8—132no—
470C2×C8—132no—
471C2×C8—132no—
472C2×C8—132no—
473C2×C8—132no—
474C2×C8—132no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—268no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8—261no—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C4⋊C4—262no—
495C4⋊C4—262no—
496C4⋊C4—262no—
497C4⋊C4—262no—
498C4⋊C4—262no—
499C4⋊C4—262no—
500C4⋊C4—262no—
501C4⋊C4—262no—
502C4⋊C4—521no—
503C4⋊C4—521no—
504C4⋊C4—521no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—262no—
509C22⋊C4—262no—
510C22⋊C4—521no—
511C42—132no—
512C42—132no—
513C42—261no—
514C42—261no—
515C42—261no—
516C42—261no—
517C42—261no—
518C42—261no—
519C42—261no—
520C42—261no—
521C42—261no—
522C42—261no—
523C13#2160611yescharacteristic, complemented, Sylow
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23Q8×#26112yes—
531C23C23×#26111yescharacteristic
532Q8—44no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—131no—
550C2×C4—131no—
551C2×C4—131no—
552C2×C4—131no—
553C2×C4—131no—
554C2×C4—131no—
555C2×C4—132no—
556C2×C4—132no—
557C2×C4—132no—
558C2×C4—262no—
559C2×C4—262no—
560C2×C4—262no—
561C2×C4—262no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—261no—
569C2×C4—261no—
570C2×C4—261no—
571C2×C4—261no—
572C2×C4—261no—
573C2×C4—261no—
574C2×C4—261no—
575C2×C4—261no—
576C2×C4—261no—
577C2×C4—261no—
578C2×C4—261no—
579C2×C4—261no—
580C2×C4#8823812yes—
581C2×C4D4×#26112yes—
582C8—134no—
583C8—262no—
584C8—262no—
585C8—262no—
586C8—521no—
587C8—521no—
588C8—521no—
589C8—521no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#8823811yescharacteristic
598C22#1171962111yescharacteristic
599C22C2×D4×#26111yescharacteristic
600C22C2×Q8×#26111yescharacteristic
601C22D13⋊#13111yescharacteristic
602C22#1171962911yescharacteristic
603C22#41×#26111yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—131no—
611C4—131no—
612C4—131no—
613C4—131no—
614C4—262no—
615C4—262no—
616C4—262no—
617C4—521no—
618C4—521no—
619C4—521no—
620C4—521no—
621C2—42no—
622C2C22⋊Q8×#26111yescharacteristic
623C2#1313156811yescharacteristic
624C2#1313158111yescharacteristic
625C2#1313163111yescharacteristic
626C2#1313163311yescharacteristic
627C2#1313165011yescharacteristic
628C2#1313165611yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC13×#393C4
prime radicalC24×C13C22×C4
cosocleC22×Dic13C24
hypercenter#393#261
hypocenterC13#21606
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#11719158C2semidirect
3. #13131005C2semidirect
4. #13130935C2semidirect
5. #13130895C2semidirect
6. #13130894C2semidirect
10. #13130635C2semidirect
11. #13130635C2semidirect
12. #13130628C2semidirect
65. C2×#6122Q8semidirect
523. C13#21606semidirect

Maximal subgroups

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

subgroupindex
C2×#117191582
C2×#117190622
#131310052
#131309352
#131308952
#131308942
#131307802
#131307792
#131306942
#131306352
#131306282
#131305782
#131181032
#2160613