Finite Group Databasebeta

← #15865999

Subgroups — #15865999

Order 3328.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117487C211yesmaximal, characteristic, complemented
2#13117197C211yesmaximal, characteristic
3#13115479C211yesmaximal, characteristic, complemented
4#13115472C211yesmaximal, characteristic, complemented
5#13115430C211yesmaximal, characteristic, complemented
6#13115409C211yesmaximal, characteristic, complemented
7#13115172C211yesmaximal, characteristic
8#13115172C211yesmaximal, characteristic
9#13115099C211yesmaximal, characteristic
10#13115085C211yesmaximal, characteristic
11#13114733C211yesmaximal, characteristic, complemented
12C4×#88075C211yesmaximal, characteristic, complemented
13C4×#88072C211yesmaximal, characteristic
14#13114589C211yesmaximal, characteristic, complemented
15C2×C13×#440C211yesmaximal, characteristic
16C22×C13×SD16C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×D4×Dic13C2211yescharacteristic
20C2×D4×Dic13C2211yescharacteristic
21C13×#440C418yescomplemented
22C2×C13×#107C2211yescharacteristic
23C2×C13×#103C2211yescharacteristic
24C2×C13×#102C2211yescharacteristic
25C2×C13×#97C2211yescharacteristic
26#11718891C2211yescharacteristic
27#11718891C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C4×C22⋊Dic13—21no—
33C2×#88071—21no—
34C2×#88058C2211yescharacteristic
35C2×#88057C2211yescharacteristic
36#11718646C2211yescharacteristic
37#11718646C2211yescharacteristic
38#11718642—21no—
39#11718642—21no—
40C4⋊C4×Dic13C2211yescharacteristic
41C4⋊C4×Dic13C2211yescharacteristic
42C22⋊C4×Dic13—21no—
43C22⋊C4×Dic13—21no—
44#11718598—21no—
45#11718597—21no—
46C4×C4⋊Dic13C2211yescharacteristic
47C4×C4⋊Dic13C2211yescharacteristic
48#11718593C2211yescharacteristic
49C2×C4×#1055C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C23×Dic13—21no—
64C23×Dic13—21no—
65C2×C13×SD16—22no—
66C2×C13×SD16—22no—
67C2×C13×SD16C2×C414yes—
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C4×Q8×C13—22no—
71C4×Q8×C13C2×C414yescomplemented
72C4×D4×C13—22no—
73C4×D4×C13C2×C414yes—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C42×C13—21no—
79C2×C42×C13C2311yescharacteristic
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82D4×Dic13—22no—
83D4×Dic13—22no—
84D4×Dic13—22no—
85D4×Dic13—22no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C13×#107C2×C414yescomplemented
101C13×#103C2×C414yescomplemented
102C13×#102C2×C414yes—
103C13×#97C2×C414yescomplemented
104#88075—22no—
105#88075—22no—
106#88072—22no—
107#88072—22no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071D411yescharacteristic
119#88071D411yescharacteristic
120#88071D411yescharacteristic
121#88071D411yescharacteristic
122#88058—22no—
123#88057—22no—
124C4×#1055—22no—
125#21125—131nomaximal, Sylow
126C24×C13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13C22×C412yes—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13C22×C412yes—
137C2×D4×C13#4112yes—
138C2×D4×C13#4112yes—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—24no—
143C22×C4×C13—81no—
144C22×C4×C13—81no—
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2411yescharacteristic
148C22×Dic13—21no—
149C22×Dic13—21no—
150C22×Dic13—24no—
151C22×Dic13—24no—
152C22×Dic13—21no—
153C22×Dic13—21no—
154C22×Dic13—81no—
155C22×Dic13—81no—
156C22×Dic13—81no—
157C22×Dic13—81no—
158C22×Dic13C2×D411yescharacteristic
159C22×Dic13C2×D411yescharacteristic
160C22×Dic13C2×D411yescharacteristic
161C22×Dic13C2×D411yescharacteristic
162C13×SD16—28no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13C22×C412yes—
168C2×C8×C13C22×C412yes—
169C13×C4⋊C4—24no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4C22×C412yes—
173C13×C4⋊C4C22×C412yes—
174C13×C22⋊C4—24no—
175C42×C13—24no—
176C42×C13C22×C412yes—
177C42×C13#4112yes—
178C22⋊Dic13—24no—
179C22⋊Dic13—24no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—81no—
187C4⋊Dic13—81no—
188C4⋊Dic13#4112yes—
189C4⋊Dic13#4112yes—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—81no—
195C4×Dic13—81no—
196C4×Dic13—81no—
197C4×Dic13—81no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#440D1311yescharacteristic
205#2320—131no—
206#2275—131no—
207#1928—131no—
208#1927—131no—
209#1921—131no—
210#1918—131no—
211#1877—131no—
212#1877—131no—
213#1864—131no—
214#1862—131no—
215#1776—131no—
216C4×#103—131no—
217C4×#102—131no—
218#1746—131no—
219C23×C13—22no—
220C23×C13—22no—
221C23×C13—22no—
222C23×C13—41no—
223C23×C13—41no—
224C23×C13—41no—
225C23×C13—41no—
226C23×C13C22×D411yescharacteristic
227Q8×C13—22no—
228Q8×C13—22no—
229Q8×C13—41no—
230Q8×C13—41no—
231D4×C13—22no—
232D4×C13—22no—
233D4×C13—22no—
234D4×C13#11714yes—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—84no—
249C2×C4×C13C4×D412yes—
250C2×C4×C13C4×D412yes—
251C2×C4×C13C23×C411yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—22no—
262C2×Dic13—22no—
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—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281C2×Dic13—84no—
282C2×Dic13—84no—
283C8×C13—22no—
284C8×C13—22no—
285#1055—41no—
286#1055—41no—
287C22×SD16C2×D1311yescharacteristic
288C2×C4×Q8C2×D1311yescharacteristic
289C2×C4×D4—131no—
290C2×C4×D4—131no—
291C2×C4×D4C2×D1311yescharacteristic
292#440Dic1318yescomplemented
293C2×#107—131no—
294C2×#107C2×D1311yescharacteristic
295C2×#106—131no—
296C2×#103—131no—
297C2×#103—131no—
298C2×#103C2×D1311yescharacteristic
299C2×#102—131no—
300C2×#102—131no—
301C2×#102C2×D1311yescharacteristic
302C2×#97C2×D1311yescharacteristic
303C2×C4×C8—131no—
304#391—131no—
305#391—131no—
306#389—131no—
307#389—131no—
308#387—261no—
309#387—261no—
310#383—261no—
311#382—261no—
312C4×C4⋊C4—131no—
313C4×C4⋊C4—131no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—131no—
316C4×C22⋊C4—261no—
317C4×C22⋊C4—261no—
318C4×C22⋊C4—261no—
319#376—131no—
320C2×#95—261no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#336—131no—
328#336—131no—
329#336—131no—
330#336—131no—
331C22×C13—24no—
332C22×C13—42no—
333C22×C13—42no—
334C22×C13—42no—
335C22×C13—42no—
336C22×C13—42no—
337C22×C13—42no—
338C22×C13C2×C4×D411yescharacteristic
339C22×C13#52511yescharacteristic
340C22×C13#52511yescharacteristic
341C22×C13#57611yescharacteristic
342C22×C13#57711yescharacteristic
343C22×C13#57711yescharacteristic
344C22×C13#57811yescharacteristic
345C4×C13—22no—
346C4×C13—22no—
347C4×C13—42no—
348C4×C13—42no—
349C4×C13—42no—
350C4×C13C2×#11711yescharacteristic
351C4×C13C4×#4111yescharacteristic
352C4×C13#52911yescharacteristic
353C4×C13#52911yescharacteristic
354Dic13—41no—
355Dic13—42no—
356Dic13—41no—
357Dic13—41no—
358Dic13—42no—
359Dic13—41no—
360Dic13—81no—
361Dic13—81no—
362C22×Q8C22×D1311yescharacteristic
363C22×D4C22×D1311yescharacteristic
364C23×C4—21no—
365C23×C4—261no—
366C23×C4—261no—
367C2×SD16—22no—
368C2×SD16—22no—
369C2×SD16C2×Dic1314yes—
370C22×C8—131no—
371C22×C8—131no—
372C22×C8C22×D1311yescharacteristic
373C22×C8C22×D1311yescharacteristic
374C4×Q8—22no—
375C4×Q8C2×Dic1314yescomplemented
376C4×D4—22no—
377C4×D4—262no—
378C4×D4—262no—
379C4×D4—262no—
380C4×D4—262no—
381C4×D4C2×Dic1314yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4—261no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4C22×D1311yescharacteristic
390C2×C4⋊C4C22×D1311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—261no—
393C2×C22⋊C4—261no—
394C2×C42—21no—
395C2×C42—131no—
396C2×C42—131no—
397C2×C42—261no—
398C2×C42—261no—
399C2×C42—261no—
400C2×C42—261no—
401C2×C42C22×D1311yescharacteristic
402#107—262no—
403#107C2×Dic1314yescomplemented
404#106—262no—
405#103—262no—
406#103—262no—
407#103C2×Dic1314yescomplemented
408#102—262no—
409#102—262no—
410#102C2×Dic1314yes—
411#97C2×Dic1314yescomplemented
412C4×C8—262no—
413#95—131no—
414#95—131no—
415#95—131no—
416#95—131no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424#95—261no—
425#95—261no—
426#95—261no—
427C2×C13—44no—
428C2×C13#229011yescharacteristic
429C2×C13#294611yescharacteristic
430C2×C13#294711yescharacteristic
431C2×C13#311811yescharacteristic
432C2×C13#312011yescharacteristic
433C2×C13#312311yescharacteristic
434C2×C13#312411yescharacteristic
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C22×Dic1312yes—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4#612712yes—
446C2×D4#612712yes—
447C2×D4C22×Dic1312yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—264no—
461C22×C4—264no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—1041no—
465C22×C4—1041no—
466C22×C4—1041no—
467C22×C4—1041no—
468C22×C4D4×D1311yescharacteristic
469C22×C4D4×D1311yescharacteristic
470C22×C4C23×D1311yescharacteristic
471SD16—28no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—261no—
482C2×C8C22×Dic1312yes—
483C2×C8C22×Dic1312yes—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—132no—
488C4⋊C4—132no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—1041no—
496C4⋊C4—1041no—
497C4⋊C4C22×Dic1312yes—
498C4⋊C4C22×Dic1312yes—
499C22⋊C4—24no—
500C22⋊C4—264no—
501C22⋊C4—264no—
502C42—24no—
503C42—261no—
504C42—261no—
505C42—261no—
506C42—261no—
507C42—1041no—
508C42—1041no—
509C42—1041no—
510C42—1041no—
511C42#612912yes—
512C42C22×Dic1312yes—
513C13#2112511yescharacteristic, complemented, Sylow
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D1311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#8818014yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—261no—
545C2×C4—261no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—262no—
557C2×C4—262no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—1044no—
569C2×C4—1044no—
570C2×C4D4×Dic1312yes—
571C2×C4D4×Dic1312yes—
572C2×C4C2×#612711yescharacteristic
573C2×C4C2×#612711yescharacteristic
574C2×C4C2×#612911yescharacteristic
575C2×C4#41×D1311yescharacteristic
576C2×C4#41×D1311yescharacteristic
577C2×C4C23×Dic1311yescharacteristic
578C8—22no—
579C8—22no—
580C8—521no—
581C8—521no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22#1171925211yescharacteristic
590C22#1171925211yescharacteristic
591C22#1171947911yescharacteristic
592C22#1171948011yescharacteristic
593C22#1171948311yescharacteristic
594C22#1171948411yescharacteristic
595C22C2×D4×Dic1311yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—521no—
602C4—522no—
603C4—521no—
604C4—521no—
605C4—522no—
606C4—521no—
607C4—1041no—
608C4—1041no—
609C4#1171924311yescharacteristic
610C4#1171924311yescharacteristic
611C4C2×#8818011yescharacteristic
612C4#41×Dic1311yescharacteristic
613C2—44no—
614C2#1311728111yescharacteristic
615C2#1312366911yescharacteristic
616C2#1312368011yescharacteristic
617C2#1312502411yescharacteristic
618C2#1312505411yescharacteristic
619C2#1312508711yescharacteristic
620C2#1312510211yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#440C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#440D13
hypocenterC13#21125
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #13117487C2semidirect
3. #13115479C2semidirect
4. #13115472C2semidirect
5. #13115430C2semidirect
6. #13115409C2semidirect
11. #13114733C2semidirect
12. C4×#88075C2semidirect
14. #13114589C2semidirect
21. C13×#440C4semidirect
71. C4×Q8×C13C2×C4semidirect
100. C13×#107C2×C4semidirect
101. C13×#103C2×C4semidirect
103. C13×#97C2×C4semidirect
292. #440Dic13semidirect
375. C4×Q8C2×Dic13semidirect
403. #107C2×Dic13semidirect
407. #103C2×Dic13semidirect
411. #97C2×Dic13semidirect
513. C13#21125semidirect

Maximal subgroups

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

subgroupindex
#131174872
#131171972
#131154792
#131154722
#131154302
#131154092
#131151722
#131150992
#131150852
#131147332
C4×#880752
C4×#880722
#131145892
C2×C13×#4402
#2112513