Finite Group Databasebeta

← #15897270

Subgroups — #15897270

Order 3328.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719182C211yesmaximal, characteristic
2C2×#11718681C211yesmaximal, characteristic, complemented
3C2×#11718688C211yesmaximal, characteristic, complemented
4#13116025C212yesmaximal, complemented
5#13115837C212yesmaximal, complemented
6#13115788C211yesmaximal, characteristic, complemented
7#13115763C211yesmaximal, characteristic, complemented
8#13115173C212yesmaximal, complemented
9#13115096C212yesmaximal, complemented
10#13114816C211yesmaximal, characteristic, complemented
11C4×C13×C22⋊Q8C211yesmaximal, characteristic, complemented
12C22×#6127—21no—
13C2×C13×C22⋊Q8C2211yescharacteristic
14C2×C4×Q8×C13C2211yescharacteristic
15C22×C42×C13C2211yescharacteristic
16C22×#6106C2212yes—
17C2×#88184—21no—
18C2×#88184C2212yes—
19#11719182—24no—
20#11719182—24no—
21C2×#88129—21no—
22C2×#88128C2211yescharacteristic
23C2×C4×D52—21no—
24C2×C4×D52C2212yes—
25C2×#88123C2211yescharacteristic
26C13×#391C2211yescharacteristic
27C13×#389C2211yescharacteristic
28C13×#387—21no—
29C13×#385—21no—
30C13×#380C2211yescharacteristic
31C4×C13×C4⋊C4—21no—
32C4×C13×C4⋊C4C2211yescharacteristic
33C4×C13×C22⋊C4—21no—
34C2×#88075C2212yes—
35C2×C22⋊#1055C2212yes—
36#11718688—24no—
37#11718681—24no—
38C2×#88049C2212yes—
39C2×#88049C2212yes—
40C2×#88049C2212yes—
41C2×#88048C2212yes—
42C2×C4⋊#1055C2211yescharacteristic
43C2×C4⋊#1055C2211yescharacteristic
44#11718121C2212yes—
45#11718121C2212yes—
46#11718116C2212yes—
47C22×Q8×C13C2311yescharacteristic
48C23×C4×C13D411yescharacteristic
49C23×C4×C13D411yescharacteristic
50C23×C4×C13C2311yescharacteristic
51C22×C22⋊D13—22no—
52C2×#6127—28no—
53C2×#6127—44no—
54C2×#6127—44no—
55C22×D52—21no—
56C22×D52—21no—
57C22×D52C2312yes—
58C22×C4×D13—22no—
59C22×Dic26—21no—
60C13×C22⋊Q8—22no—
61C13×C22⋊Q8—22no—
62C4×Q8×C13—22no—
63C4×Q8×C13—22no—
64C2×C13×C4⋊C4—21no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C22⋊C4—21no—
71C2×C13×C22⋊C4—21no—
72C2×C42×C13—22no—
73C2×C42×C13—22no—
74C2×C42×C13—21no—
75C2×C42×C13—21no—
76C2×C42×C13D411yescharacteristic
77C2×C42×C13D411yescharacteristic
78C2×C42×C13C2311yescharacteristic
79C2×C42×C13C2311yescharacteristic
80C2×#6106—24no—
81C2×#6106—24no—
82C2×#6106—42no—
83#88184—28no—
84#88184—24no—
85#88184—24no—
86C2×#6103—22no—
87C2×#6103—22no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13C2312yes—
90C22×#1055C2312yes—
91C22×#1055C2312yes—
92#88129—28no—
93#88128—22no—
94#88128D414yes—
95C4×D52—28no—
96C4×D52—24no—
97C4×D52—24no—
98#88123—22no—
99#88123D414yes—
100C13×#95—21no—
101C13×#95—21no—
102C13×#95—21no—
103C13×#95—21no—
104C13×#95—21no—
105#88075—24no—
106C22⋊#1055—42no—
107#88049—24no—
108#88049—24no—
109#88049—42no—
110#88048—24no—
111C4⋊#1055—22no—
112C4⋊#1055—22no—
113#24154—131nomaximal, Sylow
114C24×C13C2×D411yescharacteristic
115C23×D13—22no—
116C2×Q8×C13—21no—
117C2×Q8×C13—21no—
118C2×Q8×C13—21no—
119C2×Q8×C13—21no—
120C2×Q8×C13—21no—
121C2×Q8×C13#4112yescomplemented
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—21no—
125C22×C4×C13—21no—
126C22×C4×C13—22no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—41no—
136C22×C4×C13—41no—
137C22×C4×C13SD1614yescomplemented
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2411yescharacteristic
144C2×C22⋊D13—28no—
145C2×C22⋊D13—44no—
146C2×C22⋊D13—44no—
147C22×Dic13—22no—
148#6127—432no—
149C2×D52—22no—
150C2×D52—22no—
151C2×D52—22no—
152C2×D52—22no—
153C2×D52—24no—
154C2×D52—24no—
155C2×D52—22no—
156C2×D52—42no—
157C2×D52—42no—
158C2×D52—42no—
159C2×D52—42no—
160C2×D52C2×D414yes—
161C2×C4×D13—28no—
162C2×C4×D13—44no—
163C2×C4×D13—44no—
164C2×Dic26—24no—
165C2×Dic26—42no—
166C2×Dic26—42no—
167C13×C4⋊C4—24no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—42no—
174C13×C4⋊C4—42no—
175C13×C4⋊C4#4112yescomplemented
176C13×C22⋊C4—24no—
177C13×C22⋊C4—42no—
178C42×C13—22no—
179C42×C13—22no—
180C42×C13—21no—
181C42×C13—21no—
182C42×C13—42no—
183C42×C13—42no—
184C42×C13C2×D412yes—
185C42×C13C2×D412yes—
186#6106—48no—
187#6103—28no—
188#6103—28no—
189C4⋊Dic13—24no—
190C4⋊Dic13—22no—
191C4⋊Dic13C2×D414yes—
192C2×#1055—22no—
193C2×#1055—22no—
194C2×#1055—22no—
195C2×#1055—22no—
196C2×#1055—22no—
197C2×#1055—22no—
198C2×#532—131no—
199C2×#475—131no—
200C2×#460—131no—
201C4×C22⋊Q8D1311yescharacteristic, complemented
202#2034—132no—
203#2000—132no—
204#1988—131no—
205#1982—131no—
206#1877—132no—
207#1864—132no—
208#1797—131no—
209C23×C13—21no—
210C23×C13—21no—
211C23×C13—21no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13—21no—
215C23×C13C2×SD1612yes—
216C23×C13C22×D411yescharacteristic
217C22×D13—24no—
218C22×D13—42no—
219C22×D13—42no—
220C22×D13—42no—
221C22×D13—42no—
222C22×D13—42no—
223C22×D13—42no—
224Q8×C13—22no—
225Q8×C13—22no—
226Q8×C13—41no—
227Q8×C13—41no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—28no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—44no—
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—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13#12012yes—
257C2×C4×C13#12012yes—
258C2×C4×C13C2×SD1612yes—
259C2×C4×C13C2×SD1612yes—
260C2×C4×C13AGL(1,ℤ/8)12yes—
261C2×C4×C13C22×D411yescharacteristic
262C2×C4×C13C22×D411yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C13H(5,𝔽2)11yescharacteristic
266C2×C4×C132−1+411yescharacteristic
267C22⋊D13—416no—
268C2×Dic13—24no—
269C2×Dic13—42no—
270C2×Dic13—42no—
271D52—24no—
272D52—24no—
273D52—48no—
274D52—42no—
275D52—42no—
276D52—48no—
277C4×D13—416no—
278Dic26—48no—
279#1055—42no—
280#1055—42no—
281C22×#41—261no—
282C22×SD16—132no—
283#532—264no—
284#532—264no—
285C2×C4⋊Q8—131no—
286C2×#127—131no—
287C2×#124—261no—
288C2×C22⋊Q8C2×D1311yescharacteristic
289C2×#121—132no—
290C2×#121—261no—
291C2×C4×Q8C2×D1311yescharacteristic
292C2×C4×D4—132no—
293C2×C4×D4—261no—
294C22×C42C2×D1311yescharacteristic
295#475—264no—
296#460—264no—
297C2×#106—132no—
298C2×C4⋊C8—131no—
299C2×C4⋊C8—131no—
300C2×#103—132no—
301C2×#102—132no—
302C2×#102—132no—
303C2×#102—132no—
304C2×C22⋊C8—132no—
305#391C2×D1311yescharacteristic
306#389C2×D1311yescharacteristic
307#387—21no—
308#385—21no—
309#380C2×D1311yescharacteristic
310C4×C4⋊C4—21no—
311C4×C4⋊C4C2×D1311yescharacteristic
312C4×C22⋊C4—21no—
313#340—132no—
314#340—132no—
315#336—132no—
316C22×C13—22no—
317C22×C13—22no—
318C22×C13—22no—
319C22×C13—41no—
320C22×C13—41no—
321C22×C13—41no—
322C22×C13—41no—
323C22×C13C2×#12011yescharacteristic
324C22×C13#52511yescharacteristic
325C22×C13#53711yescharacteristic
326C22×C13C22×SD1611yescharacteristic
327C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C13#57511yescharacteristic
329C22×C13#57511yescharacteristic
330C2×D13—44no—
331C2×D13—44no—
332C2×D13—44no—
333C2×D13—82no—
334C2×D13—82no—
335C2×D13—82no—
336C2×D13—82no—
337C4×C13—22no—
338C4×C13—22no—
339C4×C13—42no—
340C4×C13—41no—
341C4×C13—42no—
342C4×C13—41no—
343C4×C13—81no—
344C4×C13—81no—
345C4×C13#45014yes—
346C4×C13#54611yescharacteristic
347C4×C13#54711yescharacteristic
348C4×C13#54811yescharacteristic
349C4×C13#55011yescharacteristic
350Dic13—44no—
351C2×#41—268no—
352C2×#41—524no—
353C2×#41—524no—
354C22×Q8—261no—
355C22×Q8C22×D1311yescharacteristic
356C22×D4—132no—
357C22×D4—262no—
358C22×D4—261no—
359C22×D4—261no—
360C23×C4—262no—
361C23×C4C22⋊D1311yescharacteristic
362C23×C4C22⋊D1311yescharacteristic
363C23×C4C22×D1311yescharacteristic
364C2×SD16—264no—
365C2×SD16—264no—
366C2×SD16—522no—
367C22×C8—132no—
368C22×C8—132no—
369C4⋊Q8—134no—
370C4⋊Q8—262no—
371#127—134no—
372#127—262no—
373#124—268no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376#121—264no—
377#121—264no—
378#121—268no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—264no—
382C4×D4—264no—
383C4×D4—268no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—132no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4C22×D1311yescharacteristic
390C2×C4⋊C4C22×D1311yescharacteristic
391C2×C4⋊C4C22×D1311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—262no—
395C2×C22⋊C4—262no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42C22⋊D1311yescharacteristic
401C2×C42C22⋊D1311yescharacteristic
402C2×C42C22×D1311yescharacteristic
403C2×C42C22×D1311yescharacteristic
404#106—264no—
405C4⋊C8—262no—
406C4⋊C8—262no—
407#103—264no—
408#102—264no—
409#102—264no—
410#102—522no—
411C22⋊C8—522no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417C2×C13—42no—
418C2×C13#240711yescharacteristic
419C2×C13C2×#45011yescharacteristic
420C2×C13#300511yescharacteristic
421C2×C13#310311yescharacteristic
422C2×C13#311511yescharacteristic
423C2×C13#324611yescharacteristic
424C2×C13#325111yescharacteristic
425D13—84no—
426C24—262no—
427C24C2×C22⋊D1311yescharacteristic
428#41—5232no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—264no—
435C2×Q8—522no—
436C2×Q8—522no—
437C2×Q8#612712yescomplemented
438C2×D4—134no—
439C2×D4—268no—
440C2×D4—264no—
441C2×D4—264no—
442C2×D4—262no—
443C2×D4—262no—
444C2×D4—262no—
445C2×D4—262no—
446C2×D4—262no—
447C2×D4—522no—
448C2×D4—522no—
449C2×D4—524no—
450C2×D4—524no—
451C2×D4—522no—
452C2×D4—522no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—268no—
469C22×C4—262no—
470C22×C4—524no—
471C22×C4—524no—
472C22×C4#610614yescomplemented
473C22×C4C2×C22⋊D1311yescharacteristic
474C22×C4C2×C22⋊D1311yescharacteristic
475C22×C4C2×C22⋊D1311yescharacteristic
476C22×C4C2×C22⋊D1311yescharacteristic
477C22×C4C2×C22⋊D1311yescharacteristic
478C22×C4C23×D1311yescharacteristic
479SD16—528no—
480C2×C8—262no—
481C2×C8—262no—
482C2×C8—262no—
483C2×C8—262no—
484C2×C8—262no—
485C2×C8—262no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—134no—
495C4⋊C4—264no—
496C4⋊C4—262no—
497C4⋊C4#612712yescomplemented
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—268no—
501C22⋊C4—268no—
502C42—22no—
503C42—22no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—42no—
508C42D4×D1312yes—
509C42D4×D1312yes—
510C13#2415411yescharacteristic, complemented, Sylow
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—264no—
518C23—522no—
519C23—522no—
520C23—522no—
521C23—522no—
522C23—522no—
523C23—522no—
524C23C2×#610612yes—
525C23C22×C22⋊D1311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—528no—
531D4—264no—
532D4—264no—
533D4—528no—
534D4—5216no—
535D4—528no—
536D4—522no—
537D4—522no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—264no—
567C2×C4—5216no—
568C2×C4—522no—
569C2×C4—522no—
570C2×C4#8819112yes—
571C2×C4#8819112yes—
572C2×C4C2×#610612yes—
573C2×C4C2×#610612yes—
574C2×C4#8820312yes—
575C2×C4C2×#612711yescharacteristic
576C2×C4C2×#612711yescharacteristic
577C2×C4C2×D4×D1311yescharacteristic
578C2×C4C2×D4×D1311yescharacteristic
579C2×C4#8825611yescharacteristic
580C2×C4#8825711yescharacteristic
581C8—522no—
582C8—522no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—524no—
591C22—524no—
592C22—524no—
593C22—1042no—
594C22—1042no—
595C22—1042no—
596C22—1042no—
597C22#1171949611yescharacteristic
598C22C2×#8819111yescharacteristic
599C22C22×#610611yescharacteristic
600C22C2×#8820311yescharacteristic
601C22#1171952711yescharacteristic
602C22#1171952711yescharacteristic
603C22#1171953811yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—524no—
613C4#1171870714yes—
614C4#1171925511yescharacteristic
615C4#1171925611yescharacteristic
616C4#1171925711yescharacteristic
617C4#1171925811yescharacteristic
618C2—42no—
619C2—1044no—
620C2#1311818111yescharacteristic
621C2C2×#1171870711yescharacteristic
622C2#1312407611yescharacteristic
623C2#1312486711yescharacteristic
624C2#1312498711yescharacteristic
625C2#1312614811yescharacteristic
626C2#1312619811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC4×C13×C22⋊Q8C2
socleC23×C13C22×D4
prime radicalC2×#11719182C2
hypercenterC4×C22⋊Q8D13
hypocenterC13#24154
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11718681C2semidirect
3. C2×#11718688C2semidirect
4. #13116025C2semidirect
5. #13115837C2semidirect
6. #13115788C2semidirect
7. #13115763C2semidirect
8. #13115173C2semidirect
9. #13115096C2semidirect
10. #13114816C2semidirect
11. C4×C13×C22⋊Q8C2semidirect
121. C2×Q8×C13#41semidirect
137. C22×C4×C13SD16semidirect
175. C13×C4⋊C4#41semidirect
201. C4×C22⋊Q8D13semidirect
437. C2×Q8#6127semidirect
472. C22×C4#6106semidirect
497. C4⋊C4#6127semidirect
510. C13#24154semidirect

Maximal subgroups

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

subgroupindex
C2×#117191822
C2×#117186812
C2×#117186882
#131160252
#131158372
#131157882
#131157632
#131151732
#131150962
#131148162
C4×C13×C22⋊Q82
#2415413