Finite Group Databasebeta

← #15867826

Subgroups — #15867826

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718661C211yesmaximal, characteristic
2#13117445C211yesmaximal, characteristic
3#13115604C211yesmaximal, characteristic, complemented
4#13115604C211yesmaximal, characteristic, complemented
5#13115599C211yesmaximal, characteristic, complemented
6#13115594C211yesmaximal, characteristic, complemented
7#13115035C211yesmaximal, characteristic
8#13115035C211yesmaximal, characteristic
9#13115029C211yesmaximal, characteristic
10#13115024C211yesmaximal, characteristic
11#13114734C211yesmaximal, characteristic, complemented
12#13114734C211yesmaximal, characteristic, complemented
13C4×#88047C211yesmaximal, characteristic, complemented
14C4×#88048C211yesmaximal, characteristic, complemented
15C13×#2274C211yesmaximal, characteristic
16C2×C13×#117C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C4×C13×C4⋊C4—21no—
20C4×C13×C4⋊C4—21no—
21C4×C13×C4⋊C4C2211yescharacteristic
22C4×C13×C4⋊C4C2211yescharacteristic
23C4×C13×C4⋊C4C2211yescharacteristic
24C4×C13×C4⋊C4C2211yescharacteristic
25C4×C13×C22⋊C4—21no—
26C4×C13×C22⋊C4—21no—
27#11718873C2211yescharacteristic
28#11718873C2211yescharacteristic
29C2×C22⋊#1055C2211yescharacteristic
30C2×C22⋊#1055C2211yescharacteristic
31#11718661C418yescomplemented
32#11718642—21no—
33#11718642—21no—
34C2×#88048C2211yescharacteristic
35C2×#88048C2211yescharacteristic
36C2×#88048C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42#11718612—21no—
43#11718612—21no—
44#11718597C2211yescharacteristic
45#11718596C2211yescharacteristic
46C4×C4⋊Dic13C2211yescharacteristic
47C4×C4⋊Dic13C2211yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×#88043C2211yescharacteristic
51C2×C4×#1055C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C23×C4×C13—21no—
61C23×C4×C13C2311yescharacteristic
62C13×#117—22no—
63C13×#117—22no—
64C13×#117C2×C414yes—
65C13×#117Q814yes—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4—22no—
68C2×C13×C4⋊C4—22no—
69C2×C13×C4⋊C4D411yescharacteristic
70C2×C13×C4⋊C4D411yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C13×C22⋊C4D411yescharacteristic
77C2×C13×C22⋊C4D411yescharacteristic
78C2×C42×C13—21no—
79C2×C42×C13—21no—
80C2×C42×C13—21no—
81C2×C42×C13—21no—
82C2×C42×C13—21no—
83C2×C42×C13C2311yescharacteristic
84C2×C42×C13C2311yescharacteristic
85C2×C22⋊Dic13—21no—
86C2×C22⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C13×#95—21no—
100C13×#95—21no—
101C13×#95—21no—
102C13×#95—21no—
103#88071—21no—
104#88071—21no—
105#88071—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109C22⋊#1055C2×C414yes—
110C22⋊#1055C2×C414yes—
111#88048—22no—
112#88048—22no—
113#88048—22no—
114#88048—22no—
115#88047—22no—
116#88047—22no—
117#88047—22no—
118#88047—22no—
119C4⋊#1055C2×C414yescomplemented
120C4⋊#1055C2×C414yescomplemented
121#88043C2×C414yescomplemented
122C4×#1055C2×C414yescomplemented
123#21367—131nomaximal, Sylow
124C24×C13C2×D411yescharacteristic
125C22×C4×C13—21no—
126C22×C4×C13—21no—
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—24no—
134C22×C4×C13—81no—
135C22×C4×C13—81no—
136C22×C4×C13C22×C412yes—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×Q812yes—
143C22×C4×C13C2411yescharacteristic
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C13×C4⋊C4—24no—
149C13×C4⋊C4—21no—
150C13×C4⋊C4—21no—
151C13×C4⋊C4—21no—
152C13×C4⋊C4—21no—
153C13×C4⋊C4C4⋊C414yes—
154C13×C4⋊C4C4⋊C414yes—
155C13×C4⋊C4#4112yes—
156C13×C4⋊C4#4112yes—
157C13×C4⋊C4#4112yes—
158C13×C4⋊C4#4112yes—
159C13×C22⋊C4—24no—
160C13×C22⋊C4C4⋊C414yes—
161C13×C22⋊C4C4⋊C414yes—
162C42×C13—24no—
163C42×C13—81no—
164C42×C13—81no—
165C42×C13—81no—
166C42×C13—81no—
167C42×C13C22×C412yes—
168C42×C13C22×C412yes—
169C42×C13C2×Q812yes—
170C42×C13C2×Q812yes—
171C22⋊Dic13—81no—
172C22⋊Dic13—81no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—81no—
182C4⋊Dic13—81no—
183C4×Dic13—81no—
184C4×Dic13—81no—
185C2×#1055—21no—
186C2×#1055—21no—
187C2×#1055—21no—
188C2×#1055—21no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055C22×C412yes—
194C2×#1055C22×C412yes—
195C2×#1055C22×C412yes—
196C2×#1055C22×C412yes—
197C2×#433—131no—
198#2314—131no—
199#2274D1311yescharacteristic
200#1949—131no—
201#1949—131no—
202#1948—131no—
203#1947—131no—
204#1854—131no—
205#1854—131no—
206#1853—131no—
207#1852—131no—
208#1776—131no—
209#1776—131no—
210C4×#107—131no—
211C4×#106—131no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13—21no—
215C23×C13—21no—
216C23×C13—21no—
217C23×C13—21no—
218C23×C13C2×C4⋊C412yes—
219C23×C13C22×D411yescharacteristic
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—84no—
247C2×C4×C13C2×C4⋊C412yes—
248C2×C4×C13C2×C4⋊C412yes—
249C2×C4×C13C2×C4⋊C412yes—
250C2×C4×C13C2×C4⋊C412yes—
251C2×C4×C13C2×C4⋊C412yes—
252C2×C4×C13C23×C411yescharacteristic
253C2×C4×C13C22×Q811yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270#1055—22no—
271#1055—22no—
272#1055—22no—
273#1055—22no—
274C2×#117C2×D1311yescharacteristic
275C2×#117C2×D1311yescharacteristic
276C22×C4⋊C4C2×D1311yescharacteristic
277#433—138no—
278C2×#107—131no—
279C2×#107—131no—
280C2×#107—131no—
281C2×#107—131no—
282C2×#106—131no—
283C2×#106—131no—
284C2×#106—131no—
285C2×#106—131no—
286C2×C4⋊C8—131no—
287C2×C4⋊C8—131no—
288C2×C22⋊C8—131no—
289C2×C22⋊C8—131no—
290C2×#97—131no—
291C2×C4×C8—131no—
292#388—261no—
293#388—261no—
294#387—261no—
295#387—261no—
296#384—131no—
297#382—131no—
298#380—131no—
299#380—131no—
300C4×C4⋊C4—21no—
301C4×C4⋊C4—21no—
302C4×C4⋊C4—131no—
303C4×C4⋊C4—131no—
304C4×C4⋊C4C2×D1311yescharacteristic
305C4×C4⋊C4C2×D1311yescharacteristic
306C4×C4⋊C4C2×D1311yescharacteristic
307C4×C4⋊C4C2×D1311yescharacteristic
308C4×C22⋊C4—21no—
309C4×C22⋊C4—21no—
310#340—131no—
311#340—131no—
312#340—131no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318C22×C13—22no—
319C22×C13—22no—
320C22×C13—22no—
321C22×C13—41no—
322C22×C13—41no—
323C22×C13—41no—
324C22×C13—41no—
325C22×C13C22×C4⋊C411yescharacteristic
326C22×C13#52511yescharacteristic
327C22×C13#52511yescharacteristic
328C22×C13#57611yescharacteristic
329C22×C13#57711yescharacteristic
330C22×C13#57711yescharacteristic
331C22×C13#57811yescharacteristic
332C4×C13—22no—
333C4×C13—22no—
334C4×C13—22no—
335C4×C13—22no—
336C4×C13—22no—
337C4×C13—42no—
338C4×C13—42no—
339C4×C13—42no—
340C4×C13—42no—
341C4×C13C4×#4111yescharacteristic
342C4×C13C4×#4111yescharacteristic
343C4×C13#53311yescharacteristic
344C4×C13#53311yescharacteristic
345Dic13—81no—
346Dic13—81no—
347Dic13—81no—
348Dic13—81no—
349C23×C4—21no—
350C23×C4C22×D1311yescharacteristic
351C22×C8—131no—
352C22×C8—131no—
353C22×C8—131no—
354C22×C8—131no—
355#117—22no—
356#117—22no—
357#117Dic2614yes—
358#117C4×D1314yes—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—131no—
365C2×C4⋊C4—131no—
366C2×C4⋊C4—261no—
367C2×C4⋊C4—261no—
368C2×C4⋊C4C22⋊D1311yescharacteristic
369C2×C4⋊C4C22⋊D1311yescharacteristic
370C2×C4⋊C4C22×D1311yescharacteristic
371C2×C4⋊C4C22×D1311yescharacteristic
372C2×C4⋊C4C22×D1311yescharacteristic
373C2×C4⋊C4C22×D1311yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—261no—
376C2×C22⋊C4—261no—
377C2×C22⋊C4C22⋊D1311yescharacteristic
378C2×C22⋊C4C22⋊D1311yescharacteristic
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—261no—
385C2×C42—261no—
386C2×C42C22×D1311yescharacteristic
387C2×C42C22×D1311yescharacteristic
388#107—262no—
389#107—262no—
390#107—262no—
391#107—262no—
392#106—262no—
393#106—262no—
394#106—262no—
395#106—262no—
396C4⋊C8—134no—
397C4⋊C8—134no—
398C22⋊C8—134no—
399C22⋊C8—134no—
400#97—134no—
401C4×C8—134no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—261no—
407#95—261no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95—261no—
412C2×C13—42no—
413C2×C13#229611yescharacteristic
414C2×C13#291211yescharacteristic
415C2×C13#291311yescharacteristic
416C2×C13#311811yescharacteristic
417C2×C13#311911yescharacteristic
418C2×C13#312011yescharacteristic
419C2×C13#312111yescharacteristic
420C24C2×C22⋊D1311yescharacteristic
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—261no—
433C22×C4—261no—
434C22×C4—261no—
435C22×C4—261no—
436C22×C4C2×Dic2612yes—
437C22×C4C2×C4×D1312yes—
438C22×C4C2×C22⋊D1311yescharacteristic
439C22×C4C2×C22⋊D1311yescharacteristic
440C22×C4C2×C22⋊D1311yescharacteristic
441C22×C4C2×C22⋊D1311yescharacteristic
442C22×C4C2×C22⋊D1311yescharacteristic
443C22×C4C23×D1311yescharacteristic
444C2×C8—132no—
445C2×C8—132no—
446C2×C8—132no—
447C2×C8—132no—
448C2×C8—261no—
449C2×C8—261no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—261no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—261no—
462C4⋊C4—261no—
463C4⋊C4—261no—
464C4⋊C4—261no—
465C4⋊C4—261no—
466C4⋊C4—261no—
467C4⋊C4—261no—
468C4⋊C4—261no—
469C4⋊C4—1041no—
470C4⋊C4—1041no—
471C4⋊C4#610114yes—
472C4⋊C4#610114yes—
473C4⋊C4#612712yes—
474C4⋊C4#612712yes—
475C4⋊C4#612712yes—
476C4⋊C4#612712yes—
477C22⋊C4—24no—
478C22⋊C4—1041no—
479C22⋊C4—1041no—
480C22⋊C4#610114yes—
481C22⋊C4#610114yes—
482C42—24no—
483C42—81no—
484C42—81no—
485C42—81no—
486C42—81no—
487C42—1041no—
488C42—1041no—
489C42C2×Dic2612yes—
490C42C2×Dic2612yes—
491C42C2×C4×D1312yes—
492C42C2×C4×D1312yes—
493C13#2136711yescharacteristic, complemented, Sylow
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×#610112yes—
501C23C22×C22⋊D1311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—521no—
530C2×C4—521no—
531C2×C4—521no—
532C2×C4—521no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—521no—
537C2×C4—521no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4C2×#610112yes—
542C2×C4C2×#610112yes—
543C2×C4C2×#610112yes—
544C2×C4C2×#610112yes—
545C2×C4C2×#610112yes—
546C2×C4C22×Dic2611yescharacteristic
547C2×C4C22×C4×D1311yescharacteristic
548C2×C4C2×#612711yescharacteristic
549C2×C4C2×#612711yescharacteristic
550C2×C4C2×#612711yescharacteristic
551C2×C4C2×#612711yescharacteristic
552C8—262no—
553C8—262no—
554C8—262no—
555C8—262no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×#610111yescharacteristic
564C22#1171949611yescharacteristic
565C22#1171949611yescharacteristic
566C22#1171954011yescharacteristic
567C22#1171954111yescharacteristic
568C22#1171954211yescharacteristic
569C22#1171954311yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—1041no—
580C4—1041no—
581C4—1041no—
582C4—1041no—
583C4#1171917511yescharacteristic
584C4#1171917511yescharacteristic
585C4C4×#612711yescharacteristic
586C4C4×#612711yescharacteristic
587C2—42no—
588C2#1311731911yescharacteristic
589C2#1312342911yescharacteristic
590C2#1312343411yescharacteristic
591C2#1312502011yescharacteristic
592C2#1312503511yescharacteristic
593C2#1312505011yescharacteristic
594C2#1312506511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2274C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#2274D13
hypocenterC13#21367
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #13115604C2semidirect
4. #13115604C2semidirect
5. #13115599C2semidirect
6. #13115594C2semidirect
11. #13114734C2semidirect
12. #13114734C2semidirect
13. C4×#88047C2semidirect
14. C4×#88048C2semidirect
31. #11718661C4semidirect
119. C4⋊#1055C2×C4semidirect
120. C4⋊#1055C2×C4semidirect
121. #88043C2×C4semidirect
122. C4×#1055C2×C4semidirect
493. C13#21367semidirect

Maximal subgroups

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

subgroupindex
C2×#117186612
#131174452
#131156042
#131155992
#131155942
#131150352
#131150292
#131150242
#131147342
C4×#880472
C4×#880482
C13×#22742
#2136713