Finite Group Databasebeta

← #15925849

Subgroups — #15925849

Order 3328.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718683C211yesmaximal, characteristic, complemented
2C2×#11718656C211yesmaximal, characteristic
3#13118560C211yesmaximal, characteristic
4#13116384C211yesmaximal, characteristic, complemented
5#13116332C211yesmaximal, characteristic, complemented
6#13116231C211yesmaximal, characteristic, complemented
7#13116174C211yesmaximal, characteristic, complemented
8#13115604C211yesmaximal, characteristic, complemented
9#13115414C211yesmaximal, characteristic, complemented
10#13115270C211yesmaximal, characteristic
11#13115144C211yesmaximal, characteristic
12#13115104C211yesmaximal, characteristic, complemented
13#13115035C211yesmaximal, characteristic
14#13115021C211yesmaximal, characteristic
15C13×#2362C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×#88124C2211yescharacteristic
22C13×#391C2211yescharacteristic
23C13×#389C2211yescharacteristic
24C13×#388—21no—
25C13×#388—21no—
26C13×#387—21no—
27C4×C13×C4⋊C4—21no—
28C4×C13×C4⋊C4C2211yescharacteristic
29C4×C13×C22⋊C4C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34#11718873C2211yescharacteristic
35C2×C22⋊#1055C2211yescharacteristic
36#11718683—24no—
37#11718656—24no—
38C2×#88048C2211yescharacteristic
39C2×#88048C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44#11718607—21no—
45#11718603—21no—
46#11718597C2211yescharacteristic
47#11718589C2211yescharacteristic
48#11718589C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4⋊#1055C2211yescharacteristic
51#11718321—21no—
52#11718318—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C23×Dic13—21no—
64C13×C22⋊Q8—22no—
65C13×C22⋊Q8Q814yes—
66C4×Q8×C13—22no—
67C4×Q8×C13—41no—
68C13×#117—22no—
69C13×#117—41no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4D411yescharacteristic
79C2×C13×C22⋊C4D411yescharacteristic
80C2×C42×C13—21no—
81C2×C42×C13—21no—
82C2×C42×C13—21no—
83C2×C42×C13C2311yescharacteristic
84C2×C42×C13C2311yescharacteristic
85C2×C22⋊Dic13—21no—
86C2×C4⋊Dic13—22no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—41no—
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×#6101—21no—
97C2×#6099—22no—
98C2×#6099—22no—
99C2×#6099—22no—
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104#88124—22no—
105#88124—41no—
106C13×#95—21no—
107C13×#95—21no—
108C13×#95—21no—
109C13×#95—21no—
110C13×#95D411yescharacteristic
111C13×#95D411yescharacteristic
112#88075—22no—
113#88075—22no—
114#88075—22no—
115#88075—41no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121C22⋊#1055—22no—
122#88048—22no—
123#88048—41no—
124#88048D414yes—
125#88047—22no—
126#88047—22no—
127#88047—22no—
128C4⋊#1055—22no—
129C4⋊#1055—22no—
130#26367—131nomaximal, Sylow
131C24×C13C2×D411yescharacteristic
132C2×Q8×C13—21no—
133C2×Q8×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×Q8×C13—21no—
137C2×Q8×C13C2×Q812yes—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—21no—
145C22×C4×C13—21no—
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×Q812yes—
152C22×C4×C13#4112yes—
153C22×C4×C13C2411yescharacteristic
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—42no—
159C22×Dic13—42no—
160C22×Dic13—81no—
161C13×C4⋊C4—24no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—42no—
167C13×C4⋊C4—42no—
168C13×C4⋊C4C2×D412yes—
169C13×C4⋊C4C2×Q812yes—
170C13×C22⋊C4—24no—
171C13×C22⋊C4—41no—
172C13×C22⋊C4—41no—
173C13×C22⋊C4—42no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13—21no—
177C42×C13—21no—
178C42×C13—42no—
179C42×C13—42no—
180C42×C13—42no—
181C22⋊Dic13—42no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—42no—
190C4⋊Dic13C2×D412yes—
191#6101—42no—
192#6101—42no—
193#6101—81no—
194#6101—81no—
195#6099—28no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055C2×D412yes—
207C2×#1055#4112yes—
208C2×#479—131no—
209C2×#428—131no—
210#2442—131no—
211#2362D1311yescharacteristic
212#2097—131no—
213#2087—131no—
214#2068—131no—
215#2059—131no—
216#1949—131no—
217#1919—131no—
218#1894—131no—
219#1872—131no—
220#1865—131no—
221#1854—131no—
222#1851—131no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13C22⋊Q812yes—
230C23×C13C22×D411yescharacteristic
231Q8×C13—22no—
232Q8×C13—22no—
233Q8×C13—41no—
234Q8×C13—41no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—22no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13C22⋊Q812yes—
266C2×C4×C13C22×D411yescharacteristic
267C2×C4×C13C22×Q811yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13H(5,𝔽2)11yescharacteristic
271C2×C4×C132−1+411yescharacteristic
272C2×Dic13—22no—
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—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—84no—
285#1055—22no—
286#1055—22no—
287#1055—41no—
288#1055—41no—
289C22×M16—131no—
290C2×#125—131no—
291C2×C22⋊Q8C2×D1311yescharacteristic
292C2×C4×Q8C2×D1311yescharacteristic
293C2×#117C2×D1311yescharacteristic
294C22×C4⋊C4—131no—
295#479—264no—
296#428—264no—
297C2×#107—131no—
298C2×#107—131no—
299C2×#107—131no—
300C2×#106—131no—
301C2×#106—131no—
302C2×#106—131no—
303C2×C4⋊C8—131no—
304C2×C4⋊C8—131no—
305C2×#103—131no—
306C2×#103—131no—
307C2×#103—131no—
308C2×#103—131no—
309C2×C22⋊C8—131no—
310#400—261no—
311#398—261no—
312#391C2×D1311yescharacteristic
313#389—131no—
314#389—131no—
315#389C2×D1311yescharacteristic
316#388—21no—
317#388—21no—
318#387—21no—
319#385—261no—
320#382—131no—
321#380—131no—
322#380—261no—
323C4×C4⋊C4—21no—
324C4×C4⋊C4C2×D1311yescharacteristic
325C4×C22⋊C4C2×D1311yescharacteristic
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#340—131no—
331#340—131no—
332#336—131no—
333#336—131no—
334C22×C13—22no—
335C22×C13—22no—
336C22×C13—22no—
337C22×C13—41no—
338C22×C13—41no—
339C22×C13—41no—
340C22×C13—41no—
341C22×C13C2×C22⋊Q811yescharacteristic
342C22×C13D4211yescharacteristic
343C22×C13#54711yescharacteristic
344C22×C13#57511yescharacteristic
345C22×C13#57511yescharacteristic
346C22×C13#57711yescharacteristic
347C22×C13#57811yescharacteristic
348C4×C13—22no—
349C4×C13—42no—
350C4×C13—41no—
351C4×C13—41no—
352C4×C13—42no—
353C4×C13—41no—
354C4×C13—42no—
355C4×C13—41no—
356C4×C13—81no—
357C4×C13#52511yescharacteristic
358C4×C13#53911yescharacteristic
359C4×C13D4×Q811yescharacteristic
360C4×C13#55411yescharacteristic
361Dic13—42no—
362Dic13—81no—
363Dic13—81no—
364Dic13—81no—
365C22×Q8C22×D1311yescharacteristic
366C23×C4—261no—
367C23×C4C22×D1311yescharacteristic
368C2×M16—262no—
369C2×M16—262no—
370C2×M16—262no—
371C22×C8—131no—
372C22×C8—131no—
373C22×C8—131no—
374C22×C8—131no—
375#125—262no—
376#125—521no—
377C22⋊Q8—22no—
378C22⋊Q8Dic2614yes—
379C4×Q8—22no—
380C4×Q8—41no—
381#117—22no—
382#117—41no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—262no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4—261no—
394C2×C4⋊C4—261no—
395C2×C4⋊C4—261no—
396C2×C4⋊C4—521no—
397C2×C4⋊C4C22×D1311yescharacteristic
398C2×C4⋊C4C22×D1311yescharacteristic
399C2×C4⋊C4C22×D1311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—261no—
403C2×C22⋊C4C22⋊D1311yescharacteristic
404C2×C22⋊C4C22⋊D1311yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42C22×D1311yescharacteristic
409C2×C42C22×D1311yescharacteristic
410#107—262no—
411#107—262no—
412#107—262no—
413#106—134no—
414#106—262no—
415#106—521no—
416C4⋊C8—262no—
417C4⋊C8—262no—
418#103—262no—
419#103—262no—
420#103—262no—
421#103—521no—
422C22⋊C8—262no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—261no—
428#95—261no—
429#95—261no—
430#95—261no—
431#95—261no—
432#95C22⋊D1311yescharacteristic
433#95C22⋊D1311yescharacteristic
434C2×C13—42no—
435C2×C13#244911yescharacteristic
436C2×C13#308011yescharacteristic
437C2×C13#308211yescharacteristic
438C2×C13#326911yescharacteristic
439C2×C13#327511yescharacteristic
440C2×C13#334911yescharacteristic
441C2×C13#335111yescharacteristic
442C24C2×C22⋊D1311yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×Dic2612yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—261no—
461C22×C4—522no—
462C22×C4—522no—
463C22×C4—1041no—
464C22×C4C2×Dic2612yes—
465C22×C4#612712yes—
466C22×C4C2×C22⋊D1311yescharacteristic
467C22×C4C2×C22⋊D1311yescharacteristic
468C22×C4C2×C22⋊D1311yescharacteristic
469C22×C4C2×C22⋊D1311yescharacteristic
470C22×C4C2×C22⋊D1311yescharacteristic
471C22×C4C23×D1311yescharacteristic
472M16—268no—
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—261no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8—261no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—132no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—261no—
498C4⋊C4—261no—
499C4⋊C4—261no—
500C4⋊C4—522no—
501C4⋊C4—522no—
502C4⋊C4—522no—
503C4⋊C4—1041no—
504C4⋊C4—1041no—
505C4⋊C4C2×Dic2612yes—
506C4⋊C4D4×D1312yes—
507C22⋊C4—24no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—522no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—42no—
518C42—42no—
519C13#2636711yescharacteristic, complemented, Sylow
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#8817812yes—
527C23C22×C22⋊D1311yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
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—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—262no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—521no—
574C2×C4—1044no—
575C2×C4#8817812yes—
576C2×C4C22×Dic2611yescharacteristic
577C2×C4C2×#612711yescharacteristic
578C2×C4C2×D4×D1311yescharacteristic
579C2×C4#8825111yescharacteristic
580C2×C4#41×D1311yescharacteristic
581C2×C4#8825711yescharacteristic
582C8—262no—
583C8—262no—
584C8—521no—
585C8—521no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#8817811yescharacteristic
594C22D4×C22⋊D1311yescharacteristic
595C22#1171952711yescharacteristic
596C22#1171952711yescharacteristic
597C22#1171952911yescharacteristic
598C22#1171954211yescharacteristic
599C22#1171954311yescharacteristic
600C4—22no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—522no—
610C4—1041no—
611C4—1041no—
612C4—1041no—
613C4D4×Dic2611yescharacteristic
614C4#1171924411yescharacteristic
615C4#1171925211yescharacteristic
616C4#1171926911yescharacteristic
617C2—42no—
618C2#1311863911yescharacteristic
619C2#1312465811yescharacteristic
620C2#1312467811yescharacteristic
621C2#1312638711yescharacteristic
622C2#1312646811yescharacteristic
623C2#1312736011yescharacteristic
624C2#1312739611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2362C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2362D13
hypocenterC13#26367
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#11718683C2semidirect
4. #13116384C2semidirect
5. #13116332C2semidirect
6. #13116231C2semidirect
7. #13116174C2semidirect
8. #13115604C2semidirect
9. #13115414C2semidirect
12. #13115104C2semidirect
519. C13#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#117186832
C2×#117186562
#131185602
#131163842
#131163322
#131162312
#131161742
#131156042
#131154142
#131152702
#131151442
#131151042
#131150352
#131150212
C13×#23622
#2636713