Finite Group Databasebeta

← #15931124

Subgroups — #15931124

Order 3328.

726 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
1#13119177C211yesmaximal, characteristic
2#13116158C211yesmaximal, characteristic, complemented
3#13116036C211yesmaximal, characteristic
4#13115987C211yesmaximal, characteristic, complemented
5#13115844C211yesmaximal, characteristic
6#13115793C211yesmaximal, characteristic, complemented
7#13115511C211yesmaximal, characteristic, complemented
8#13115477C211yesmaximal, characteristic, complemented
9#13115444C211yesmaximal, characteristic
10#13115404C211yesmaximal, characteristic, complemented
11#13115177C211yesmaximal, characteristic, complemented
12#13115148C211yesmaximal, characteristic
13#13114931C211yesmaximal, characteristic
14C4×#88075C211yesmaximal, characteristic, complemented
15C13×#2362C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C22×#6107C2211yescharacteristic
20C2×#88178C2211yescharacteristic
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
33#11718873C2211yescharacteristic
34C2×C22⋊#1055C2211yescharacteristic
35C2×C22⋊#1055C2211yescharacteristic
36C2×#88050C2211yescharacteristic
37C2×#88050C2211yescharacteristic
38C2×#88050C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40#11718613—21no—
41#11718605—21no—
42C4×C4⋊Dic13C2211yescharacteristic
43#11718590C2211yescharacteristic
44#11718590C2211yescharacteristic
45C2×C4⋊#1055C2211yescharacteristic
46C2×C4×#1055C2211yescharacteristic
47#11718319—21no—
48#11718311—21no—
49#11718121C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C22×Q8×C13C2311yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C22×Dic26C2311yescharacteristic
62C13×C22⋊Q8—22no—
63C13×C22⋊Q8—22no—
64C4×Q8×C13—22no—
65C4×Q8×C13—41no—
66C13×#117—22no—
67C13×#117—41no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C22⋊C4—21no—
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×C13C2311yescharacteristic
82C2×C42×C13C2311yescharacteristic
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13—21no—
85C2×#6107—22no—
86C2×#6107—41no—
87C2×#6107—41no—
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91#88178—22no—
92#88178—41no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×C4×Dic13—21no—
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101#88124—22no—
102#88124—41no—
103C13×#95—21no—
104C13×#95—21no—
105C13×#95—21no—
106C13×#95—21no—
107C13×#95D411yescharacteristic
108C13×#95D411yescharacteristic
109#88075—22no—
110#88075—22no—
111#88075—41no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118C22⋊#1055—41no—
119C22⋊#1055—41no—
120#88050—22no—
121#88050—22no—
122#88050—41no—
123#88047—22no—
124C4⋊#1055—41no—
125C4×#1055—41no—
126#26732—131nomaximal, Sylow
127C24×C13C2×D411yescharacteristic
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13—21no—
133C2×Q8×C13#4112yes—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
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×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2411yescharacteristic
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—21no—
154C2×Dic26—21no—
155C2×Dic26—21no—
156C2×Dic26—21no—
157C2×Dic26—21no—
158C2×Dic26—21no—
159C2×Dic26—21no—
160C13×C4⋊C4—24no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—42no—
166C13×C4⋊C4—42no—
167C13×C4⋊C4#4112yes—
168C13×C4⋊C4#4112yes—
169C13×C22⋊C4—24no—
170C13×C22⋊C4—22no—
171C13×C22⋊C4—41no—
172C13×C22⋊C4—42no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13—21no—
177C42×C13—42no—
178C42×C13—42no—
179C42×C13—42no—
180C22⋊Dic13—42no—
181C22⋊Dic13—42no—
182#6107—44no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13#4112yes—
189#6101—42no—
190#6101—42no—
191#6101—42no—
192C4×Dic13—42no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
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—
205#2489—131no—
206#2362D1311yescharacteristic
207#2056—131no—
208#2036—131no—
209#2026—131no—
210#2001—131no—
211#1989—131no—
212#1933—131no—
213#1928—131no—
214#1923—131no—
215#1918—131no—
216#1878—131no—
217#1873—131no—
218#1825—131no—
219C4×#103—131no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13C22×D411yescharacteristic
228Q8×C13—22no—
229Q8×C13—22no—
230Q8×C13—41no—
231Q8×C13—41no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—22no—
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—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—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13#13512yes—
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C13C2×#4111yescharacteristic
266C2×C4×C13C2×#4111yescharacteristic
267C2×C4×C13H(5,𝔽2)11yescharacteristic
268C2×C4×C13H(5,𝔽2)11yescharacteristic
269C2×Dic13—22no—
270C2×Dic13—41no—
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—
281Dic26—41no—
282Dic26—41no—
283Dic26—41no—
284Dic26—41no—
285#1055—41no—
286#1055—41no—
287#1055—41no—
288#1055—41no—
289C22×Q16—131no—
290C2×#125—131no—
291C2×C22⋊Q8—131no—
292C2×C22⋊Q8C2×D1311yescharacteristic
293C2×C4×Q8C2×D1311yescharacteristic
294C2×#117C2×D1311yescharacteristic
295C2×#107—131no—
296C2×C4⋊C8—131no—
297C2×#103—131no—
298C2×#103—131no—
299C2×#103—131no—
300C2×#103—131no—
301C2×#103—131no—
302C2×#103—131no—
303C2×C22⋊C8—131no—
304C2×C22⋊C8—131no—
305C2×C4×C8—131no—
306#400—261no—
307#397—261no—
308#391—131no—
309#391—131no—
310#391C2×D1311yescharacteristic
311#389C2×D1311yescharacteristic
312#388—21no—
313#388—21no—
314#388—261no—
315#387—21no—
316#382—261no—
317#380—131no—
318C4×C4⋊C4—21no—
319C4×C4⋊C4—131no—
320C4×C4⋊C4C2×D1311yescharacteristic
321C4×C22⋊C4C2×D1311yescharacteristic
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#336—131no—
331#336—131no—
332C22×C13—41no—
333C22×C13—41no—
334C22×C13—41no—
335C22×C13—41no—
336C22×C13—41no—
337C22×C13—41no—
338C22×C13—41no—
339C22×C13#52511yescharacteristic
340C22×C13#54611yescharacteristic
341C22×C13#54611yescharacteristic
342C22×C13C2×#13511yescharacteristic
343C22×C13#57511yescharacteristic
344C22×C13#57811yescharacteristic
345C22×C13#57811yescharacteristic
346C4×C13—22no—
347C4×C13—42no—
348C4×C13—41no—
349C4×C13—42no—
350C4×C13—41no—
351C4×C13—41no—
352C4×C13—42no—
353C4×C13—41no—
354C4×C13—81no—
355C4×C13#52911yescharacteristic
356C4×C13#53811yescharacteristic
357C4×C13#55511yescharacteristic
358C4×C13#55911yescharacteristic
359Dic13—42no—
360Dic13—81no—
361Dic13—81no—
362Dic13—81no—
363C22×Q8—131no—
364C22×Q8C22×D1311yescharacteristic
365C23×C4C22×D1311yescharacteristic
366C2×Q16—262no—
367C2×Q16—521no—
368C2×Q16—521no—
369C22×C8—131no—
370C22×C8—131no—
371C22×C8—131no—
372C22×C8—131no—
373#125—262no—
374#125—521no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—262no—
378C22⋊Q8—521no—
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—261no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4C22×D1311yescharacteristic
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—261no—
398C2×C22⋊C4—261no—
399C2×C22⋊C4C22⋊D1311yescharacteristic
400C2×C22⋊C4C22⋊D1311yescharacteristic
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—261no—
405C2×C42C22×D1311yescharacteristic
406C2×C42C22×D1311yescharacteristic
407#107—262no—
408C4⋊C8—521no—
409#103—262no—
410#103—262no—
411#103—262no—
412#103—262no—
413#103—521no—
414#103—521no—
415C22⋊C8—521no—
416C22⋊C8—521no—
417C4×C8—521no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—261no—
423#95—261no—
424#95—261no—
425#95—261no—
426#95—261no—
427#95—261no—
428#95C22⋊D1311yescharacteristic
429#95C22⋊D1311yescharacteristic
430C2×C13—81no—
431C2×C13#245911yescharacteristic
432C2×C13#310211yescharacteristic
433C2×C13#312711yescharacteristic
434C2×C13#330011yescharacteristic
435C2×C13#330211yescharacteristic
436C2×C13#331511yescharacteristic
437C2×C13#332711yescharacteristic
438C24C2×C22⋊D1311yescharacteristic
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—261no—
445C2×Q8—261no—
446C2×Q8—261no—
447C2×Q8—261no—
448C2×Q8—261no—
449C2×Q8—261no—
450C2×Q8#612712yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—261no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—261no—
465C22×C4C2×C22⋊D1311yescharacteristic
466C22×C4C2×C22⋊D1311yescharacteristic
467C22×C4C2×C22⋊D1311yescharacteristic
468C22×C4C2×C22⋊D1311yescharacteristic
469C22×C4C2×C22⋊D1311yescharacteristic
470C22×C4C23×D1311yescharacteristic
471Q16—524no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
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—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—132no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—522no—
498C4⋊C4—522no—
499C4⋊C4—522no—
500C4⋊C4#612712yes—
501C4⋊C4#612912yes—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—522no—
507C22⋊C4—522no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—522no—
516C13#2673211yescharacteristic, complemented, Sylow
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C22×C22⋊D1311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—521no—
530Q8—521no—
531Q8—521no—
532Q8—521no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
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—21no—
550C2×C4—41no—
551C2×C4—41no—
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—262no—
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—521no—
575C2×C4#8820412yes—
576C2×C4C2×#612711yescharacteristic
577C2×C4C2×#612711yescharacteristic
578C2×C4C2×#612911yescharacteristic
579C2×C4#8825111yescharacteristic
580C2×C4#41×D1311yescharacteristic
581C2×C4#8825611yescharacteristic
582C8—521no—
583C8—521no—
584C8—521no—
585C8—521no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1171949611yescharacteristic
594C22#1171951111yescharacteristic
595C22C2×#8820411yescharacteristic
596C22#1171952711yescharacteristic
597C22#1171953411yescharacteristic
598C22#1171954311yescharacteristic
599C22#1171954311yescharacteristic
600C4—22no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—522no—
610C4—1041no—
611C4—1041no—
612C4—1041no—
613C4#1171924311yescharacteristic
614C4#1171926411yescharacteristic
615C4#1171926511yescharacteristic
616C4#1171926811yescharacteristic
617C2—81no—
618C2#1311876711yescharacteristic
619C2#1312485411yescharacteristic
620C2#1312513611yescharacteristic
621C2#1312672311yescharacteristic
622C2#1312674711yescharacteristic
623C2#1312693211yescharacteristic
624C2#1312709811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2362C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2362D13
hypocenterC13#26732
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
2. #13116158C2semidirect
4. #13115987C2semidirect
6. #13115793C2semidirect
7. #13115511C2semidirect
8. #13115477C2semidirect
10. #13115404C2semidirect
11. #13115177C2semidirect
14. C4×#88075C2semidirect
516. C13#26732semidirect

Maximal subgroups

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

subgroupindex
#131191772
#131161582
#131160362
#131159872
#131158442
#131157932
#131155112
#131154772
#131154442
#131154042
#131151772
#131151482
#131149312
C4×#880752
C13×#23622
#2673213