Finite Group Databasebeta

← #15926779

Subgroups — #15926779

Order 3328.

784 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#13118538C211yesmaximal, characteristic
2#13116399C211yesmaximal, characteristic, complemented
3#13116337C211yesmaximal, characteristic, complemented
4#13116217C211yesmaximal, characteristic, complemented
5#13115977C211yesmaximal, characteristic, complemented
6#13115788C211yesmaximal, characteristic, complemented
7#13115407C211yesmaximal, characteristic, complemented
8#13115270C211yesmaximal, characteristic
9#13115144C211yesmaximal, characteristic
10#13115090C211yesmaximal, characteristic, complemented
11#13115035C211yesmaximal, characteristic
12#13115021C211yesmaximal, characteristic
13#13114841C211yesmaximal, characteristic, complemented
14#13114557C211yesmaximal, characteristic
15C13×#2362C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×#88132—21no—
22C2×#88123C2211yescharacteristic
23C13×#391C2211yescharacteristic
24C13×#389C2211yescharacteristic
25C13×#388—21no—
26C13×#388—21no—
27C13×#387—21no—
28C4×C13×C4⋊C4—21no—
29C4×C13×C4⋊C4C2211yescharacteristic
30C4×C13×C22⋊C4C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35#11718873C2211yescharacteristic
36C2×C22⋊#1055C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39#11718611—21no—
40#11718596C2211yescharacteristic
41#11718589C2211yescharacteristic
42#11718589C2211yescharacteristic
43C2×C4⋊#1055C2211yescharacteristic
44C2×C4⋊#1055C2211yescharacteristic
45#11718322—21no—
46#11718317—21no—
47#11718121C2211yescharacteristic
48#11718121C2211yescharacteristic
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
61C23×Dic13—21no—
62C22×Dic26—21no—
63C13×C22⋊Q8—22no—
64C13×C22⋊Q8Q814yescomplemented
65C4×Q8×C13—22no—
66C4×Q8×C13—41no—
67C13×#117—22no—
68C13×#117—41no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4D411yescharacteristic
78C2×C13×C22⋊C4D411yescharacteristic
79C2×C42×C13—21no—
80C2×C42×C13—21no—
81C2×C42×C13—21no—
82C2×C42×C13C2311yescharacteristic
83C2×C42×C13C2311yescharacteristic
84C2×C22⋊Dic13—21no—
85C2×C4⋊Dic13—22no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13—41no—
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×#6101—21no—
96C2×#6101—21no—
97C2×#6099—22no—
98C2×#6099—41no—
99C2×#6099—41no—
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104#88132—44no—
105#88123—22no—
106#88123—41no—
107C13×#95—21no—
108C13×#95—21no—
109C13×#95—21no—
110C13×#95—21no—
111C13×#95D411yescharacteristic
112C13×#95D411yescharacteristic
113#88075—22no—
114#88075—41no—
115#88075—41no—
116#88075—41no—
117#88071—21no—
118#88071—21no—
119#88071—21no—
120C22⋊#1055—22no—
121#88048—41no—
122#88047—22no—
123C4⋊#1055—41no—
124C4⋊#1055—41no—
125#26429—131nomaximal, Sylow
126C24×C13C2×D411yescharacteristic
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13C2×Q812yes—
133C22×C4×C13—21no—
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×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×Q812yes—
147C22×C4×C13#4112yes—
148C22×C4×C13C2411yescharacteristic
149C22×Dic13—21no—
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—42no—
154C22×Dic13—42no—
155C22×Dic13—81no—
156C2×Dic26—42no—
157C2×Dic26—42no—
158C2×Dic26—81no—
159C13×C4⋊C4—24no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—42no—
165C13×C4⋊C4—42no—
166C13×C4⋊C4C2×Q812yes—
167C13×C4⋊C4#4112yes—
168C13×C22⋊C4—24no—
169C13×C22⋊C4—22no—
170C13×C22⋊C4—22no—
171C13×C22⋊C4—42no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13—42no—
177C42×C13—42no—
178C42×C13—42no—
179C22⋊Dic13—42no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—42no—
188C4⋊Dic13—42no—
189C4⋊Dic13—42no—
190C4⋊Dic13—81no—
191C4⋊Dic13#4112yes—
192#6101—42no—
193#6101—42no—
194#6099—44no—
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—
205C2×#1055—21no—
206C2×#1055—21no—
207#2440—131no—
208#2362D1311yescharacteristic
209#2100—131no—
210#2088—131no—
211#2066—131no—
212#2024—131no—
213#1988—131no—
214#1918—131no—
215#1894—131no—
216#1872—131no—
217#1863—131no—
218#1854—131no—
219#1851—131no—
220#1803—131no—
221#1736—131no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13C22⋊Q812yes—
229C23×C13C22×D411yescharacteristic
230Q8×C13—22no—
231Q8×C13—22no—
232Q8×C13—41no—
233Q8×C13—41no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—22no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
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×C13C22⋊Q812yes—
263C2×C4×C13AGL(1,ℤ/8)12yes—
264C2×C4×C13#13712yes—
265C2×C4×C13C22×Q811yescharacteristic
266C2×C4×C13C2×#4111yescharacteristic
267C2×C4×C13C2×#4111yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C13H(5,𝔽2)11yescharacteristic
270C2×C4×C13H(5,𝔽2)11yescharacteristic
271C2×Dic13—22no—
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—41no—
282C2×Dic13—41no—
283C2×Dic13—84no—
284Dic26—84no—
285#1055—41no—
286#1055—41no—
287#1055—41no—
288#1055—41no—
289C22×M16—131no—
290C2×C4⋊Q8—131no—
291C2×C22⋊Q8—261no—
292C2×C22⋊Q8C2×D1311yescharacteristic
293C2×C4×Q8C2×D1311yescharacteristic
294C2×#117C2×D1311yescharacteristic
295C22×C4⋊C4—131no—
296C2×#107—131no—
297C2×#106—131no—
298C2×C4⋊C8—131no—
299C2×C4⋊C8—131no—
300C2×#103—131no—
301C2×#103—131no—
302C2×#103—131no—
303C2×#103—131no—
304C2×C22⋊C8—131no—
305#398—261no—
306#395—261no—
307#391C2×D1311yescharacteristic
308#389—131no—
309#389—131no—
310#389C2×D1311yescharacteristic
311#388—21no—
312#388—21no—
313#387—21no—
314#385—261no—
315#384—131no—
316#380—131no—
317C4×C4⋊C4—21no—
318C4×C4⋊C4C2×D1311yescharacteristic
319C4×C22⋊C4C2×D1311yescharacteristic
320#340—131no—
321#340—131no—
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—22no—
333C22×C13—22no—
334C22×C13—22no—
335C22×C13—41no—
336C22×C13—41no—
337C22×C13—41no—
338C22×C13—41no—
339C22×C13C2×C22⋊Q811yescharacteristic
340C22×C13#54611yescharacteristic
341C22×C13#54611yescharacteristic
342C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
343C22×C13C2×#13711yescharacteristic
344C22×C13#57711yescharacteristic
345C22×C13#57811yescharacteristic
346C4×C13—22no—
347C4×C13—22no—
348C4×C13—22no—
349C4×C13—41no—
350C4×C13—41no—
351C4×C13—42no—
352C4×C13—42no—
353C4×C13—42no—
354C4×C13—81no—
355C4×C13#52911yescharacteristic
356C4×C13#53811yescharacteristic
357C4×C13#55411yescharacteristic
358C4×C13#55411yescharacteristic
359Dic13—42no—
360Dic13—81no—
361Dic13—81no—
362Dic13—81no—
363C22×Q8—261no—
364C22×Q8C22×D1311yescharacteristic
365C23×C4—261no—
366C23×C4C22×D1311yescharacteristic
367C2×M16—262no—
368C2×M16—521no—
369C2×M16—521no—
370C22×C8—131no—
371C22×C8—131no—
372C22×C8—131no—
373C22×C8—131no—
374C4⋊Q8—262no—
375C4⋊Q8—521no—
376C22⋊Q8—22no—
377C22⋊Q8—524no—
378C22⋊Q8Dic2614yescomplemented
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—261no—
397C2×C4⋊C4—521no—
398C2×C4⋊C4C22×D1311yescharacteristic
399C2×C4⋊C4C22×D1311yescharacteristic
400C2×C4⋊C4C22×D1311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—261no—
404C2×C22⋊C4C22⋊D1311yescharacteristic
405C2×C22⋊C4C22⋊D1311yescharacteristic
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42C22×D1311yescharacteristic
410C2×C42C22×D1311yescharacteristic
411#107—262no—
412#106—521no—
413C4⋊C8—521no—
414C4⋊C8—521no—
415#103—262no—
416#103—521no—
417#103—521no—
418#103—521no—
419C22⋊C8—262no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—261no—
425#95—261no—
426#95—261no—
427#95C22⋊D1311yescharacteristic
428#95C22⋊D1311yescharacteristic
429C2×C13—42no—
430C2×C13#245011yescharacteristic
431C2×C13#307911yescharacteristic
432C2×C13#308211yescharacteristic
433C2×C13#331011yescharacteristic
434C2×C13#331611yescharacteristic
435C2×C13#332811yescharacteristic
436C2×C13#333011yescharacteristic
437C24C2×C22⋊D1311yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—522no—
444C2×Q8—522no—
445C2×Q8—1041no—
446C2×Q8C2×Dic2612yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—261no—
456C22×C4—261no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—522no—
460C22×C4—522no—
461C22×C4—1041no—
462C22×C4C2×Dic2612yes—
463C22×C4#612712yes—
464C22×C4C2×C22⋊D1311yescharacteristic
465C22×C4C2×C22⋊D1311yescharacteristic
466C22×C4C2×C22⋊D1311yescharacteristic
467C22×C4C2×C22⋊D1311yescharacteristic
468C22×C4C2×C22⋊D1311yescharacteristic
469C22×C4C23×D1311yescharacteristic
470M16—524no—
471C2×C8—261no—
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—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—132no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—261no—
498C4⋊C4—522no—
499C4⋊C4—522no—
500C4⋊C4—522no—
501C4⋊C4—522no—
502C4⋊C4—522no—
503C4⋊C4—1041no—
504C4⋊C4C2×Dic2612yes—
505C4⋊C4#612912yes—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—522no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—42no—
517C42—42no—
518C13#2642911yescharacteristic, complemented, Sylow
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#8817812yes—
526C23C22×C22⋊D1311yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531Q8—1044no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
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—262no—
561C2×C4—521no—
562C2×C4—521no—
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—1044no—
573C2×C4#8817812yes—
574C2×C4#8820312yes—
575C2×C4#8820512yes—
576C2×C4C22×Dic2611yescharacteristic
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—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#8817811yescharacteristic
594C22#1171951111yescharacteristic
595C22C2×#8820311yescharacteristic
596C22C2×#8820511yescharacteristic
597C22#1171953411yescharacteristic
598C22#1171954211yescharacteristic
599C22#1171954311yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—522no—
610C4—1041no—
611C4—1041no—
612C4—1041no—
613C4#1171924311yescharacteristic
614C4#1171924411yescharacteristic
615C4#1171924811yescharacteristic
616C4#1171926511yescharacteristic
617C2—42no—
618C2#1311865411yescharacteristic
619C2#1312465111yescharacteristic
620C2#1312467811yescharacteristic
621C2#1312685111yescharacteristic
622C2#1312694711yescharacteristic
623C2#1312711311yescharacteristic
624C2#1312714111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2362C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2362D13
hypocenterC13#26429
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. #13116399C2semidirect
3. #13116337C2semidirect
4. #13116217C2semidirect
5. #13115977C2semidirect
6. #13115788C2semidirect
7. #13115407C2semidirect
10. #13115090C2semidirect
13. #13114841C2semidirect
64. C13×C22⋊Q8Q8semidirect
378. C22⋊Q8Dic26semidirect
518. C13#26429semidirect

Maximal subgroups

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

subgroupindex
#131185382
#131163992
#131163372
#131162172
#131159772
#131157882
#131154072
#131152702
#131151442
#131150902
#131150352
#131150212
#131148412
#131145572
C13×#23622
#2642913