Finite Group Databasebeta

← #15926554

Subgroups — #15926554

Order 3328.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13118538C211yesmaximal, characteristic
2#13116399C211yesmaximal, characteristic, complemented
3#13116332C211yesmaximal, characteristic, complemented
4#13116231C211yesmaximal, characteristic, complemented
5#13115977C211yesmaximal, characteristic, complemented
6#13115784C211yesmaximal, characteristic, complemented
7#13115428C211yesmaximal, characteristic, complemented
8#13115260C211yesmaximal, characteristic
9#13115144C211yesmaximal, characteristic
10#13115090C211yesmaximal, characteristic, complemented
11#13115024C211yesmaximal, characteristic
12#13115016C211yesmaximal, characteristic
13#13114841C211yesmaximal, characteristic, complemented
14C2×#11718121C211yesmaximal, characteristic
15C13×#2360C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C23×#1055C2211yescharacteristic
21C2×#88132—21no—
22C2×#88123C2211yescharacteristic
23C13×#391C2211yescharacteristic
24C13×#389—21no—
25C13×#389C2211yescharacteristic
26C13×#387—21no—
27C13×#385—21no—
28C4×C13×C4⋊C4C2211yescharacteristic
29C4×C13×C22⋊C4—21no—
30C4×C13×C22⋊C4C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35#11718873C2211yescharacteristic
36C2×C22⋊#1055C2211yescharacteristic
37C2×#88047C2211yescharacteristic
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×C13—21no—
61C23×C4×C13C2311yescharacteristic
62C23×Dic13—21no—
63C22×Dic26—21no—
64C13×C22⋊Q8—22no—
65C13×C22⋊Q8Q814yescomplemented
66C4×Q8×C13—22no—
67C4×Q8×C13—41no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—22no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—41no—
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4D411yescharacteristic
79C2×C13×C22⋊C4D411yescharacteristic
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—
97C22×#1055—22no—
98C22×#1055—41no—
99C22×#1055—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#88047—22no—
122#88047—41no—
123C4⋊#1055—41no—
124C4⋊#1055—41no—
125#26414—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×C13—42no—
142C22×C4×C13—42no—
143C22×C4×C13—81no—
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×Q812yes—
150C22×C4×C13#4112yes—
151C22×C4×C13C2411yescharacteristic
152C22×Dic13—21no—
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13—42no—
157C22×Dic13—42no—
158C22×Dic13—81no—
159C2×Dic26—42no—
160C2×Dic26—42no—
161C2×Dic26—81no—
162C13×C4⋊C4—24no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—42no—
168C13×C4⋊C4—42no—
169C13×C4⋊C4—81no—
170C13×C4⋊C4C2×Q812yes—
171C13×C4⋊C4#4112yes—
172C13×C22⋊C4—24no—
173C13×C22⋊C4—22no—
174C13×C22⋊C4Q1614yes—
175C42×C13—21no—
176C42×C13—21no—
177C42×C13—21no—
178C42×C13—21no—
179C42×C13—42no—
180C42×C13—81no—
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⋊Dic13—42no—
191C4⋊Dic13—42no—
192C4⋊Dic13—81no—
193C4⋊Dic13#4112yes—
194#6101—42no—
195#6101—42no—
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—
207C2×#1055—21no—
208C2×#1055—44no—
209#2440—131no—
210#2360D1311yescharacteristic
211#2100—131no—
212#2087—131no—
213#2068—131no—
214#2024—131no—
215#1987—131no—
216#1921—131no—
217#1892—131no—
218#1872—131no—
219#1863—131no—
220#1852—131no—
221#1850—131no—
222#1803—131no—
223C2×#340—131no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13C22⋊Q812yes—
229C23×C13C2×Q1612yes—
230C23×C13AGL(1,ℤ/8)12yes—
231C23×C13C22×D411yescharacteristic
232Q8×C13—22no—
233Q8×C13—22no—
234Q8×C13—41no—
235Q8×C13—41no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—22no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
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—41no—
263C2×C4×C13—41no—
264C2×C4×C13—84no—
265C2×C4×C13C22⋊Q812yes—
266C2×C4×C13C2×Q1612yes—
267C2×C4×C13C2×Q1612yes—
268C2×C4×C13C22×Q811yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13C2×#4111yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×C4×C13H(5,𝔽2)11yescharacteristic
273C2×C4×C13H(5,𝔽2)11yescharacteristic
274C2×Dic13—22no—
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—41no—
285C2×Dic13—41no—
286C2×Dic13—84no—
287Dic26—84no—
288#1055—41no—
289#1055—41no—
290#1055—41no—
291#1055—41no—
292C23×C8—131no—
293C2×C4⋊Q8—131no—
294C2×C22⋊Q8—261no—
295C2×C22⋊Q8C2×D1311yescharacteristic
296C2×C4×Q8C2×D1311yescharacteristic
297C22×C4⋊C4—131no—
298C22×C4⋊C4C2×D1311yescharacteristic
299C2×#107—131no—
300C2×#107—131no—
301C2×C4⋊C8—131no—
302C2×C4⋊C8—131no—
303C2×#103—131no—
304C2×#103—131no—
305C2×#103—131no—
306C2×#103—131no—
307C2×C22⋊C8—131no—
308#398—261no—
309#395—261no—
310#391C2×D1311yescharacteristic
311#389—21no—
312#389—131no—
313#389—131no—
314#389C2×D1311yescharacteristic
315#387—21no—
316#385—21no—
317#385—261no—
318#384—131no—
319#380—131no—
320C4×C4⋊C4C2×D1311yescharacteristic
321C4×C22⋊C4—21no—
322C4×C22⋊C4C2×D1311yescharacteristic
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#340—131no—
331#340—131no—
332#340—131no—
333#336—131no—
334#336—131no—
335C22×C13—22no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—22no—
340C22×C13—22no—
341C22×C13#47714yes—
342C22×C13C2×C22⋊Q811yescharacteristic
343C22×C13#54611yescharacteristic
344C22×C13#54611yescharacteristic
345C22×C13C22×Q1611yescharacteristic
346C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
347C22×C13#57711yescharacteristic
348C22×C13#57711yescharacteristic
349C4×C13—22no—
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—42no—
353C4×C13—42no—
354C4×C13—41no—
355C4×C13—42no—
356C4×C13—41no—
357C4×C13—81no—
358C4×C13#52911yescharacteristic
359C4×C13#53811yescharacteristic
360C4×C13#55411yescharacteristic
361C4×C13#55411yescharacteristic
362Dic13—42no—
363Dic13—81no—
364Dic13—81no—
365Dic13—81no—
366C22×Q8—261no—
367C22×Q8C22×D1311yescharacteristic
368C23×C4—21no—
369C23×C4—261no—
370C23×C4C22×D1311yescharacteristic
371C22×C8—131no—
372C22×C8—131no—
373C22×C8—131no—
374C22×C8—131no—
375C22×C8—262no—
376C22×C8—521no—
377C22×C8—521no—
378C4⋊Q8—262no—
379C4⋊Q8—521no—
380C22⋊Q8—22no—
381C22⋊Q8—524no—
382C22⋊Q8Dic2614yescomplemented
383C4×Q8—22no—
384C4×Q8—41no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—41no—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—131no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—262no—
396C2×C4⋊C4—261no—
397C2×C4⋊C4—261no—
398C2×C4⋊C4—261no—
399C2×C4⋊C4—261no—
400C2×C4⋊C4—261no—
401C2×C4⋊C4—261no—
402C2×C4⋊C4—521no—
403C2×C4⋊C4C22×D1311yescharacteristic
404C2×C4⋊C4C22×D1311yescharacteristic
405C2×C4⋊C4C22×D1311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—261no—
408C2×C22⋊C4C22⋊D1311yescharacteristic
409C2×C22⋊C4C22⋊D1311yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42C22×D1311yescharacteristic
413C2×C42C22×D1311yescharacteristic
414#107—262no—
415#107—521no—
416C4⋊C8—521no—
417C4⋊C8—521no—
418#103—262no—
419#103—521no—
420#103—521no—
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#95C22⋊D1311yescharacteristic
431#95C22⋊D1311yescharacteristic
432C2×C13—24no—
433C2×C13#245011yescharacteristic
434C2×C13C2×#47711yescharacteristic
435C2×C13#308411yescharacteristic
436C2×C13#329911yescharacteristic
437C2×C13#329911yescharacteristic
438C2×C13#330611yescharacteristic
439C2×C13#331011yescharacteristic
440C24C2×C22⋊D1311yescharacteristic
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—522no—
447C2×Q8—522no—
448C2×Q8—1041no—
449C2×Q8C2×Dic2612yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—261no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—261no—
465C22×C4—522no—
466C22×C4—522no—
467C22×C4—1041no—
468C22×C4C2×Dic2612yes—
469C22×C4#612712yes—
470C22×C4C2×C22⋊D1311yescharacteristic
471C22×C4C2×C22⋊D1311yescharacteristic
472C22×C4C2×C22⋊D1311yescharacteristic
473C22×C4C2×C22⋊D1311yescharacteristic
474C22×C4C2×C22⋊D1311yescharacteristic
475C22×C4C23×D1311yescharacteristic
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—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8—524no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—132no—
498C4⋊C4—261no—
499C4⋊C4—261no—
500C4⋊C4—261no—
501C4⋊C4—261no—
502C4⋊C4—261no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—522no—
506C4⋊C4—522no—
507C4⋊C4—522no—
508C4⋊C4—522no—
509C4⋊C4—522no—
510C4⋊C4—1041no—
511C4⋊C4C2×Dic2612yes—
512C4⋊C4#612912yes—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—522no—
516C22⋊C4#610714yes—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—42no—
522C42—81no—
523C13#2641411yescharacteristic, complemented, Sylow
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#8817812yes—
529C23C2×#610712yes—
530C23#8820312yes—
531C23C22×C22⋊D1311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—1044no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
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—84no—
566C2×C4—262no—
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—521no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—1044no—
579C2×C4#8817812yes—
580C2×C4C2×#610712yes—
581C2×C4C2×#610712yes—
582C2×C4C22×Dic2611yescharacteristic
583C2×C4C2×#612711yescharacteristic
584C2×C4C2×#612911yescharacteristic
585C2×C4#8825111yescharacteristic
586C2×C4#41×D1311yescharacteristic
587C2×C4#8825611yescharacteristic
588C8—521no—
589C8—521no—
590C8—521no—
591C8—521no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1171868214yes—
599C22C2×#8817811yescharacteristic
600C22#1171951111yescharacteristic
601C22C22×#610711yescharacteristic
602C22C2×#8820311yescharacteristic
603C22#1171953411yescharacteristic
604C22#1171954111yescharacteristic
605C22#1171954211yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—522no—
616C4—1041no—
617C4—1041no—
618C4—1041no—
619C4#1171924311yescharacteristic
620C4#1171924411yescharacteristic
621C4#1171924811yescharacteristic
622C4#1171926511yescharacteristic
623C2—24no—
624C2#1311865411yescharacteristic
625C2C2×#1171868211yescharacteristic
626C2#1312469811yescharacteristic
627C2#1312670611yescharacteristic
628C2#1312671211yescharacteristic
629C2#1312679711yescharacteristic
630C2#1312685111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2360C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2360D13
hypocenterC13#26414
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #13116399C2semidirect
3. #13116332C2semidirect
4. #13116231C2semidirect
5. #13115977C2semidirect
6. #13115784C2semidirect
7. #13115428C2semidirect
10. #13115090C2semidirect
13. #13114841C2semidirect
65. C13×C22⋊Q8Q8semidirect
382. C22⋊Q8Dic26semidirect
523. C13#26414semidirect

Maximal subgroups

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

subgroupindex
#131185382
#131163992
#131163322
#131162312
#131159772
#131157842
#131154282
#131152602
#131151442
#131150902
#131150242
#131150162
#131148412
C2×#117181212
C13×#23602
#2641413