Finite Group Databasebeta

← #15926532

Subgroups — #15926532

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#13118539C211yesmaximal, characteristic
2#13117841C211yesmaximal, characteristic
3#13116396C211yesmaximal, characteristic, complemented
4#13116335C211yesmaximal, characteristic, complemented
5#13116212C211yesmaximal, characteristic, complemented
6#13115976C211yesmaximal, characteristic, complemented
7#13115786C211yesmaximal, characteristic, complemented
8#13115416C211yesmaximal, characteristic, complemented
9#13115257C211yesmaximal, characteristic
10#13115141C211yesmaximal, characteristic
11#13115101C211yesmaximal, characteristic, complemented
12#13115013C211yesmaximal, characteristic
13#13114840C211yesmaximal, characteristic, complemented
14C2×#11718121C211yesmaximal, characteristic
15C13×#1853C211yesmaximal, characteristic
16C22×C13×C4⋊C4C2211yescharacteristic
17C22×C4⋊Dic13C2211yescharacteristic
18C2×#88178C2211yescharacteristic
19C23×#1055C2211yescharacteristic
20C2×#88142C2211yescharacteristic
21C2×#88141C2211yescharacteristic
22C2×#88132—21no—
23C2×C13×#106C2211yescharacteristic
24C2×C13×#106C2211yescharacteristic
25C2×C13×C22⋊C8C2211yescharacteristic
26C13×#380C2211yescharacteristic
27#11718871—21no—
28C4×C22⋊Dic13—21no—
29C4×C22⋊Dic13C2211yescharacteristic
30C2×#88056C2211yescharacteristic
31C2×#88056C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×#88054C2211yescharacteristic
34#11718646C2211yescharacteristic
35#11718639—21no—
36#11718637C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38#11718635C2211yescharacteristic
39#11718634C2211yescharacteristic
40#11718634C2211yescharacteristic
41#11718633—21no—
42C2×#88050C2211yescharacteristic
43C2×#88050C2211yescharacteristic
44#11718607—21no—
45#11718318—21no—
46#11718317—21no—
47C13×#340C2211yescharacteristic
48C13×#340C2211yescharacteristic
49#11718143C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59C23×C4×C13—21no—
60C23×C4×C13C2311yescharacteristic
61C23×Dic13—21no—
62C22×Dic26—21no—
63C22×Dic26C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C2×C13×C4⋊C4—22no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—41no—
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C22⋊C4—21no—
74C2×C22⋊Dic13—21no—
75C2×C22⋊Dic13D411yescharacteristic
76C2×C22⋊Dic13D411yescharacteristic
77C2×C4⋊Dic13—22no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13—41no—
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84#88178—22no—
85#88178Q814yescomplemented
86C2×#6101—21no—
87C2×#6101—21no—
88C2×#6101—21no—
89C2×#6101—21no—
90C2×#6101—21no—
91C2×#6101—21no—
92C2×#6101—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C22×#1055—22no—
98C22×#1055—41no—
99C22×#1055—41no—
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102#88142—22no—
103#88142—41no—
104#88141—22no—
105#88141—41no—
106#88132—44no—
107C13×#106—22no—
108C13×#106—41no—
109C13×C22⋊C8—22no—
110C13×#95—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071D411yescharacteristic
118#88071D411yescharacteristic
119#88056—22no—
120#88056—41no—
121#88054—41no—
122#88054—41no—
123#88050—41no—
124#88050—41no—
125#26412—131nomaximal, Sylow
126C24×C13C2×D411yescharacteristic
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—21no—
134C22×C4×C13—21no—
135C22×C4×C13—42no—
136C22×C4×C13—42no—
137C22×C4×C13—81no—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×Q812yes—
140C22×C4×C13#4112yes—
141C22×C4×C13C2411yescharacteristic
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—42no—
147C22×Dic13—42no—
148C22×Dic13—81no—
149C22×Dic13C2×D411yescharacteristic
150C22×Dic13C2×D411yescharacteristic
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C2×Dic26—21no—
154C2×Dic26—21no—
155C2×Dic26—21no—
156C2×Dic26—21no—
157C2×Dic26—21no—
158C2×Dic26—42no—
159C2×Dic26—42no—
160C2×Dic26—81no—
161C2×Dic26C2×Q812yes—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—42no—
176C13×C4⋊C4#4112yes—
177C13×C22⋊C4—42no—
178C22⋊Dic13—24no—
179C22⋊Dic13—22no—
180C22⋊Dic13SD1614yes—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—42no—
186C4⋊Dic13—42no—
187C4⋊Dic13C2×Q812yes—
188C4⋊Dic13#4112yes—
189#6101—24no—
190#6101—42no—
191#6101—42no—
192#6101—42no—
193#6101—42no—
194#6101—81no—
195#6101—81no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—42no—
201C4×Dic13—81no—
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#2360—131no—
211#2099—131no—
212#2088—131no—
213#2066—131no—
214#2024—131no—
215#1988—131no—
216#1919—131no—
217#1892—131no—
218#1872—131no—
219#1865—131no—
220#1853D1311yescharacteristic
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×SD1612yes—
230C23×C13#13712yes—
231C23×C13C22×D411yescharacteristic
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
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—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—84no—
251C2×C4×C13C22⋊Q812yes—
252C2×C4×C13C22×Q811yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13H(5,𝔽2)11yescharacteristic
257C2×C4×C13H(5,𝔽2)11yescharacteristic
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—22no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
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—84no—
281C2×Dic13C2×SD1612yes—
282C2×Dic13C2×SD1612yes—
283Dic26—22no—
284Dic26—22no—
285Dic26—41no—
286Dic26—41no—
287Dic26—84no—
288C8×C13—41no—
289C8×C13—41no—
290#1055—41no—
291#1055—41no—
292C23×C8—131no—
293C2×C4⋊Q8—131no—
294C2×C22⋊Q8—131no—
295C2×C22⋊Q8—261no—
296C2×C4×Q8—131no—
297C22×C4⋊C4—131no—
298C22×C4⋊C4C2×D1311yescharacteristic
299C2×#106C2×D1311yescharacteristic
300C2×#106C2×D1311yescharacteristic
301C2×C4⋊C8—131no—
302C2×C4⋊C8—131no—
303C2×#103—131no—
304C2×#103—131no—
305C2×#103—131no—
306C2×#103—131no—
307C2×C22⋊C8C2×D1311yescharacteristic
308#398—261no—
309#395—261no—
310#391—131no—
311#389—131no—
312#389—131no—
313#389—131no—
314#389—261no—
315#387—261no—
316#385—261no—
317#385—261no—
318#384—131no—
319#380C2×D1311yescharacteristic
320C4×C4⋊C4—131no—
321C4×C22⋊C4—131no—
322C4×C22⋊C4—261no—
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#340C2×D1311yescharacteristic
332#340C2×D1311yescharacteristic
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#47514yes—
342C22×C13C2×C22⋊Q811yescharacteristic
343C22×C13#54611yescharacteristic
344C22×C13#54611yescharacteristic
345C22×C13C22×SD1611yescharacteristic
346C22×C13C2×#13711yescharacteristic
347C22×C13#57611yescharacteristic
348C22×C13#57811yescharacteristic
349C4×C13—22no—
350C4×C13—42no—
351C4×C13—81no—
352C4×C13—81no—
353C4×C13—81no—
354C4×C13#52911yescharacteristic
355C4×C13#53811yescharacteristic
356C4×C13#55411yescharacteristic
357C4×C13#55411yescharacteristic
358Dic13—22no—
359Dic13—22no—
360Dic13—42no—
361Dic13—42no—
362Dic13—42no—
363Dic13—41no—
364Dic13—41no—
365Dic13—81no—
366C22×Q8—131no—
367C22×Q8—261no—
368C23×C4—21no—
369C23×C4—261no—
370C23×C4C22×D1311yescharacteristic
371C22×C8—131no—
372C22×C8—131no—
373C22×C8—262no—
374C22×C8—521no—
375C22×C8—521no—
376C22×C8C22×D1311yescharacteristic
377C22×C8C22×D1311yescharacteristic
378C4⋊Q8—262no—
379C4⋊Q8—521no—
380C22⋊Q8—134no—
381C22⋊Q8—262no—
382C22⋊Q8—524no—
383C4×Q8—262no—
384C4×Q8—521no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—41no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—262no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4—261no—
394C2×C4⋊C4—261no—
395C2×C4⋊C4—261no—
396C2×C4⋊C4—261no—
397C2×C4⋊C4—261no—
398C2×C4⋊C4—261no—
399C2×C4⋊C4—261no—
400C2×C4⋊C4—261no—
401C2×C4⋊C4—521no—
402C2×C4⋊C4C22×D1311yescharacteristic
403C2×C4⋊C4C22×D1311yescharacteristic
404C2×C4⋊C4C22×D1311yescharacteristic
405C2×C4⋊C4C22×D1311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—131no—
408C2×C22⋊C4—131no—
409C2×C22⋊C4—261no—
410C2×C42—131no—
411C2×C42—131no—
412C2×C42—261no—
413C2×C42—261no—
414#106—22no—
415#106—41no—
416C4⋊C8—521no—
417C4⋊C8—521no—
418#103—262no—
419#103—521no—
420#103—521no—
421#103—521no—
422C22⋊C8—22no—
423#95—21no—
424#95—131no—
425#95—131no—
426#95—261no—
427#95—261no—
428#95—261no—
429#95—261no—
430#95—261no—
431#95—261no—
432C2×C13—24no—
433C2×C13#245011yescharacteristic
434C2×C13C2×#47511yescharacteristic
435C2×C13#308311yescharacteristic
436C2×C13#329511yescharacteristic
437C2×C13#329811yescharacteristic
438C2×C13#331211yescharacteristic
439C2×C13#331411yescharacteristic
440C24D4×D1311yescharacteristic
441C2×Q8—132no—
442C2×Q8—261no—
443C2×Q8—261no—
444C2×Q8—261no—
445C2×Q8—261no—
446C2×Q8—261no—
447C2×Q8—522no—
448C2×Q8—522no—
449C2×Q8—1041no—
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—131no—
462C22×C4—131no—
463C22×C4—131no—
464C22×C4—131no—
465C22×C4—261no—
466C22×C4—261no—
467C22×C4—261no—
468C22×C4—261no—
469C22×C4—522no—
470C22×C4—522no—
471C22×C4—1041no—
472C22×C4#612712yes—
473C22×C4D4×D1311yescharacteristic
474C22×C4Q8×D1312yes—
475C22×C4C23×D1311yescharacteristic
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8—261no—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8—524no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—132no—
498C4⋊C4—132no—
499C4⋊C4—264no—
500C4⋊C4—261no—
501C4⋊C4—261no—
502C4⋊C4—261no—
503C4⋊C4—261no—
504C4⋊C4—522no—
505C4⋊C4—522no—
506C4⋊C4—522no—
507C4⋊C4—522no—
508C4⋊C4—522no—
509C4⋊C4—522no—
510C4⋊C4—1041no—
511C4⋊C4—1041no—
512C4⋊C4#612912yes—
513C22⋊C4—42no—
514C22⋊C4—134no—
515C22⋊C4—264no—
516C22⋊C4—262no—
517C42—261no—
518C42—261no—
519C42—261no—
520C42—261no—
521C42—522no—
522C42—1041no—
523C13#2641211yescharacteristic, complemented, Sylow
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#8815012yes—
529C23SD16×D1312yes—
530C23#8816912yes—
531C23C2×D4×D1311yescharacteristic
532Q8—262no—
533Q8—262no—
534Q8—521no—
535Q8—521no—
536Q8—1044no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—132no—
557C2×C4—132no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—262no—
565C2×C4—262no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—261no—
569C2×C4—261no—
570C2×C4—262no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—521no—
574C2×C4—521no—
575C2×C4—521no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—521no—
579C2×C4—521no—
580C2×C4—1044no—
581C2×C4#8815012yes—
582C2×C4C2×#612711yescharacteristic
583C2×C4C2×#612911yescharacteristic
584C2×C4#8825111yescharacteristic
585C2×C4C2×Q8×D1311yescharacteristic
586C2×C4#41×D1311yescharacteristic
587C2×C4#8825611yescharacteristic
588C8—41no—
589C8—41no—
590C8—521no—
591C8—521no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1171851514yes—
599C22C2×#8815011yescharacteristic
600C22#1171929611yescharacteristic
601C22#1171935911yescharacteristic
602C22C2×SD16×D1311yescharacteristic
603C22C2×#8816911yescharacteristic
604C22#1171947911yescharacteristic
605C22#1171948411yescharacteristic
606C4—22no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—262no—
612C4—262no—
613C4—522no—
614C4—521no—
615C4—522no—
616C4—522no—
617C4—521no—
618C4—1041no—
619C4#1171924311yescharacteristic
620C4#1171926511yescharacteristic
621C4#1171934011yescharacteristic
622C4#1171934111yescharacteristic
623C2—24no—
624C2#1311866211yescharacteristic
625C2C2×#1171851511yescharacteristic
626C2#1312468611yescharacteristic
627C2#1312665811yescharacteristic
628C2#1312669411yescharacteristic
629C2#1312688511yescharacteristic
630C2#1312692111yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1853C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1853D13
hypocenterC13#26412
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
3. #13116396C2semidirect
4. #13116335C2semidirect
5. #13116212C2semidirect
6. #13115976C2semidirect
7. #13115786C2semidirect
8. #13115416C2semidirect
11. #13115101C2semidirect
13. #13114840C2semidirect
85. #88178Q8semidirect
523. C13#26412semidirect

Maximal subgroups

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

subgroupindex
#131185392
#131178412
#131163962
#131163352
#131162122
#131159762
#131157862
#131154162
#131152572
#131151412
#131151012
#131150132
#131148402
C2×#117181212
C13×#18532
#2641213