Finite Group Databasebeta

← #15867262

Subgroups — #15867262

Order 3328.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117314C211yesmaximal, characteristic, complemented
2#13117301C211yesmaximal, characteristic
3#13115719C211yesmaximal, characteristic
4#13115692C211yesmaximal, characteristic
5#13115575C211yesmaximal, characteristic, complemented
6#13115570C211yesmaximal, characteristic, complemented
7#13115535C211yesmaximal, characteristic
8#13115528C211yesmaximal, characteristic, complemented
9#13115521C211yesmaximal, characteristic
10#13115514C211yesmaximal, characteristic, complemented
11#13114733C211yesmaximal, characteristic, complemented
12#13114666C211yesmaximal, characteristic, complemented
13#13114661C211yesmaximal, characteristic
14#97×Dic13C211yesmaximal, characteristic, complemented
15C2×C13×#440C211yesmaximal, characteristic
16C22×C13×SD16C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×#88180—21no—
20C13×#440C418yescomplemented
21C2×C13×#107C2211yescharacteristic
22C2×C13×#103C2211yescharacteristic
23C2×C13×#102C2211yescharacteristic
24C2×C13×#97C2211yescharacteristic
25#11718891C2211yescharacteristic
26#11718891C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29#11718882C2211yescharacteristic
30#11718882C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C4×C22⋊Dic13—21no—
34C2×#88058C2211yescharacteristic
35C2×#88057C2211yescharacteristic
36C2×#88055C2211yescharacteristic
37C2×#88055C2211yescharacteristic
38C2×C8×Dic13C2211yescharacteristic
39C2×C8×Dic13C2211yescharacteristic
40#11718646C2211yescharacteristic
41#11718646C2211yescharacteristic
42#11718646C2211yescharacteristic
43#11718646C2211yescharacteristic
44#11718642—21no—
45#11718642—21no—
46#11718612—21no—
47#11718612—21no—
48C4×C4⋊Dic13—21no—
49C4×C4⋊Dic13C2211yescharacteristic
50C4×C4⋊Dic13C2211yescharacteristic
51#11718593—21no—
52C42×Dic13C2211yescharacteristic
53C2×#88043C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C2×C13×SD16—22no—
64C2×C13×SD16—22no—
65C2×C13×SD16C2×C414yes—
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C4×Q8×C13—22no—
69C4×Q8×C13C2×C414yes—
70C4×D4×C13—22no—
71C4×D4×C13C2×C414yes—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C42×C13—21no—
77C2×C42×C13C2311yescharacteristic
78C2×C22⋊Dic13—21no—
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82#88180—44no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4×Dic13—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13D411yescharacteristic
94C2×C4×Dic13D411yescharacteristic
95C2×C4×Dic13D411yescharacteristic
96C2×C4×Dic13D411yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C13×#107C2×C414yes—
102C13×#103C2×C414yes—
103C13×#102C2×C414yes—
104C13×#97C2×C414yes—
105#88075—22no—
106#88075—22no—
107#88072—22no—
108#88072—22no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88058—22no—
120#88057—22no—
121#88055—22no—
122#88055—22no—
123C8×Dic13—22no—
124C8×Dic13—22no—
125#88043—22no—
126#21288—131nomaximal, Sylow
127C24×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13—21no—
133C2×Q8×C13C22×C412yes—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—21no—
137C2×D4×C13—21no—
138C2×D4×C13—21no—
139C2×D4×C13C22×C412yes—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—24no—
144C22×C4×C13—42no—
145C22×C4×C13—42no—
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2411yescharacteristic
149C22×Dic13—21no—
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C22×Dic13C2×D411yescharacteristic
157C13×SD16—28no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13C22×C412yes—
161C2×C8×C13C22×C412yes—
162C2×C8×C13#4112yes—
163C2×C8×C13#4112yes—
164C13×C4⋊C4—24no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4C22×C412yes—
168C13×C4⋊C4C22×C412yes—
169C13×C22⋊C4—24no—
170C42×C13—24no—
171C42×C13C22×C412yes—
172C42×C13#4112yes—
173C22⋊Dic13—42no—
174C22⋊Dic13—42no—
175C22⋊Dic13—42no—
176C22⋊Dic13—42no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—42no—
186C4⋊Dic13—42no—
187C4⋊Dic13—42no—
188C4⋊Dic13—42no—
189C4×Dic13—21no—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—22no—
194C4×Dic13—22no—
195C4×Dic13—41no—
196C4×Dic13—41no—
197C4×Dic13—42no—
198C4×Dic13—42no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055#4112yes—
204C2×#1055#4112yes—
205C2×#440D1311yescharacteristic
206#2295—131no—
207#2293—131no—
208#1973—131no—
209#1968—131no—
210#1943—131no—
211#1942—131no—
212#1936—131no—
213#1935—131no—
214#1934—131no—
215#1933—131no—
216#1776—131no—
217#1763—131no—
218#1762—131no—
219C4×#97—131no—
220C23×C13—22no—
221C23×C13—41no—
222C23×C13—41no—
223C23×C13—41no—
224C23×C13—41no—
225C23×C13—41no—
226C23×C13—41no—
227C23×C13C22×D411yescharacteristic
228Q8×C13—22no—
229Q8×C13—22no—
230Q8×C13—41no—
231Q8×C13—41no—
232D4×C13—22no—
233D4×C13—22no—
234D4×C13—41no—
235D4×C13—41no—
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—48no—
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×C13C4×D412yes—
251C2×C4×C13C4×D412yes—
252C2×C4×C13C23×C411yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—41no—
269C2×Dic13—41no—
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×Dic13AGL(1,ℤ/8)12yes—
281C2×Dic13#13712yes—
282C8×C13—22no—
283C8×C13#11714yescomplemented
284#1055—22no—
285#1055—22no—
286C22×SD16C2×D1311yescharacteristic
287C2×C4×Q8C2×D1311yescharacteristic
288C2×C4×D4C2×D1311yescharacteristic
289C2×#117—261no—
290#440Dic1318yescomplemented
291C2×#107—131no—
292C2×#107C2×D1311yescharacteristic
293C2×#106—131no—
294C2×#103—131no—
295C2×#103—131no—
296C2×#103C2×D1311yescharacteristic
297C2×#102—131no—
298C2×#102—131no—
299C2×#102C2×D1311yescharacteristic
300C2×#97—131no—
301C2×#97—131no—
302C2×#97—131no—
303C2×#97C2×D1311yescharacteristic
304C2×C4×C8—131no—
305C2×C4×C8—131no—
306#391—131no—
307#391—131no—
308#390—131no—
309#390—131no—
310#389—131no—
311#389—131no—
312#389—131no—
313#389—131no—
314#388—261no—
315#388—261no—
316#387—261no—
317#387—261no—
318C4×C4⋊C4—131no—
319C4×C4⋊C4—131no—
320C4×C4⋊C4—261no—
321C4×C22⋊C4—261no—
322#376—261no—
323C43—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330C22×C13—42no—
331C22×C13—42no—
332C22×C13—42no—
333C22×C13—81no—
334C22×C13—81no—
335C22×C13—81no—
336C22×C13—81no—
337C22×C13C2×C4×D411yescharacteristic
338C22×C13#53211yescharacteristic
339C22×C13#53211yescharacteristic
340C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C13C2×#13711yescharacteristic
342C22×C13#57511yescharacteristic
343C22×C13#57511yescharacteristic
344C4×C13—22no—
345C4×C13—22no—
346C4×C13—42no—
347C4×C13—42no—
348C4×C13—42no—
349C4×C13C2×#11711yescharacteristic
350C4×C13C4×#4111yescharacteristic
351C4×C13#52911yescharacteristic
352C4×C13#52911yescharacteristic
353Dic13—22no—
354Dic13—22no—
355Dic13—41no—
356Dic13—41no—
357Dic13—81no—
358Dic13—81no—
359Dic13—81no—
360Dic13—81no—
361C22×Q8C22×D1311yescharacteristic
362C22×D4C22×D1311yescharacteristic
363C23×C4—21no—
364C2×SD16—22no—
365C2×SD16—22no—
366C2×SD16C2×Dic1314yes—
367C22×C8—131no—
368C22×C8—131no—
369C22×C8C22×D1311yescharacteristic
370C22×C8C22×D1311yescharacteristic
371C4×Q8—22no—
372C4×Q8C2×Dic1314yes—
373C4×D4—22no—
374C4×D4C2×Dic1314yes—
375#117—524no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4C22×D1311yescharacteristic
386C2×C4⋊C4C22×D1311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—261no—
389C2×C22⋊C4—261no—
390C2×C22⋊C4—261no—
391C2×C22⋊C4—261no—
392C2×C42—21no—
393C2×C42—131no—
394C2×C42—131no—
395C2×C42—131no—
396C2×C42—131no—
397C2×C42—131no—
398C2×C42—131no—
399C2×C42—261no—
400C2×C42—261no—
401C2×C42C22×D1311yescharacteristic
402#107—262no—
403#107C2×Dic1314yes—
404#106—262no—
405#103—262no—
406#103—262no—
407#103C2×Dic1314yes—
408#102—262no—
409#102—262no—
410#102C2×Dic1314yes—
411#97—262no—
412#97—262no—
413#97—262no—
414#97C2×Dic1314yes—
415C4×C8—262no—
416C4×C8—262no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424#95—261no—
425#95—261no—
426#95—261no—
427C2×C13—82no—
428C2×C13#229311yescharacteristic
429C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
430C2×C13C4×#13711yescharacteristic
431C2×C13#318011yescharacteristic
432C2×C13#318111yescharacteristic
433C2×C13#318211yescharacteristic
434C2×C13#318311yescharacteristic
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C22×Dic1312yes—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4C22×Dic1312yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—261no—
461C22×C4—261no—
462C22×C4D4×D1311yescharacteristic
463C22×C4D4×D1311yescharacteristic
464C22×C4C23×D1311yescharacteristic
465SD16—28no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—132no—
469C2×C8—132no—
470C2×C8—261no—
471C2×C8—261no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8#612712yes—
475C2×C8#612712yes—
476C2×C8C22×Dic1312yes—
477C2×C8C22×Dic1312yes—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—261no—
482C4⋊C4—261no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—261no—
489C4⋊C4—522no—
490C4⋊C4—522no—
491C4⋊C4—522no—
492C4⋊C4—522no—
493C4⋊C4C22×Dic1312yes—
494C4⋊C4C22×Dic1312yes—
495C22⋊C4—24no—
496C22⋊C4—522no—
497C22⋊C4—522no—
498C22⋊C4—522no—
499C22⋊C4—522no—
500C42—24no—
501C42—262no—
502C42—262no—
503C42—261no—
504C42—261no—
505C42—261no—
506C42—261no—
507C42—521no—
508C42—521no—
509C42—522no—
510C42—522no—
511C42#612912yes—
512C42C22×Dic1312yes—
513C13#2128811yescharacteristic, complemented, Sylow
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D1311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—132no—
545C2×C4—132no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—521no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4D4×Dic1312yes—
569C2×C4D4×Dic1312yes—
570C2×C4C2×#612711yescharacteristic
571C2×C4C2×#612711yescharacteristic
572C2×C4C2×#612911yescharacteristic
573C2×C4#41×D1311yescharacteristic
574C2×C4#41×D1311yescharacteristic
575C2×C4C23×Dic1311yescharacteristic
576C8—22no—
577C8—262no—
578C8—262no—
579C8#8818014yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#1171925311yescharacteristic
588C22#1171925311yescharacteristic
589C22D13×AGL(1,ℤ/8)11yescharacteristic
590C22#1171947811yescharacteristic
591C22D13×#13711yescharacteristic
592C22#1171948211yescharacteristic
593C22C2×D4×Dic1311yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—262no—
600C4—262no—
601C4—521no—
602C4—521no—
603C4—1041no—
604C4—1041no—
605C4—1041no—
606C4—1041no—
607C4#1171926211yescharacteristic
608C4#1171926211yescharacteristic
609C4C2×#8818011yescharacteristic
610C4#41×Dic1311yescharacteristic
611C2—82no—
612C2#1311730111yescharacteristic
613C2AGL(1,ℤ/8)×Dic1311yescharacteristic
614C2#137×Dic1311yescharacteristic
615C2#1312553011yescharacteristic
616C2#1312554111yescharacteristic
617C2#1312555111yescharacteristic
618C2#1312556111yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#440C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#440D13
hypocenterC13#21288
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #13117314C2semidirect
5. #13115575C2semidirect
6. #13115570C2semidirect
8. #13115528C2semidirect
10. #13115514C2semidirect
11. #13114733C2semidirect
12. #13114666C2semidirect
14. #97×Dic13C2semidirect
20. C13×#440C4semidirect
283. C8×C13#117semidirect
290. #440Dic13semidirect
513. C13#21288semidirect
579. C8#88180semidirect

Maximal subgroups

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

subgroupindex
#131173142
#131173012
#131157192
#131156922
#131155752
#131155702
#131155352
#131155282
#131155212
#131155142
#131147332
#131146662
#131146612
#97×Dic132
C2×C13×#4402
#2128813