Finite Group Databasebeta

← #15867829

Subgroups — #15867829

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718501C211yesmaximal, characteristic
2#13117444C211yesmaximal, characteristic
3#13117190C211yesmaximal, characteristic
4#13115606C211yesmaximal, characteristic, complemented
5#13115604C211yesmaximal, characteristic, complemented
6#13115592C211yesmaximal, characteristic, complemented
7#13115034C211yesmaximal, characteristic
8#13115034C211yesmaximal, characteristic
9#13115030C211yesmaximal, characteristic
10#13115025C211yesmaximal, characteristic
11#13114736C211yesmaximal, characteristic, complemented
12#13114732C211yesmaximal, characteristic, complemented
13#13114728C211yesmaximal, characteristic, complemented
14#106×Dic13C211yesmaximal, characteristic, complemented
15C13×#1948C211yesmaximal, characteristic, complemented
16C2×#88146C2211yescharacteristic
17C2×C4⋊C4×D13C2211yescharacteristic
18C2×#88126C2211yescharacteristic
19C2×C13×#106C2211yescharacteristic
20C2×C13×#106C2211yescharacteristic
21C2×C13×C4⋊C8C2211yescharacteristic
22C13×#384C2211yescharacteristic
23C4×C13×C4⋊C4C2211yescharacteristic
24C2×#88060C2211yescharacteristic
25C2×#88060C2211yescharacteristic
26C2×#88058C2211yescharacteristic
27C2×#88057C2211yescharacteristic
28C2×#88055C2211yescharacteristic
29C2×C8×Dic13C2211yescharacteristic
30#11718651—21no—
31#11718651—21no—
32#11718648C2211yescharacteristic
33#11718648C2211yescharacteristic
34#11718639—21no—
35#11718639—21no—
36C4⋊C4×Dic13C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38C4⋊C4×Dic13C2211yescharacteristic
39C4⋊C4×Dic13C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44C4×#6103—21no—
45#11718597C2211yescharacteristic
46C4×C4⋊Dic13C2211yescharacteristic
47C4×#6101—21no—
48C2×C4⋊#1055C2211yescharacteristic
49#11718501C418yescomplemented
50#11718327—21no—
51#11718310—21no—
52C13×#340C2211yescharacteristic
53C13×#340C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×C4×D13—21no—
61C22×C4×D13C2311yescharacteristic
62C22×C8×C13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C2×C13×C4⋊C4—21no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4C2311yescharacteristic
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C42×C13—21no—
71C2×C42×C13C2311yescharacteristic
72C2×#6103—21no—
73C2×#6103—21no—
74C2×#6103—21no—
75C2×#6103D411yescharacteristic
76C2×#6103D411yescharacteristic
77C2×C4⋊Dic13C2311yescharacteristic
78C2×C4⋊Dic13C2311yescharacteristic
79C2×C4⋊Dic13C2311yescharacteristic
80C2×C4⋊Dic13C2311yescharacteristic
81C2×#6101—21no—
82C2×#6101D411yescharacteristic
83C2×#6101D411yescharacteristic
84C2×C4×Dic13—21no—
85C2×C4×Dic13—21no—
86C2×C4×Dic13—21no—
87C2×C4×Dic13—21no—
88C2×C4×Dic13—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13C2311yescharacteristic
91C22×#1055C2311yescharacteristic
92C22×#1055C2311yescharacteristic
93#88146—22no—
94#88146—22no—
95C4⋊C4×D13—22no—
96C4⋊C4×D13—22no—
97#88126C2×C414yes—
98#88126Q814yes—
99C13×#106—22no—
100C13×#106—22no—
101C13×C4⋊C8C2×C414yescomplemented
102C13×#95—21no—
103#88071—21no—
104#88071—21no—
105#88071—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88060C2×C414yes—
113#88060C2×C414yes—
114#88058—22no—
115#88057—22no—
116#88055C2×C414yescomplemented
117C8×Dic13C2×C414yescomplemented
118#88048—22no—
119#88047—22no—
120#88047—22no—
121#88047—22no—
122C4⋊#1055C2×C414yescomplemented
123#21367—131nomaximal, Sylow
124C23×D13C2×D411yescharacteristic
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13C2×D411yescharacteristic
130C22×C4×C13C2×D411yescharacteristic
131C22×C4×C13C2411yescharacteristic
132C22×Dic13—21no—
133C22×Dic13—21no—
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13C2×D411yescharacteristic
137C22×Dic13C2×D411yescharacteristic
138C22×Dic13C2×D411yescharacteristic
139C2×C4×D13—21no—
140C2×C4×D13—21no—
141C2×C4×D13—21no—
142C2×C4×D13—21no—
143C2×C4×D13—24no—
144C2×C4×D13—81no—
145C2×C4×D13—81no—
146C2×C4×D13C22×C412yes—
147C2×C4×D13C2×Q812yes—
148C2×C8×C13—21no—
149C2×C8×C13—21no—
150C2×C8×C13—21no—
151C2×C8×C13—21no—
152C2×C8×C13C22×C412yes—
153C2×C8×C13C22×C412yes—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—81no—
161C13×C4⋊C4—81no—
162C13×C4⋊C4#4112yes—
163C13×C4⋊C4#4112yes—
164C42×C13—81no—
165C42×C13C22×C412yes—
166C42×C13C2×Q812yes—
167#6103—24no—
168#6103—81no—
169#6103—81no—
170#6103C4⋊C414yes—
171#6103C4⋊C414yes—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13#4112yes—
179C4⋊Dic13#4112yes—
180#6101—24no—
181#6101C4⋊C414yes—
182#6101C4⋊C414yes—
183C4×Dic13—24no—
184C4×Dic13—81no—
185C4×Dic13—81no—
186C4×Dic13—81no—
187C4×Dic13—81no—
188C4×Dic13—81no—
189C4×Dic13C22×C412yes—
190C4×Dic13C2×Q812yes—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055C22×C412yes—
196C2×#1055C22×C412yes—
197C2×#433—131no—
198#2314—131no—
199#2274—131no—
200#1949—131no—
201#1949—131no—
202#1948D1311yescharacteristic, complemented
203#1947—131no—
204#1854—131no—
205#1854—131no—
206#1853—131no—
207#1852—131no—
208#1776—131no—
209#1776—131no—
210C4×#107—131no—
211C4×#106—131no—
212C23×C13C22×D411yescharacteristic
213C22×D13—21no—
214C22×D13—21no—
215C22×D13—21no—
216C22×D13—21no—
217C22×D13—21no—
218C22×D13—21no—
219C22×D13C2×C4⋊C412yes—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—22no—
225C2×C4×C13—22no—
226C2×C4×C13—41no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13C2×C4⋊C412yes—
237C2×C4×C13C2×C4⋊C412yes—
238C2×C4×C13C23×C411yescharacteristic
239C2×C4×C13C22×Q811yescharacteristic
240C2×C4×C13C2×#4111yescharacteristic
241C2×C4×C13C2×#4111yescharacteristic
242C2×C4×C13C2×#4111yescharacteristic
243C2×C4×C13C2×#4111yescharacteristic
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—41no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13C2×C4⋊C412yes—
263C2×Dic13C2×C4⋊C412yes—
264C2×Dic13C2×C4⋊C412yes—
265C4×D13—22no—
266C4×D13—22no—
267C4×D13—22no—
268C4×D13—22no—
269C4×D13—84no—
270C8×C13—22no—
271C8×C13—22no—
272#1055—22no—
273#1055—22no—
274C2×#117—131no—
275C2×#117—131no—
276C22×C4⋊C4—131no—
277#433—138no—
278C2×#107—131no—
279C2×#107—131no—
280C2×#107—131no—
281C2×#107—131no—
282C2×#106—131no—
283C2×#106—131no—
284C2×#106C2×D1311yescharacteristic
285C2×#106C2×D1311yescharacteristic
286C2×C4⋊C8—131no—
287C2×C4⋊C8C2×D1311yescharacteristic
288C2×C22⋊C8—131no—
289C2×C22⋊C8—131no—
290C2×#97—131no—
291C2×C4×C8—131no—
292#388—261no—
293#388—261no—
294#387—261no—
295#387—261no—
296#384C2×D1311yescharacteristic
297#382—131no—
298#380—131no—
299#380—131no—
300C4×C4⋊C4—131no—
301C4×C4⋊C4—131no—
302C4×C4⋊C4—131no—
303C4×C4⋊C4—131no—
304C4×C4⋊C4—131no—
305C4×C4⋊C4—261no—
306C4×C4⋊C4—261no—
307C4×C4⋊C4C2×D1311yescharacteristic
308C4×C22⋊C4—261no—
309C4×C22⋊C4—261no—
310#340—131no—
311#340—131no—
312#340—131no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340C2×D1311yescharacteristic
317#340C2×D1311yescharacteristic
318C22×C13C22×C4⋊C411yescharacteristic
319C22×C13#52511yescharacteristic
320C22×C13#52511yescharacteristic
321C22×C13#57611yescharacteristic
322C22×C13#57711yescharacteristic
323C22×C13#57711yescharacteristic
324C22×C13#57811yescharacteristic
325C2×D13—22no—
326C2×D13—22no—
327C2×D13—22no—
328C2×D13—41no—
329C2×D13—41no—
330C2×D13—41no—
331C2×D13—41no—
332C4×C13—22no—
333C4×C13—22no—
334C4×C13—42no—
335C4×C13—42no—
336C4×C13—81no—
337C4×C13—81no—
338C4×C13C4×#4111yescharacteristic
339C4×C13C4×#4111yescharacteristic
340C4×C13#53311yescharacteristic
341C4×C13#53311yescharacteristic
342Dic13—22no—
343Dic13—22no—
344Dic13—22no—
345Dic13—42no—
346Dic13—42no—
347Dic13—81no—
348Dic13—81no—
349C23×C4—131no—
350C23×C4—261no—
351C22×C8—131no—
352C22×C8—131no—
353C22×C8C22×D1311yescharacteristic
354C22×C8C22×D1311yescharacteristic
355#117—134no—
356#117—134no—
357#117—262no—
358#117—262no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—131no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—131no—
365C2×C4⋊C4—131no—
366C2×C4⋊C4—131no—
367C2×C4⋊C4—262no—
368C2×C4⋊C4—261no—
369C2×C4⋊C4—262no—
370C2×C4⋊C4C22×D1311yescharacteristic
371C2×C4⋊C4C22×D1311yescharacteristic
372C2×C4⋊C4C22×D1311yescharacteristic
373C2×C4⋊C4C22×D1311yescharacteristic
374C2×C22⋊C4—131no—
375C2×C22⋊C4—131no—
376C2×C22⋊C4—261no—
377C2×C22⋊C4—261no—
378C2×C22⋊C4—261no—
379C2×C42—21no—
380C2×C42—131no—
381C2×C42—261no—
382C2×C42—261no—
383C2×C42—261no—
384C2×C42—261no—
385C2×C42—261no—
386C2×C42—261no—
387C2×C42C22×D1311yescharacteristic
388#107—262no—
389#107—262no—
390#107—262no—
391#107—262no—
392#106—22no—
393#106—22no—
394#106—262no—
395#106—262no—
396C4⋊C8—134no—
397C4⋊C8C4×D1314yescomplemented
398C22⋊C8—134no—
399C22⋊C8—134no—
400#97—134no—
401C4×C8—134no—
402#95—21no—
403#95—261no—
404#95—261no—
405#95—261no—
406#95—261no—
407#95—261no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95—261no—
412C2×C13#229611yescharacteristic
413C2×C13#291211yescharacteristic
414C2×C13#291311yescharacteristic
415C2×C13#311811yescharacteristic
416C2×C13#311911yescharacteristic
417C2×C13#312011yescharacteristic
418C2×C13#312111yescharacteristic
419D13—42no—
420C24—131no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—131no—
426C22×C4—131no—
427C22×C4—131no—
428C22×C4—132no—
429C22×C4—132no—
430C22×C4—264no—
431C22×C4—261no—
432C22×C4—261no—
433C22×C4—261no—
434C22×C4—261no—
435C22×C4—261no—
436C22×C4—261no—
437C22×C4—261no—
438C22×C4—261no—
439C22×C4—1041no—
440C22×C4—1041no—
441C22×C4D4×D1311yescharacteristic
442C22×C4D4×D1311yescharacteristic
443C22×C4C23×D1311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—132no—
449C2×C8—132no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8C2×C4×D1312yes—
455C2×C8C2×C4×D1312yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—134no—
465C4⋊C4—134no—
466C4⋊C4—132no—
467C4⋊C4—132no—
468C4⋊C4—264no—
469C4⋊C4—261no—
470C4⋊C4—261no—
471C4⋊C4—261no—
472C4⋊C4—261no—
473C4⋊C4—261no—
474C4⋊C4—261no—
475C4⋊C4#612912yes—
476C4⋊C4#613112yes—
477C22⋊C4—134no—
478C22⋊C4—134no—
479C22⋊C4—264no—
480C22⋊C4—1041no—
481C22⋊C4—1041no—
482C42—81no—
483C42—132no—
484C42—132no—
485C42—264no—
486C42—1041no—
487C42—1041no—
488C42—1041no—
489C42—1041no—
490C42—1041no—
491C42C2×C4×D1312yes—
492C42Q8×D1312yes—
493C13#2136711yescharacteristic, complemented, Sylow
494C23—132no—
495C23—261no—
496C23—261no—
497C23—261no—
498C23—261no—
499C23—261no—
500C23—261no—
501C23C2×D4×D1311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—132no—
519C2×C4—132no—
520C2×C4—132no—
521C2×C4—261no—
522C2×C4—261no—
523C2×C4—261no—
524C2×C4—261no—
525C2×C4—261no—
526C2×C4—261no—
527C2×C4—262no—
528C2×C4—262no—
529C2×C4—262no—
530C2×C4—262no—
531C2×C4—262no—
532C2×C4—262no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—521no—
537C2×C4—521no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—1044no—
544C2×C4C4⋊C4×D1312yes—
545C2×C4C4⋊C4×D1312yes—
546C2×C4C22×C4×D1311yescharacteristic
547C2×C4C2×#612911yescharacteristic
548C2×C4C2×Q8×D1311yescharacteristic
549C2×C4C2×#613111yescharacteristic
550C2×C4#41×D1311yescharacteristic
551C2×C4#41×D1311yescharacteristic
552C8—22no—
553C8—22no—
554C8—262no—
555C8—262no—
556C22—262no—
557C22—262no—
558C22—262no—
559C22—521no—
560C22—521no—
561C22—521no—
562C22—521no—
563C22C2×C4⋊C4×D1311yescharacteristic
564C22#1171931111yescharacteristic
565C22#1171933211yescharacteristic
566C22#1171946711yescharacteristic
567C22#1171946711yescharacteristic
568C22#1171947911yescharacteristic
569C22#1171948411yescharacteristic
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—81no—
576C4—262no—
577C4—262no—
578C4—262no—
579C4—522no—
580C4—522no—
581C4—1041no—
582C4—1041no—
583C4C4×#612911yescharacteristic
584C4C4×#613111yescharacteristic
585C4#1171939311yescharacteristic
586C4#1171943311yescharacteristic
587C2—522no—
588C2#1311731711yescharacteristic
589C2#1312342711yescharacteristic
590C2#1312343511yescharacteristic
591C2#1312502511yescharacteristic
592C2#1312503411yescharacteristic
593C2#1312504711yescharacteristic
594C2#1312507211yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1948C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
cosocleC22×C4×C13C24
hypercenter#1948D13
hypocenterC13#21367
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #13115606C2semidirect
5. #13115604C2semidirect
6. #13115592C2semidirect
11. #13114736C2semidirect
12. #13114732C2semidirect
13. #13114728C2semidirect
14. #106×Dic13C2semidirect
15. C13×#1948C2semidirect
49. #11718501C4semidirect
101. C13×C4⋊C8C2×C4semidirect
116. #88055C2×C4semidirect
117. C8×Dic13C2×C4semidirect
122. C4⋊#1055C2×C4semidirect
202. #1948D13semidirect
397. C4⋊C8C4×D13semidirect
493. C13#21367semidirect

Maximal subgroups

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

subgroupindex
C2×#117185012
#131174442
#131171902
#131156062
#131156042
#131155922
#131150342
#131150302
#131150252
#131147362
#131147322
#131147282
#106×Dic132
C13×#19482
#2136713