Finite Group Databasebeta

← #16019408

Subgroups — #16019408

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718840C211yesmaximal, characteristic, complemented
2C22×#88068C211yesmaximal, characteristic, complemented
3#13126523C211yesmaximal, characteristic
4#13117105C211yesmaximal, characteristic, complemented
5#13116951C211yesmaximal, characteristic, complemented
6#13116925C211yesmaximal, characteristic, complemented
7#13116898C211yesmaximal, characteristic, complemented
8#13116878C211yesmaximal, characteristic, complemented
9#13116774C211yesmaximal, characteristic, complemented
10#13116770C211yesmaximal, characteristic, complemented
11#13116764C211yesmaximal, characteristic, complemented
12#13116753C211yesmaximal, characteristic, complemented
13#13116698C211yesmaximal, characteristic, complemented
14#13116662C211yesmaximal, characteristic, complemented
15D4×C13×Q16C211yesmaximal, characteristic, complemented
16C22×C13×Q16C2211yescharacteristic
17C22×C13×Q16C2211yescharacteristic, complemented
18D4×Q8×C13—21no—
19C2×#88158C2211yescharacteristic
20C22×D104C2211yescharacteristic
21#11719255—21no—
22C13×#494C2211yescharacteristic, complemented
23C13×#467C2211yescharacteristic
24C13×#467C2211yescharacteristic, complemented
25C13×#462—21no—
26C13×#451—21no—
27C13×#451—21no—
28C4×C13×Q16C2211yescharacteristic, complemented
29C8×D4×C13C2211yescharacteristic
30C2×#88069C2211yescharacteristic, complemented
31C2×#88069C2211yescharacteristic, complemented
32C2×#88069C2211yescharacteristic, complemented
33#11718840—24no—
34C2×#88068—24no—
35C2×#88068C2211yescharacteristic
36C2×#88068C2211yescharacteristic
37C2×#88068C2211yescharacteristic
38C2×#88068C2211yescharacteristic
39C2×#88068C2211yescharacteristic
40C2×#88068C2211yescharacteristic
41#11718777C2211yescharacteristic
42#11718776C2211yescharacteristic
43C2×#88052C2211yescharacteristic, complemented
44C22×#6090C2211yescharacteristic
45#11718505—21no—
46#11718395—21no—
47#11718392—21no—
48#11718357C2211yescharacteristic
49C4×D104C2211yescharacteristic
50#11718156C2211yescharacteristic, complemented
51#11718156C2211yescharacteristic, complemented
52#11718156C2211yescharacteristic, complemented
53C22⋊#6090C2211yescharacteristic
54C22⋊#6090C2211yescharacteristic, complemented
55#11718084C2211yescharacteristic, complemented
56#11718083C2211yescharacteristic
57#11718083C2211yescharacteristic
58#11718081C2211yescharacteristic, complemented
59C4⋊#6090C2211yescharacteristic, complemented
60#11718053C2211yescharacteristic, complemented
61C22×Q8×C13—21no—
62C22×Q8×C13—21no—
63C2×#6127—21no—
64C22×D52—21no—
65C2×C13×Q16—22no—
66C2×C13×Q16—22no—
67C2×C13×Q16—21no—
68C2×C13×Q16—21no—
69C2×C13×Q16—21no—
70C2×C13×Q16—41no—
71C2×C13×Q16—41no—
72C2×C13×Q16C2311yescharacteristic
73C2×C13×Q16C2311yescharacteristic
74C2×C13×Q16C2311yescharacteristic
75C22×C8×C13C2311yescharacteristic
76C22×C8×C13C2311yescharacteristic
77C13×C4⋊Q8—21no—
78C13×C4⋊Q8—41no—
79C13×C22⋊Q8—21no—
80C13×C22⋊Q8—21no—
81C13×C22⋊Q8—41no—
82C13×C22⋊Q8—41no—
83C4×Q8×C13—21no—
84C4×D4×C13—41no—
85C4×D4×C13D411yescharacteristic
86#88191—41no—
87#88184—21no—
88#88178—21no—
89C2×Dic52C2311yescharacteristic
90#88158—22no—
91#88158—41no—
92C2×D104—22no—
93C2×D104—21no—
94C2×D104—41no—
95C2×D104C2311yescharacteristic
96C2×D104C2311yescharacteristic
97C2×#6095—21no—
98C2×#6095—21no—
99#88148—41no—
100#88138—41no—
101C22⋊C4×D13—41no—
102#88129—21no—
103C4×D52—21no—
104C13×#107C2311yescharacteristic
105C13×C4⋊C8D411yescharacteristic
106C13×#103—21no—
107C13×#103—21no—
108C13×#103—21no—
109C13×C22⋊C8D411yescharacteristic
110C13×C22⋊C8D411yescharacteristic
111C4×C8×C13C2311yescharacteristic
112#88069—22no—
113#88069—22no—
114#88069—41no—
115#88068—22no—
116#88068—22no—
117#88068—22no—
118#88068—22no—
119#88068—41no—
120#88068D414yescomplemented
121#88061—21no—
122#88061—21no—
123#88058C2311yescharacteristic
124#88056—21no—
125#88052—22no—
126C2×#6090—22no—
127C2×#6090C2311yescharacteristic
128C2×#6090C2311yescharacteristic
129C2×#6090C2311yescharacteristic
130C2×#6090C2311yescharacteristic
131#33832—131nomaximal, Sylow
132C23×D13—41no—
133C2×Q8×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×Q8×C13—42no—
137C2×Q8×C13—42no—
138C2×Q8×C13—41no—
139C2×Q8×C13—41no—
140C2×Q8×C13—81no—
141C2×Q8×C13—81no—
142C2×D4×C13D811yescharacteristic
143C22×C4×C13—41no—
144C22×C4×C13—41no—
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C2×C22⋊D13—41no—
148C2×C22⋊D13—41no—
149#6127—42no—
150#6127—81no—
151C2×D52—21no—
152C2×D52—21no—
153C2×D52—42no—
154C2×D52—41no—
155C2×D52—81no—
156C2×C4×D13—41no—
157C2×C4×D13—41no—
158C2×Dic26—21no—
159C13×Q16—21no—
160C13×Q16—21no—
161C13×Q16—21no—
162C13×Q16—21no—
163C13×Q16—21no—
164C13×Q16—42no—
165C13×Q16—42no—
166C13×Q16—81no—
167C13×Q16C2×D412yes—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13—21no—
171C2×C8×C13—21no—
172C2×C8×C13C2×D411yescharacteristic
173C2×C8×C13C2×D411yescharacteristic
174C2×C8×C13C2×D411yescharacteristic
175C2×C8×C13C2411yescharacteristic
176C13×C4⋊C4—21no—
177C13×C4⋊C4—41no—
178C13×C4⋊C4—41no—
179C13×C4⋊C4—41no—
180C13×C4⋊C4—41no—
181C13×C4⋊C4—41no—
182C13×C4⋊C4D811yescharacteristic
183C13×C22⋊C4—41no—
184C13×C22⋊C4—41no—
185C13×C22⋊C4D811yescharacteristic
186C13×C22⋊C4D811yescharacteristic
187C42×C13—41no—
188C42×C13C2×D411yescharacteristic
189C22⋊Dic13—41no—
190#6103—41no—
191#6103—41no—
192#6103—41no—
193#6103—41no—
194C4⋊Dic13—21no—
195#6101—41no—
196Dic52—21no—
197Dic52—21no—
198D104—21no—
199D104—21no—
200D104—21no—
201D104—42no—
202D104C2×D412yes—
203#6095—42no—
204#6095—81no—
205#6090—21no—
206#6090—21no—
207#6090—21no—
208#6090C2×D412yes—
209C2×#510—131no—
210C22×SD32—131no—
211D4×Q16D1311yescharacteristic, complemented
212#3280—131no—
213#2251—131no—
214#2220—131no—
215#2216—131no—
216#2211—131no—
217#2208—131no—
218#2189—131no—
219#2188—131no—
220#2187—131no—
221#2185—131no—
222#2175—131no—
223#2168—131no—
224C23×C13C2×D811yescharacteristic
225C23×C13C2×D811yescharacteristic
226C22×D13—41no—
227C22×D13—41no—
228C22×D13—82no—
229C22×D13—82no—
230C22×D13—161no—
231Q8×C13—22no—
232Q8×C13—41no—
233Q8×C13—41no—
234Q8×C13—41no—
235Q8×C13—41no—
236Q8×C13—41no—
237Q8×C13—82no—
238Q8×C13—82no—
239D4×C13—22no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—82no—
249C2×C4×C13—82no—
250C2×C4×C13C2×D811yescharacteristic
251C2×C4×C13C2×D811yescharacteristic
252C2×C4×C13C2×D811yescharacteristic
253C2×C4×C13C2×D811yescharacteristic
254C2×C4×C13C22×D411yescharacteristic
255C22⋊D13—82no—
256C22⋊D13—161no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259D52—22no—
260D52—41no—
261D52—41no—
262D52—41no—
263D52—82no—
264C4×D13—82no—
265C4×D13—161no—
266Dic26—41no—
267Dic26—41no—
268C8×C13—21no—
269C8×C13—21no—
270C8×C13—21no—
271C8×C13C22×D411yescharacteristic
272C8×C13H(5,𝔽2)11yescharacteristic
273C2×#135—131no—
274C22×Q16C2×D1311yescharacteristic
275C22×Q16C2×D1311yescharacteristic, complemented
276C22×D8—131no—
277D4×Q8—21no—
278#546—261no—
279#510—264no—
280C2×Q32—131no—
281C2×Q32—131no—
282C2×Q32—131no—
283C2×SD32—131no—
284C2×SD32—131no—
285C2×SD32—131no—
286C2×SD32—131no—
287C2×SD32—131no—
288C2×SD32—131no—
289C2×SD32—264no—
290C2×M32—131no—
291C22×C16—131no—
292#495—131no—
293#494C2×D1311yescharacteristic, complemented
294#467—131no—
295#467C2×D1311yescharacteristic
296#467C2×D1311yescharacteristic, complemented
297#466—131no—
298#463—261no—
299#462—21no—
300#451—21no—
301#451—21no—
302#450—261no—
303#449—261no—
304C4×Q16C2×D1311yescharacteristic, complemented
305C4×D8—131no—
306C8×D4C2×D1311yescharacteristic
307#366—131no—
308C4⋊C16—131no—
309#358—131no—
310#358—131no—
311#358—131no—
312#358—131no—
313#357—131no—
314#357—131no—
315C22⋊C16—131no—
316C22⋊C16—131no—
317#346—131no—
318C22×C13—21no—
319C22×C13—21no—
320C22×C13—21no—
321C22×C13—21no—
322C22×C13—21no—
323C22×C13#51012yescomplemented
324C22×C13C22×D811yescharacteristic
325C2×D13—42no—
326C2×D13—81no—
327C2×D13—81no—
328C2×D13—81no—
329C2×D13—81no—
330C2×D13—81no—
331C2×D13—164no—
332C4×C13—21no—
333C4×C13—21no—
334C4×C13—21no—
335C4×C13—21no—
336C4×C13—42no—
337C4×C13—81no—
338C4×C13—81no—
339C4×C13—81no—
340C4×C13D4211yescharacteristic
341C4×C13#57611yescharacteristic
342Dic13—81no—
343Dic13—81no—
344C2×#41—261no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×D4—261no—
348#135—262no—
349#135—521no—
350C2×Q16—22no—
351C2×Q16—22no—
352C2×Q16—21no—
353C2×Q16—21no—
354C2×Q16—21no—
355C2×Q16—41no—
356C2×Q16—41no—
357C2×Q16—131no—
358C2×Q16C22×D1311yescharacteristic
359C2×Q16C22×D1311yescharacteristic
360C2×Q16C22×D1311yescharacteristic
361C2×SD16—261no—
362C2×SD16—261no—
363C2×D8—131no—
364C2×D8—131no—
365C2×D8—262no—
366C2×D8—261no—
367C2×D8—521no—
368C22×C8C22×D1311yescharacteristic
369C22×C8C22×D1311yescharacteristic
370C4⋊Q8—21no—
371C4⋊Q8—41no—
372#124—261no—
373#123—521no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—41no—
377C22⋊Q8—41no—
378C22⋊Q8—261no—
379#121—261no—
380#121—521no—
381#120—521no—
382C4×Q8—21no—
383C4×D4—41no—
384C4×D4—261no—
385C4×D4C22⋊D1311yescharacteristic
386C2×C22⋊C4—521no—
387Q32—262no—
388Q32—262no—
389Q32—521no—
390SD32—134no—
391SD32—262no—
392SD32—262no—
393SD32—262no—
394SD32—262no—
395SD32—521no—
396M32—262no—
397C2×C16—131no—
398C2×C16—131no—
399C2×C16—131no—
400C2×C16—131no—
401C2×C16—262no—
402#107—131no—
403#107C22×D1311yescharacteristic
404C4⋊C8C22⋊D1311yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—261no—
409#102—261no—
410#102—261no—
411C22⋊C8C22⋊D1311yescharacteristic
412C22⋊C8C22⋊D1311yescharacteristic
413C4×C8C22×D1311yescharacteristic
414C2×C13—22no—
415C2×C13—41no—
416C2×C13D4×D811yescharacteristic
417C2×C13C2×#51011yescharacteristic
418C2×C13#341611yescharacteristic
419D13—82no—
420D13—161no—
421C24—521no—
422#41—522no—
423#41—1041no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—41no—
430C2×Q8—41no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×Q8—261no—
434C2×D4—261no—
435C2×D4—261no—
436C2×D4—522no—
437C2×D4—521no—
438C2×D4—521no—
439C2×D4—521no—
440C2×D4—1041no—
441C2×D4#610411yescharacteristic
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—521no—
445C22×C4—521no—
446C22×C4C2×C22⋊D1311yescharacteristic
447C22×C4C2×C22⋊D1311yescharacteristic
448Q16—21no—
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—42no—
454Q16—42no—
455Q16—81no—
456Q16—261no—
457Q16—261no—
458Q16D4×D1312yes—
459SD16—522no—
460SD16—1041no—
461D8—132no—
462D8—261no—
463D8—261no—
464D8—261no—
465D8—522no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8C2×C22⋊D1311yescharacteristic
471C2×C8C2×C22⋊D1311yescharacteristic
472C2×C8C2×C22⋊D1311yescharacteristic
473C2×C8C23×D1311yescharacteristic
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—261no—
481C4⋊C4—521no—
482C4⋊C4#610411yescharacteristic
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—521no—
486C22⋊C4—521no—
487C22⋊C4—521no—
488C22⋊C4—521no—
489C22⋊C4—521no—
490C22⋊C4#610411yescharacteristic
491C22⋊C4#610411yescharacteristic
492C42—41no—
493C42C2×C22⋊D1311yescharacteristic
494C16—132no—
495C16—261no—
496C16—261no—
497C16—261no—
498C13#3383211yescharacteristic, complemented, Sylow
499C23—521no—
500C23—521no—
501C23—1042no—
502C23—1042no—
503C23—2081no—
504C23C2×#610411yescharacteristic
505C23C2×#610411yescharacteristic
506Q8—22no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—82no—
514Q8—521no—
515Q8—521no—
516D4—22no—
517D4—262no—
518D4—521no—
519D4—521no—
520D4—521no—
521D4—1042no—
522D4—1042no—
523D4—2081no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—1042no—
537C2×C4—2081no—
538C2×C4C2×#610411yescharacteristic
539C2×C4C2×#610411yescharacteristic
540C2×C4C2×#610411yescharacteristic
541C2×C4C2×#610411yescharacteristic
542C2×C4C22×C22⋊D1311yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D1311yescharacteristic
547C8#8825111yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—522no—
554C22—1041no—
555C22—1041no—
556C22—1041no—
557C22—1041no—
558C22—1041no—
559C22—2084no—
560C22#1171884012yescomplemented
561C22C22×#610411yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—1041no—
571C4—1041no—
572C4D4×C22⋊D1311yescharacteristic
573C4#1171954011yescharacteristic
574C2—22no—
575C2—41no—
576C2—1042no—
577C2—2081no—
578C2D4×#610411yescharacteristic
579C2C2×#1171884011yescharacteristic
580C2#1312798011yescharacteristic
subgroupG/H
centerC22C22×#6104
commutator, cosocleC2×C8×C13C24
FrattiniC2×C8C23×D13
FittingD4×C13×Q16C2
socleC22×C13C22×D8
prime radical#13126523C2
hypercenterD4×Q16D13
hypocenterC13#33832
LCSG ▷ C2×C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#11718840C2semidirect
2. C22×#88068C2semidirect
4. #13117105C2semidirect
5. #13116951C2semidirect
6. #13116925C2semidirect
7. #13116898C2semidirect
8. #13116878C2semidirect
9. #13116774C2semidirect
10. #13116770C2semidirect
11. #13116764C2semidirect
12. #13116753C2semidirect
13. #13116698C2semidirect
14. #13116662C2semidirect
15. D4×C13×Q16C2semidirect
17. C22×C13×Q16C22semidirect
22. C13×#494C22semidirect
24. C13×#467C22semidirect
28. C4×C13×Q16C22semidirect
30. C2×#88069C22semidirect
31. C2×#88069C22semidirect
32. C2×#88069C22semidirect
43. C2×#88052C22semidirect
50. #11718156C22semidirect
51. #11718156C22semidirect
52. #11718156C22semidirect
54. C22⋊#6090C22semidirect
55. #11718084C22semidirect
58. #11718081C22semidirect
59. C4⋊#6090C22semidirect
60. #11718053C22semidirect
120. #88068D4semidirect
211. D4×Q16D13semidirect
275. C22×Q16C2×D13semidirect
293. #494C2×D13semidirect
296. #467C2×D13semidirect
304. C4×Q16C2×D13semidirect
323. C22×C13#510semidirect
498. C13#33832semidirect
560. C22#11718840semidirect

Maximal subgroups

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

subgroupindex
C2×#117188402
C22×#880682
#131265232
#131171052
#131169512
#131169252
#131168982
#131168782
#131167742
#131167702
#131167642
#131167532
#131166982
#131166622
D4×C13×Q162
#3383213