Finite Group Databasebeta

← #15848320

Subgroups — #15848320

Order 3328.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125774C211yesmaximal, characteristic, complemented
2#13124316C211yesmaximal, characteristic
3#13123947C211yesmaximal, characteristic, complemented
4#13114254C212yesmaximal, complemented
5#13114247C212yesmaximal, complemented
6#13114186C212yesmaximal, complemented
7#13114179C212yesmaximal, complemented
8#13113606C211yesmaximal, characteristic, complemented
9#13113526C211yesmaximal, characteristic, complemented
10C2×#11718077C211yesmaximal, characteristic, complemented
11C13×#3076C211yesmaximal, characteristic, complemented
12C13×#533C2211yescharacteristic
13C4×C13×#41C2211yescharacteristic
14C2×#88203—21no—
15C2×#88128C2211yescharacteristic
16C13×#479—21no—
17C13×#478—21no—
18C13×#477C2211yescharacteristic, complemented
19C13×#476C2211yescharacteristic, complemented
20C13×#475C2211yescharacteristic, complemented
21C13×#474C2211yescharacteristic, complemented
22C13×#427—21no—
23C2×C13×C4⋊C8C2211yescharacteristic
24C13×#416—21no—
25#11718749C2212yes—
26#11718746C2212yes—
27#11718707—22no—
28#11718688—21no—
29#11718688—21no—
30#11718688—21no—
31#11718688C2212yes—
32Q8×#1055C2212yescomplemented
33#11718676—21no—
34#11718676C2212yes—
35D4×#1055—22no—
36D4×#1055C2212yescomplemented
37#11718662—21no—
38#11718586—21no—
39#11718586C2211yescharacteristic
40#11718585C2211yescharacteristic
41C2×C4⋊#1055—21no—
42C2×C4⋊#1055C2211yescharacteristic
43#11718139C2212yescomplemented
44#11718138C2212yescomplemented
45#11718077C2212yes—
46#11718077C2212yes—
47#11718072C2212yescomplemented
48#11718072C2212yescomplemented
49C2×C13×#41D411yescharacteristic
50C22×D52—21no—
51C22×D52—21no—
52C22×D52—21no—
53C22×C8×C13—21no—
54C13×C4⋊Q8C2311yescharacteristic
55C13×C4⋊Q8C2311yescharacteristic
56C13×#125—21no—
57C13×C22⋊Q8—21no—
58C13×C22⋊Q8—21no—
59C4×Q8×C13—21no—
60C4×Q8×C13—21no—
61C4×Q8×C13D411yescharacteristic
62C4×Q8×C13C2311yescharacteristic
63C4×D4×C13—21no—
64C4×D4×C13—21no—
65C4×D4×C13D411yescharacteristic
66C4×D4×C13C2311yescharacteristic
67C13×#117—21no—
68C13×#117—21no—
69C13×#117D411yescharacteristic
70C2×C42×C13—21no—
71C2×C42×C13C2311yescharacteristic
72#88203—44no—
73C2×#6106—22no—
74C2×#6106—22no—
75C2×#6106—22no—
76C2×#6104—22no—
77C2×#6099—21no—
78C22×#1055—21no—
79C22×#1055—22no—
80#88128—21no—
81#88128—21no—
82#88128—21no—
83#88128C2312yes—
84C13×#107—21no—
85C13×#107—21no—
86C13×#106—21no—
87C13×#106—21no—
88C13×C4⋊C8—21no—
89C13×C4⋊C8C2311yescharacteristic
90C13×C4⋊C8C2311yescharacteristic
91C13×#103—21no—
92C13×#103—21no—
93C13×#102—21no—
94C13×#102—21no—
95C22⋊#1055—22no—
96C22⋊#1055D412yes—
97#88049—22no—
98#88049—22no—
99#88049—22no—
100#88049—22no—
101C4⋊#1055—21no—
102C4⋊#1055—21no—
103C4⋊#1055—22no—
104C4⋊#1055D411yescharacteristic
105C4⋊#1055D411yescharacteristic
106C4⋊#1055C2312yes—
107C4⋊#1055C2312yes—
108C4×#1055—22no—
109C4×#1055C2312yes—
110#19409—131nomaximal, Sylow
111C23×D13—41no—
112C13×#41—22no—
113C13×#41SD1614yescomplemented
114C2×Q8×C13—21no—
115C2×Q8×C13—21no—
116C2×Q8×C13C2×D411yescharacteristic
117C2×D4×C13—21no—
118C2×D4×C13C2×D411yescharacteristic
119C22×C4×C13—21no—
120C22×C4×C13—21no—
121C22×C4×C13—21no—
122C22×C4×C13C2×D411yescharacteristic
123C22×C4×C13C2×D411yescharacteristic
124C22×C4×C13#4111yescharacteristic
125C2×D52—22no—
126C2×D52—22no—
127C2×D52—22no—
128C2×D52—42no—
129C2×D52—42no—
130C2×D52—41no—
131C2×D52—41no—
132C2×D52—41no—
133C2×D52—81no—
134C2×D52—81no—
135C2×D52—82no—
136C2×C8×C13—21no—
137C2×C8×C13—21no—
138C2×C8×C13—42no—
139C13×C4⋊C4—21no—
140C13×C4⋊C4—21no—
141C13×C4⋊C4—21no—
142C13×C4⋊C4—21no—
143C13×C4⋊C4—21no—
144C13×C4⋊C4—21no—
145C13×C4⋊C4—41no—
146C13×C4⋊C4—41no—
147C13×C4⋊C4C2×D411yescharacteristic
148C13×C4⋊C4C2×D411yescharacteristic
149C13×C22⋊C4—21no—
150C13×C22⋊C4—21no—
151C13×C22⋊C4—21no—
152C13×C22⋊C4—41no—
153C42×C13—21no—
154C42×C13—21no—
155C42×C13—21no—
156C42×C13—41no—
157C42×C13C2×D411yescharacteristic
158C42×C13C2×D411yescharacteristic
159C42×C13#4111yescharacteristic
160C42×C13C2411yescharacteristic
161#6106—44no—
162#6106—82no—
163#6106—82no—
164#6104—44no—
165#6099—42no—
166C2×#1055—22no—
167C2×#1055—22no—
168C2×#1055—82no—
169C2×#1055—81no—
170C2×#1055C2×D412yes—
171C2×#1055C2×D412yes—
172#3208—131no—
173#3076D1311yescharacteristic, complemented
174#3038—131no—
175#2988—131no—
176#1678—132no—
177#1677—132no—
178#1666—132no—
179#1665—132no—
180#1570—131no—
181#1550—131no—
182C2×#331—131no—
183C23×C13—21no—
184C23×C13#12111yescharacteristic
185C22×D13—42no—
186C22×D13—82no—
187C22×D13—82no—
188C22×D13—82no—
189Q8×C13—21no—
190Q8×C13—41no—
191Q8×C13—41no—
192Q8×C13—81no—
193Q8×C13C2×SD1612yes—
194D4×C13—24no—
195D4×C13—21no—
196D4×C13C2×SD1612yes—
197C2×C4×C13—21no—
198C2×C4×C13—21no—
199C2×C4×C13—21no—
200C2×C4×C13—24no—
201C2×C4×C13—21no—
202C2×C4×C13—21no—
203C2×C4×C13—41no—
204C2×C4×C13—41no—
205C2×C4×C13—41no—
206C2×C4×C13—41no—
207C2×C4×C13—41no—
208C2×C4×C13—81no—
209C2×C4×C13#12111yescharacteristic
210C2×C4×C13#12111yescharacteristic
211C2×C4×C13#12111yescharacteristic
212C2×C4×C13C2×SD1612yes—
213C2×C4×C13#13512yes—
214C2×C4×C13C22×D411yescharacteristic
215C2×C4×C13C22×D411yescharacteristic
216C2×C4×C13C2×#4111yescharacteristic
217D52—42no—
218D52—42no—
219D52—42no—
220D52—42no—
221D52—82no—
222D52—82no—
223D52—82no—
224D52—82no—
225C8×C13—41no—
226C8×C13—41no—
227#1055—22no—
228#1055—22no—
229#1055—42no—
230#1055—42no—
231C2×AGL(1,ℤ/8)—261no—
232#533C2×D1311yescharacteristic
233C2×#127—131no—
234C4×#41C2×D1311yescharacteristic
235#492—132no—
236#486—132no—
237#479—21no—
238#478—21no—
239#477C2×D1311yescharacteristic, complemented
240#476C2×D1311yescharacteristic, complemented
241#475C2×D1311yescharacteristic, complemented
242#474C2×D1311yescharacteristic, complemented
243#460—132no—
244#460—261no—
245#460—261no—
246#460—261no—
247#459—132no—
248#459—261no—
249#450—262no—
250C8×Q8—132no—
251C8×D4—132no—
252C8×D4—262no—
253#431—131no—
254#431—261no—
255#427—21no—
256#423—131no—
257C2×C4⋊C8—131no—
258C2×C4⋊C8—261no—
259C2×C4⋊C8C2×D1311yescharacteristic
260#418—261no—
261#416—21no—
262#334—132no—
263#334—132no—
264#333—132no—
265#332—132no—
266#331—132no—
267#331—132no—
268C22×C13—21no—
269C22×C13—21no—
270C22×C13—22no—
271C22×C13—41no—
272C22×C13—41no—
273C22×C13#46512yes—
274C22×C13C2×#12111yescharacteristic
275C2×D13—82no—
276C2×D13—82no—
277C2×D13—82no—
278C2×D13—164no—
279C4×C13—21no—
280C4×C13—22no—
281C4×C13—21no—
282C4×C13—21no—
283C4×C13—41no—
284C4×C13—41no—
285C4×C13—81no—
286C4×C13—81no—
287C4×C13#46512yes—
288C4×C13C22×SD1611yescharacteristic
289C4×C13C2×#13511yescharacteristic
290C4×C13#57611yescharacteristic
291C4×C13#57711yescharacteristic
292C2×#41C22⋊D1311yescharacteristic
293C22×D4—261no—
294C22×D4—261no—
295C22×D4—261no—
296AGL(1,ℤ/8)—524no—
297C2×SD16—262no—
298C2×SD16—262no—
299C2×SD16—262no—
300C2×D8—262no—
301C2×M16—261no—
302C22×C8—21no—
303C22×C8—262no—
304C22×C8—261no—
305C4⋊Q8C22×D1311yescharacteristic
306C4⋊Q8C22×D1311yescharacteristic
307#127—132no—
308#127—261no—
309#127—261no—
310#127—261no—
311#125—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8C22⋊D1311yescharacteristic
317C4×Q8C22×D1311yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4C22⋊D1311yescharacteristic
321C4×D4C22×D1311yescharacteristic
322#117—21no—
323#117—21no—
324#117C22⋊D1311yescharacteristic
325C2×C42—21no—
326C2×C42C22×D1311yescharacteristic
327#107—21no—
328#107—21no—
329#106—21no—
330#106—21no—
331C4⋊C8—21no—
332C4⋊C8—131no—
333C4⋊C8—131no—
334C4⋊C8—132no—
335C4⋊C8—132no—
336C4⋊C8—262no—
337C4⋊C8—261no—
338C4⋊C8—261no—
339C4⋊C8C22×D1311yescharacteristic
340C4⋊C8C22×D1311yescharacteristic
341#103—21no—
342#103—21no—
343#102—21no—
344#102—21no—
345#102—262no—
346#102—262no—
347#102—262no—
348#102—262no—
349C22⋊C8—132no—
350C22⋊C8—262no—
351C4×C8—132no—
352C4×C8—262no—
353C2×C13—22no—
354C2×C13—42no—
355C2×C13#304511yescharacteristic
356C2×C13C2×#46511yescharacteristic
357C2×C13#306811yescharacteristic
358D13—162no—
359C24—521no—
360#41—22no—
361#41#610614yescomplemented
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8C2×C22⋊D1311yescharacteristic
365C2×D4—21no—
366C2×D4—262no—
367C2×D4—262no—
368C2×D4—262no—
369C2×D4—521no—
370C2×D4—521no—
371C2×D4—521no—
372C2×D4—522no—
373C2×D4—522no—
374C2×D4—1041no—
375C2×D4—1041no—
376C2×D4—1042no—
377C2×D4C2×C22⋊D1311yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4D4×D1311yescharacteristic
382C22×C4#612911yescharacteristic
383C22×C4C2×C22⋊D1311yescharacteristic
384SD16—524no—
385SD16—1042no—
386SD16—1042no—
387D8—524no—
388M16—522no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—42no—
392C2×C8—132no—
393C2×C8—132no—
394C2×C8—262no—
395C2×C8—262no—
396C2×C8—1042no—
397C2×C8—1041no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4C2×C22⋊D1311yescharacteristic
407C4⋊C4C2×C22⋊D1311yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—41no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—41no—
416C42D4×D1311yescharacteristic
417C42#612911yescharacteristic
418C42C2×C22⋊D1311yescharacteristic
419C42C23×D1311yescharacteristic
420C13#1940911yescharacteristic, complemented, Sylow
421C23—21no—
422C23—522no—
423C23—1042no—
424C23—1042no—
425C23—1042no—
426C23#8819311yescharacteristic
427Q8—21no—
428Q8—41no—
429Q8—41no—
430Q8—81no—
431Q8C2×#610612yes—
432D4—24no—
433D4—21no—
434D4—522no—
435D4—522no—
436D4—522no—
437D4—522no—
438D4—1042no—
439D4—1042no—
440D4—1042no—
441D4—1042no—
442D4C2×#610612yes—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4#8819311yescharacteristic
456C2×C4#8819311yescharacteristic
457C2×C4#8819311yescharacteristic
458C2×C4C2×#610612yes—
459C2×C4#8820412yes—
460C2×C4C2×D4×D1311yescharacteristic
461C2×C4C2×#612911yescharacteristic
462C2×C4C22×C22⋊D1311yescharacteristic
463C8—41no—
464C8—41no—
465C8—262no—
466C8—262no—
467C8—522no—
468C8—522no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—1042no—
475C22—1042no—
476C22—1042no—
477C22—2084no—
478C22#1171870912yes—
479C22C2×#8819311yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—81no—
488C4#1171870912yes—
489C4#1171947411yescharacteristic
490C4#1171947511yescharacteristic
491C4C22×#610611yescharacteristic
492C4C2×#8820411yescharacteristic
493C2—22no—
494C2—42no—
495C2—2082no—
496C2#1312437311yescharacteristic
497C2C2×#1171870911yescharacteristic
498C2#1312456611yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3076C2
socleC22×C13C2×#121
prime radical#13124316C2
hypercenter#3076D13
hypocenterC13#19409
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #13125774C2semidirect
3. #13123947C2semidirect
4. #13114254C2semidirect
5. #13114247C2semidirect
6. #13114186C2semidirect
7. #13114179C2semidirect
8. #13113606C2semidirect
9. #13113526C2semidirect
10. C2×#11718077C2semidirect
11. C13×#3076C2semidirect
18. C13×#477C22semidirect
19. C13×#476C22semidirect
20. C13×#475C22semidirect
21. C13×#474C22semidirect
32. Q8×#1055C22semidirect
36. D4×#1055C22semidirect
43. #11718139C22semidirect
44. #11718138C22semidirect
47. #11718072C22semidirect
48. #11718072C22semidirect
113. C13×#41SD16semidirect
173. #3076D13semidirect
239. #477C2×D13semidirect
240. #476C2×D13semidirect
241. #475C2×D13semidirect
242. #474C2×D13semidirect
361. #41#6106semidirect
420. C13#19409semidirect

Maximal subgroups

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

subgroupindex
#131257742
#131243162
#131239472
#131142542
#131142472
#131141862
#131141792
#131136062
#131135262
C2×#117180772
C13×#30762
#1940913