Finite Group Databasebeta

← #15845782

Subgroups — #15845782

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125293C211yesmaximal, characteristic
2#13124867C211yesmaximal, characteristic
3Q8×#6099C211yesmaximal, characteristic, complemented
4#13114328C212yesmaximal, complemented
5#13114310C212yesmaximal, complemented
6#13114204C212yesmaximal, complemented
7#13114186C212yesmaximal, complemented
8#13113586C211yesmaximal, characteristic, complemented
9#13113511C211yesmaximal, characteristic
10#13113508C211yesmaximal, characteristic, complemented
11C2×C13×#475C211yesmaximal, characteristic, complemented
12C2×C13×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8×C13C2211yescharacteristic
14C2×#88196—21no—
15C2×#88196—21no—
16#11719182C2211yescharacteristic
17C13×#475—21no—
18C13×#475—21no—
19C13×#475C2212yes—
20C13×#475C2212yes—
21C2×C13×#106—21no—
22C2×C13×C4⋊C8C2211yescharacteristic
23C2×C13×#103—21no—
24#11718888—21no—
25#11718753C2212yes—
26#11718747—22no—
27#11718746C2212yes—
28#11718688C2211yescharacteristic
29#11718688C2211yescharacteristic
30C4×#6106—21no—
31C4×#6106—21no—
32#11718685—22no—
33#11718685C2212yes—
34Q8×#1055C2212yes—
35#11718681C2211yescharacteristic
36#11718681C2211yescharacteristic
37#11718629—21no—
38#11718586C2211yescharacteristic
39#11718585—21no—
40#11718585C2211yescharacteristic
41C4×#6099—21no—
42C4×#6099C2211yescharacteristic
43#11718139C2212yes—
44#11718139C2212yes—
45#11718078C2212yes—
46#11718077C2212yes—
47#11718072C2212yes—
48#11718071C2212yes—
49C22×Q8×C13—21no—
50C22×Q8×C13D411yescharacteristic
51C2×#6127—21no—
52C22×C8×C13—21no—
53C13×C4⋊Q8—21no—
54C13×C4⋊Q8—21no—
55C13×C4⋊Q8—21no—
56C13×C4⋊Q8C2312yes—
57C4×Q8×C13—21no—
58C4×Q8×C13—21no—
59C4×Q8×C13D411yescharacteristic
60C4×Q8×C13D411yescharacteristic
61C4×Q8×C13C2311yescharacteristic
62C4×Q8×C13C2311yescharacteristic
63C2×C13×C4⋊C4—21no—
64C2×C13×C4⋊C4—21no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4D411yescharacteristic
67C2×C42×C13—21no—
68C2×C42×C13C2311yescharacteristic
69C2×#6107—22no—
70C2×#6107—22no—
71#88196—44no—
72#88196—44no—
73C2×#6106—21no—
74C2×#6106—21no—
75C2×#6106—22no—
76C2×#6106—22no—
77#88184—21no—
78#88184—21no—
79C2×#6099—21no—
80C2×#6099D411yescharacteristic
81C2×#6099D411yescharacteristic
82#88129—21no—
83#88128C2311yescharacteristic
84C4×D52—21no—
85C4×D52—21no—
86#88123C2311yescharacteristic
87C13×#106—22no—
88C13×#106—22no—
89C13×C4⋊C8—21no—
90C13×C4⋊C8C2312yes—
91C13×#103—22no—
92C13×#103—22no—
93#88075—21no—
94#88075—21no—
95C22⋊#1055—21no—
96#88049—21no—
97#88049—21no—
98#88048—21no—
99#88048—21no—
100C4⋊#1055—21no—
101C4⋊#1055—22no—
102C4⋊#1055—22no—
103C4⋊#1055C2311yescharacteristic
104C4⋊#1055C2311yescharacteristic
105C4⋊#1055C2312yes—
106#88043—22no—
107#88043D412yes—
108C4×#1055—21no—
109C4×#1055—22no—
110C4×#1055C2312yes—
111#19171—131nomaximal, Sylow
112C2×Q8×C13—21no—
113C2×Q8×C13—22no—
114C2×Q8×C13—22no—
115C2×Q8×C13—41no—
116C2×Q8×C13—41no—
117C2×Q8×C13—42no—
118C2×Q8×C13—42no—
119C2×Q8×C13C2×D411yescharacteristic
120C2×Q8×C13C2×D411yescharacteristic
121C22×C4×C13—21no—
122C22×C4×C13—21no—
123C22×C4×C13—41no—
124C22×C4×C13C2×D411yescharacteristic
125C22×C4×C13C2×D411yescharacteristic
126C22×C4×C13#4111yescharacteristic
127C2×C22⋊D13—41no—
128#6127—42no—
129#6127—42no—
130C2×D52—21no—
131C2×D52—21no—
132C2×D52—21no—
133C2×C4×D13—41no—
134C2×Dic26—21no—
135C2×C8×C13—22no—
136C2×C8×C13—42no—
137C13×C4⋊C4—21no—
138C13×C4⋊C4—21no—
139C13×C4⋊C4—21no—
140C13×C4⋊C4—21no—
141C13×C4⋊C4—21no—
142C13×C4⋊C4—22no—
143C13×C4⋊C4—22no—
144C13×C4⋊C4—41no—
145C13×C4⋊C4—41no—
146C13×C4⋊C4C2×D411yescharacteristic
147C13×C4⋊C4C2×D411yescharacteristic
148C42×C13—21no—
149C42×C13—21no—
150C42×C13—21no—
151C42×C13—21no—
152C42×C13C2×D411yescharacteristic
153C42×C13C2×D411yescharacteristic
154C42×C13#4111yescharacteristic
155C42×C13C2411yescharacteristic
156#6107—44no—
157#6107—44no—
158#6106—44no—
159#6106—42no—
160#6106—42no—
161#6106—44no—
162#6103—41no—
163#6103—41no—
164C4⋊Dic13—21no—
165C4⋊Dic13—21no—
166#6099—22no—
167#6099—42no—
168#6099SD1614yescomplemented
169C2×#1055—21no—
170C2×#1055—21no—
171C2×#1055—22no—
172C2×#1055C2×D412yes—
173C2×#1055C2×D412yes—
174#3152—131no—
175#3103—131no—
176C2×#475D1311yescharacteristic, complemented
177Q8×M16—131no—
178#1690—132no—
179#1686—132no—
180#1670—132no—
181#1666—132no—
182#1565—131no—
183#1547—131no—
184#1546—131no—
185C23×C13#12111yescharacteristic
186C22×D13—41no—
187Q8×C13—21no—
188Q8×C13—21no—
189Q8×C13—21no—
190Q8×C13—42no—
191Q8×C13—42no—
192Q8×C13—42no—
193Q8×C13—82no—
194Q8×C13#13712yescomplemented
195C2×C4×C13—21no—
196C2×C4×C13—21no—
197C2×C4×C13—21no—
198C2×C4×C13—21no—
199C2×C4×C13—21no—
200C2×C4×C13—21no—
201C2×C4×C13—41no—
202C2×C4×C13—42no—
203C2×C4×C13—42no—
204C2×C4×C13—81no—
205C2×C4×C13—82no—
206C2×C4×C13#12111yescharacteristic
207C2×C4×C13#12111yescharacteristic
208C2×C4×C13#12111yescharacteristic
209C2×C4×C13C2×SD1612yes—
210C2×C4×C13AGL(1,ℤ/8)12yes—
211C2×C4×C13C22×D411yescharacteristic
212C2×C4×C13C22×D411yescharacteristic
213C2×C4×C13C2×#4111yescharacteristic
214C22⋊D13—82no—
215C2×Dic13—41no—
216D52—41no—
217D52—41no—
218D52—41no—
219D52—41no—
220D52—81no—
221C4×D13—82no—
222Dic26—41no—
223Dic26—41no—
224C8×C13—42no—
225#1055—22no—
226#1055—42no—
227#1055—41no—
228#1055—41no—
229#1055C2×SD1614yes—
230C2×#137—261no—
231C2×#137—261no—
232#532—131no—
233C2×C4⋊Q8C2×D1311yescharacteristic
234C2×C4×Q8C2×D1311yescharacteristic
235#497—262no—
236#494—132no—
237#492—132no—
238#481—261no—
239#475—21no—
240#475—21no—
241#475—131no—
242#475—131no—
243#475C2×D1312yes—
244#475C2×D1312yes—
245#460—131no—
246#460—131no—
247#448—261no—
248#446—132no—
249#446—262no—
250C8×Q8—132no—
251C4×SD16—261no—
252C4×SD16—261no—
253#431—131no—
254C2×#106—21no—
255#423—131no—
256#423—261no—
257C2×C4⋊C8C2×D1311yescharacteristic
258C2×#103—21no—
259C4×M16—131no—
260C4×M16—261no—
261#335—132no—
262#334—132no—
263#332—132no—
264#332—132no—
265#332—132no—
266#331—132no—
267C22×C13—21no—
268C22×C13—21no—
269C22×C13#46012yes—
270C22×C13C2×#12111yescharacteristic
271C2×D13—81no—
272C2×D13—81no—
273C2×D13—81no—
274C4×C13—21no—
275C4×C13—22no—
276C4×C13—21no—
277C4×C13—21no—
278C4×C13—41no—
279C4×C13—41no—
280C4×C13—41no—
281C4×C13—82no—
282C4×C13#46012yes—
283C4×C13C22×SD1611yescharacteristic
284C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C13C2×#13711yescharacteristic
286C4×C13#57511yescharacteristic
287Dic13—81no—
288C2×#41—261no—
289C22×Q8—21no—
290C22×Q8C22⋊D1311yescharacteristic
291#137—524no—
292#137—524no—
293C2×Q16—262no—
294C2×Q16—262no—
295C2×SD16—262no—
296C2×SD16—262no—
297C2×SD16—261no—
298C2×SD16—261no—
299C2×M16—131no—
300C2×M16—131no—
301C2×M16—261no—
302C22×C8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—131no—
307C4⋊Q8C22×D1312yes—
308#127—131no—
309#124—261no—
310#121—261no—
311#121—261no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8C22⋊D1311yescharacteristic
315C4×Q8C22⋊D1311yescharacteristic
316C4×Q8C22×D1311yescharacteristic
317C4×Q8C22×D1311yescharacteristic
318C4×D4—261no—
319C4×D4—261no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4C22⋊D1311yescharacteristic
324C2×C42—21no—
325C2×C42C22×D1311yescharacteristic
326#106—22no—
327#106—22no—
328#106—261no—
329#106—261no—
330C4⋊C8—21no—
331C4⋊C8—132no—
332C4⋊C8—131no—
333C4⋊C8—131no—
334C4⋊C8—262no—
335C4⋊C8—262no—
336C4⋊C8—261no—
337C4⋊C8C22×D1312yes—
338#103—22no—
339#103—22no—
340#103—261no—
341#103—261no—
342#102—261no—
343#102—261no—
344C22⋊C8—261no—
345#97—132no—
346#97—262no—
347C4×C8—132no—
348C4×C8—262no—
349C4×C8—261no—
350C2×C13—22no—
351C2×C13C2×#46011yescharacteristic
352C2×C13#305611yescharacteristic
353C2×C13#306911yescharacteristic
354D13—161no—
355#41—522no—
356#41—522no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—261no—
365C2×Q8C2×C22⋊D1311yescharacteristic
366C2×Q8C2×C22⋊D1311yescharacteristic
367C2×D4—261no—
368C2×D4—261no—
369C2×D4—261no—
370C2×D4—521no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—521no—
375C22×C4D4×D1311yescharacteristic
376C22×C4#612911yescharacteristic
377C22×C4C2×C22⋊D1311yescharacteristic
378Q16—524no—
379Q16—524no—
380SD16—524no—
381SD16—524no—
382SD16—522no—
383SD16—522no—
384M16—134no—
385M16—262no—
386M16—522no—
387C2×C8—22no—
388C2×C8—42no—
389C2×C8—132no—
390C2×C8—132no—
391C2×C8—262no—
392C2×C8—261no—
393C2×C8—261no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—261no—
404C4⋊C4—261no—
405C4⋊C4C2×C22⋊D1311yescharacteristic
406C4⋊C4C2×C22⋊D1311yescharacteristic
407C22⋊C4—521no—
408C22⋊C4—521no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42D4×D1311yescharacteristic
414C42#612911yescharacteristic
415C42C2×C22⋊D1311yescharacteristic
416C42C23×D1311yescharacteristic
417C13#1917111yescharacteristic, complemented, Sylow
418C23—521no—
419C23#8819311yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—82no—
427Q8—521no—
428Q8—521no—
429Q8#8819612yescomplemented
430D4—521no—
431D4—521no—
432D4—521no—
433D4—521no—
434D4—1041no—
435D4—1042no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—521no—
448C2×C4—1042no—
449C2×C4SD16×D1312yes—
450C2×C4#8816812yes—
451C2×C4#8819311yescharacteristic
452C2×C4#8819311yescharacteristic
453C2×C4#8819311yescharacteristic
454C2×C4C2×D4×D1311yescharacteristic
455C2×C4C2×#612911yescharacteristic
456C2×C4C22×C22⋊D1311yescharacteristic
457C8—42no—
458C8—134no—
459C8—262no—
460C8—522no—
461C8—521no—
462C8—521no—
463C22—21no—
464C22—21no—
465C22—1041no—
466C22—1041no—
467C22—1041no—
468C22#1171882412yes—
469C22C2×#8819311yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—1041no—
479C4#1171882412yes—
480C4C2×SD16×D1311yescharacteristic
481C4C2×#8816811yescharacteristic
482C4C2×#8819611yescharacteristic
483C4#1171952711yescharacteristic
484C2—22no—
485C2—2081no—
486C2C2×#1171882411yescharacteristic
487C2#1312446211yescharacteristic
488C2#1312457011yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×C13×#475C2
socleC22×C13C2×#121
prime radical#11719182C22
hypercenterC2×#475D13
hypocenterC13#19171
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. Q8×#6099C2semidirect
4. #13114328C2semidirect
5. #13114310C2semidirect
6. #13114204C2semidirect
7. #13114186C2semidirect
8. #13113586C2semidirect
10. #13113508C2semidirect
11. C2×C13×#475C2semidirect
168. #6099SD16semidirect
176. C2×#475D13semidirect
194. Q8×C13#137semidirect
417. C13#19171semidirect
429. Q8#88196semidirect

Maximal subgroups

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

subgroupindex
#131252932
#131248672
Q8×#60992
#131143282
#131143102
#131142042
#131141862
#131135862
#131135112
#131135082
C2×C13×#4752
#1917113