Finite Group Databasebeta

← #15845821

Subgroups — #15845821

Order 3328.

680 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#13125212C211yesmaximal, characteristic
2C2×#11718688C211yesmaximal, characteristic
3Q8×#6099C211yesmaximal, characteristic, complemented
4#13114406C212yesmaximal, complemented
5#13114378C212yesmaximal, complemented
6#13114282C212yesmaximal, complemented
7#13114254C212yesmaximal, complemented
8#13113589C211yesmaximal, characteristic, complemented
9#13113526C211yesmaximal, characteristic, complemented
10#13113503C211yesmaximal, characteristic
11C13×#3104C211yesmaximal, characteristic, complemented
12C13×#533C2211yescharacteristic
13C2×C4×Q8×C13C2211yescharacteristic
14C22×#6106—21no—
15C2×#88128C2211yescharacteristic
16C13×#484—21no—
17C13×#477C2211yescharacteristic, complemented
18C13×#477C2211yescharacteristic, complemented
19C13×#462C2211yescharacteristic, complemented
20C13×#462C2211yescharacteristic, complemented
21C13×#451—21no—
22C4×C13×Q16—21no—
23C4×C13×Q16—21no—
24C13×#431C2211yescharacteristic
25#11718749C2212yes—
26#11718749C2212yes—
27#11718728—22no—
28#11718688—21no—
29#11718688—21no—
30#11718688C2212yes—
31#11718688C2212yes—
32#11718685—22no—
33#11718685C2212yescomplemented
34Q8×#1055C2212yescomplemented
35#11718662—21no—
36#11718662—21no—
37C2×#88049—21no—
38#11718585—21no—
39#11718585C2211yescharacteristic
40C2×C4⋊#1055C2211yescharacteristic
41C4×#6099—21no—
42C4×#6099C2211yescharacteristic
43#11718139C2212yescomplemented
44#11718138C2212yescomplemented
45#11718077C2212yes—
46#11718077C2212yes—
47#11718072C2212yescomplemented
48#11718071C2212yescomplemented
49C22×Q8×C13D411yescharacteristic
50C22×D52—21no—
51C22×D52—21no—
52C22×D52—21no—
53C2×C13×Q16—21no—
54C2×C13×Q16—21no—
55C13×C4⋊Q8C2311yescharacteristic
56C13×C4⋊Q8C2311yescharacteristic
57C13×#125—21no—
58C13×C22⋊Q8—21no—
59C13×C22⋊Q8—21no—
60C4×Q8×C13—21no—
61C4×Q8×C13—21no—
62C4×Q8×C13—21no—
63C4×Q8×C13—21no—
64C4×Q8×C13D411yescharacteristic
65C4×Q8×C13D411yescharacteristic
66C4×Q8×C13C2311yescharacteristic
67C4×Q8×C13C2311yescharacteristic
68C13×#117—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4D411yescharacteristic
71C2×C42×C13—21no—
72C2×C42×C13C2311yescharacteristic
73C2×#6106—22no—
74C2×#6106—22no—
75C2×#6106—44no—
76C2×#6099—21no—
77C2×#6099D411yescharacteristic
78C2×#6099D411yescharacteristic
79C22×#1055—21no—
80#88128—21no—
81#88128—21no—
82#88128—21no—
83#88128C2312yes—
84C13×#107—21no—
85C13×#107—21no—
86C13×C4⋊C8C2311yescharacteristic
87C13×C4⋊C8C2311yescharacteristic
88C13×#103—21no—
89C13×#103—21no—
90C13×#103—21no—
91C13×#103—21no—
92C13×C22⋊C8—21no—
93C4×C8×C13—21no—
94#88049—22no—
95#88049—22no—
96#88049—22no—
97#88049—22no—
98#88049—22no—
99#88049—22no—
100C4⋊#1055—21no—
101C4⋊#1055—21no—
102C4⋊#1055—22no—
103C4⋊#1055—22no—
104C4⋊#1055C2312yes—
105C4⋊#1055C2312yes—
106#88043—22no—
107#88043D412yes—
108C4×#1055—22no—
109C4×#1055C2312yes—
110#19174—131nomaximal, Sylow
111C23×D13—41no—
112C2×Q8×C13—21no—
113C2×Q8×C13—22no—
114C2×Q8×C13—21no—
115C2×Q8×C13—21no—
116C2×Q8×C13—41no—
117C2×Q8×C13C2×D411yescharacteristic
118C2×Q8×C13C2×D411yescharacteristic
119C22×C4×C13—21no—
120C22×C4×C13—21no—
121C22×C4×C13C2×D411yescharacteristic
122C22×C4×C13C2×D411yescharacteristic
123C22×C4×C13#4111yescharacteristic
124C2×D52—22no—
125C2×D52—22no—
126C2×D52—22no—
127C2×D52—42no—
128C2×D52—42no—
129C2×D52—42no—
130C2×D52—42no—
131C2×D52—41no—
132C2×D52—41no—
133C2×D52—41no—
134C2×D52—82no—
135C13×Q16—42no—
136C13×Q16—42no—
137C2×C8×C13—21no—
138C2×C8×C13—21no—
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—21no—
146C13×C4⋊C4—21no—
147C13×C4⋊C4—21no—
148C13×C4⋊C4—41no—
149C13×C4⋊C4—41no—
150C13×C4⋊C4C2×D411yescharacteristic
151C13×C4⋊C4C2×D411yescharacteristic
152C13×C22⋊C4—41no—
153C42×C13—21no—
154C42×C13—21no—
155C42×C13—21no—
156C42×C13—21no—
157C42×C13—41no—
158C42×C13C2×D411yescharacteristic
159C42×C13C2×D411yescharacteristic
160C42×C13#4111yescharacteristic
161C42×C13C2411yescharacteristic
162#6106—44no—
163#6106—44no—
164#6099—22no—
165#6099—22no—
166#6099—42no—
167C2×#1055—22no—
168C2×#1055—22no—
169C2×#1055—42no—
170C2×#1055C2×D412yes—
171C2×#1055C2×D412yes—
172#3138—131no—
173#3104D1311yescharacteristic, complemented
174C2×#460—131no—
175Q8×M16—131no—
176#1702—132no—
177#1698—132no—
178#1682—132no—
179#1678—132no—
180#1565—131no—
181#1550—131no—
182#1545—131no—
183C23×C13#12111yescharacteristic
184C22×D13—42no—
185C22×D13—82no—
186C22×D13—82no—
187C22×D13—82no—
188Q8×C13—21no—
189Q8×C13—21no—
190Q8×C13—21no—
191Q8×C13—42no—
192Q8×C13—41no—
193Q8×C13—41no—
194Q8×C13—81no—
195Q8×C13AGL(1,ℤ/8)12yes—
196C2×C4×C13—21no—
197C2×C4×C13—21no—
198C2×C4×C13—21no—
199C2×C4×C13—21no—
200C2×C4×C13—21no—
201C2×C4×C13—21no—
202C2×C4×C13—41no—
203C2×C4×C13—42no—
204C2×C4×C13—41no—
205C2×C4×C13—41no—
206C2×C4×C13—81no—
207C2×C4×C13#12111yescharacteristic
208C2×C4×C13#12111yescharacteristic
209C2×C4×C13#12111yescharacteristic
210C2×C4×C13#13512yes—
211C2×C4×C13#13712yes—
212C2×C4×C13C22×D411yescharacteristic
213C2×C4×C13C22×D411yescharacteristic
214C2×C4×C13C2×#4111yescharacteristic
215D52—42no—
216D52—42no—
217D52—42no—
218D52—42no—
219D52—82no—
220D52—82no—
221D52—82no—
222D52—82no—
223C8×C13—41no—
224C8×C13—41no—
225#1055—22no—
226#1055—22no—
227#1055—42no—
228#1055—42no—
229C22×SD16—261no—
230#533C2×D1311yescharacteristic
231C2×#127—131no—
232C2×C4×Q8C2×D1311yescharacteristic
233#490—262no—
234#486—132no—
235#486—132no—
236#484—21no—
237#477C2×D1311yescharacteristic, complemented
238#477C2×D1311yescharacteristic, complemented
239#462C2×D1311yescharacteristic, complemented
240#462C2×D1311yescharacteristic, complemented
241#460—132no—
242#460—132no—
243#460—261no—
244#460—261no—
245#451—21no—
246#446—132no—
247#446—262no—
248C8×Q8—132no—
249C4×Q16—21no—
250C4×Q16—21no—
251#431C2×D1311yescharacteristic
252#423—131no—
253#423—261no—
254C2×C4⋊C8—131no—
255#418—261no—
256#418—261no—
257C2×#102—261no—
258C4×M16—131no—
259C4×M16—261no—
260#335—132no—
261#334—132no—
262#333—132no—
263#332—132no—
264#331—132no—
265#331—132no—
266C22×C13—21no—
267C22×C13—21no—
268C22×C13#46412yes—
269C22×C13C2×#12111yescharacteristic
270C2×D13—82no—
271C2×D13—82no—
272C2×D13—82no—
273C2×D13—164no—
274C4×C13—21no—
275C4×C13—22no—
276C4×C13—21no—
277C4×C13—21no—
278C4×C13—41no—
279C4×C13—41no—
280C4×C13—41no—
281C4×C13—81no—
282C4×C13—81no—
283C4×C13#46412yes—
284C4×C13C2×#13511yescharacteristic
285C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
286C4×C13C2×#13711yescharacteristic
287C4×C13#57511yescharacteristic
288C22×Q8C22⋊D1311yescharacteristic
289C22×D4—261no—
290C22×D4—261no—
291C22×D4—261no—
292C2×Q16—21no—
293C2×Q16—21no—
294C2×SD16—262no—
295C2×SD16—262no—
296C2×SD16—524no—
297C2×M16—131no—
298C2×M16—131no—
299C2×M16—261no—
300C22×C8—261no—
301C4⋊Q8C22×D1311yescharacteristic
302C4⋊Q8C22×D1311yescharacteristic
303#127—132no—
304#127—261no—
305#127—261no—
306#127—261no—
307#125—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8C22⋊D1311yescharacteristic
315C4×Q8C22⋊D1311yescharacteristic
316C4×Q8C22×D1311yescharacteristic
317C4×Q8C22×D1311yescharacteristic
318#117—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C22⋊D1311yescharacteristic
321C2×C42—21no—
322C2×C42C22×D1311yescharacteristic
323#107—21no—
324#107—21no—
325C4⋊C8—132no—
326C4⋊C8—132no—
327C4⋊C8—262no—
328C4⋊C8—262no—
329C4⋊C8—261no—
330C4⋊C8—261no—
331C4⋊C8C22×D1311yescharacteristic
332C4⋊C8C22×D1311yescharacteristic
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337#102—262no—
338#102—262no—
339#102—262no—
340#102—262no—
341#102—262no—
342#102—262no—
343C22⋊C8—21no—
344#97—132no—
345#97—262no—
346C4×C8—21no—
347C4×C8—132no—
348C4×C8—262no—
349C2×C13—22no—
350C2×C13C2×#46411yescharacteristic
351C2×C13#305511yescharacteristic
352C2×C13#306811yescharacteristic
353D13—162no—
354C24—521no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—41no—
360C2×Q8C2×C22⋊D1311yescharacteristic
361C2×Q8C2×C22⋊D1311yescharacteristic
362C2×D4—262no—
363C2×D4—262no—
364C2×D4—262no—
365C2×D4—522no—
366C2×D4—522no—
367C2×D4—522no—
368C2×D4—522no—
369C2×D4—521no—
370C2×D4—521no—
371C2×D4—521no—
372C2×D4—1042no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4D4×D1311yescharacteristic
376C22×C4#612911yescharacteristic
377C22×C4C2×C22⋊D1311yescharacteristic
378Q16—42no—
379Q16—42no—
380SD16—524no—
381SD16—524no—
382M16—262no—
383M16—262no—
384M16—522no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—132no—
388C2×C8—132no—
389C2×C8—262no—
390C2×C8—262no—
391C2×C8—522no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4C2×C22⋊D1311yescharacteristic
404C4⋊C4C2×C22⋊D1311yescharacteristic
405C22⋊C4—41no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—41no—
411C42D4×D1311yescharacteristic
412C42#612911yescharacteristic
413C42C2×C22⋊D1311yescharacteristic
414C42C23×D1311yescharacteristic
415C13#1917411yescharacteristic, complemented, Sylow
416C23—522no—
417C23—1042no—
418C23—1042no—
419C23—1042no—
420C23#8819311yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—41no—
426Q8—41no—
427Q8—81no—
428Q8#8820312yes—
429D4—522no—
430D4—522no—
431D4—522no—
432D4—522no—
433D4—1042no—
434D4—1042no—
435D4—1042no—
436D4—1042no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4#8817212yes—
449C2×C4#8817312yes—
450C2×C4#8819311yescharacteristic
451C2×C4#8819311yescharacteristic
452C2×C4#8819311yescharacteristic
453C2×C4C2×D4×D1311yescharacteristic
454C2×C4C2×#612911yescharacteristic
455C2×C4C22×C22⋊D1311yescharacteristic
456C8—41no—
457C8—41no—
458C8—262no—
459C8—262no—
460C8—522no—
461C8—522no—
462C22—21no—
463C22—21no—
464C22—1042no—
465C22—1042no—
466C22—1042no—
467C22—2084no—
468C22#1171884612yes—
469C22C2×#8819311yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—81no—
478C4—81no—
479C4#1171884612yes—
480C4C2×#8817211yescharacteristic
481C4C2×#8817311yescharacteristic
482C4C2×#8820311yescharacteristic
483C4#1171952711yescharacteristic
484C2—22no—
485C2—2082no—
486C2C2×#1171884611yescharacteristic
487C2#1312445711yescharacteristic
488C2#1312456611yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3104C2
socleC22×C13C2×#121
prime radicalC2×#88128C22
hypercenter#3104D13
hypocenterC13#19174
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. Q8×#6099C2semidirect
4. #13114406C2semidirect
5. #13114378C2semidirect
6. #13114282C2semidirect
7. #13114254C2semidirect
8. #13113589C2semidirect
9. #13113526C2semidirect
11. C13×#3104C2semidirect
17. C13×#477C22semidirect
18. C13×#477C22semidirect
19. C13×#462C22semidirect
20. C13×#462C22semidirect
33. #11718685C22semidirect
34. Q8×#1055C22semidirect
43. #11718139C22semidirect
44. #11718138C22semidirect
47. #11718072C22semidirect
48. #11718071C22semidirect
173. #3104D13semidirect
237. #477C2×D13semidirect
238. #477C2×D13semidirect
239. #462C2×D13semidirect
240. #462C2×D13semidirect
415. C13#19174semidirect

Maximal subgroups

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

subgroupindex
#131252122
C2×#117186882
Q8×#60992
#131144062
#131143782
#131142822
#131142542
#131135892
#131135262
#131135032
C13×#31042
#1917413