Finite Group Databasebeta

← #15846693

Subgroups — #15846693

Order 3328.

744 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13124629C211yesmaximal, characteristic, complemented
2#13124296C211yesmaximal, characteristic
3C2×C8×D52C211yesmaximal, characteristic, complemented
4#13114256C212yesmaximal, complemented
5#13114242C212yesmaximal, complemented
6#13114208C212yesmaximal, complemented
7#13114174C212yesmaximal, complemented
8#13113592C211yesmaximal, characteristic, complemented
9#13113526C211yesmaximal, characteristic, complemented
10#13113513C211yesmaximal, characteristic, complemented
11C13×#3167C211yesmaximal, characteristic, complemented
12C13×#533C2211yescharacteristic
13C13×#532C2211yescharacteristic
14C2×#88203—21no—
15C22×C8×D13—21no—
16C2×C4×D52C2211yescharacteristic
17C13×#498C2211yescharacteristic, complemented
18C13×#492C2211yescharacteristic, complemented
19C13×#487C2211yescharacteristic, complemented
20C13×#486C2211yescharacteristic, complemented
21C13×#465—21no—
22C13×#465—21no—
23C4×C13×SD16—21no—
24C4×C13×SD16—21no—
25C2×C4×C8×C13C2211yescharacteristic
26#11718893—21no—
27C2×#88060—21no—
28#11718748C2212yes—
29#11718745C2212yes—
30#11718744C2212yes—
31#11718733C2212yes—
32#11718727—22no—
33#11718718—22no—
34#11718662—21no—
35#11718656—21no—
36#11718585C2211yescharacteristic
37#11718585C2211yescharacteristic
38C2×C4⋊#1055C2211yescharacteristic
39C8×D52—21no—
40C8×D52—21no—
41C8×D52C2212yes—
42C8×D52C2212yes—
43#11718139C2212yescomplemented
44#11718139C2212yescomplemented
45#11718138C2212yescomplemented
46#11718136C2212yescomplemented
47#11718049C2212yescomplemented
48#11718049C2212yescomplemented
49C2×C13×#41—21no—
50C22×D52D411yescharacteristic
51C22×C4×D13—21no—
52C2×C13×SD16—21no—
53C2×C13×SD16—21no—
54C22×C8×C13D411yescharacteristic
55C22×C8×C13D411yescharacteristic
56C13×C4⋊Q8C2311yescharacteristic
57C13×C4⋊Q8C2311yescharacteristic
58C13×C4⋊Q8C2311yescharacteristic
59C13×#127C2311yescharacteristic
60C13×#125—21no—
61C13×#124—21no—
62C13×C22⋊Q8—21no—
63C13×C22⋊Q8—21no—
64C13×#121—21no—
65C13×#121—21no—
66C4×Q8×C13—21no—
67C4×Q8×C13—21no—
68C4×D4×C13—21no—
69C4×D4×C13—21no—
70C13×#117—21no—
71C2×C42×C13C2311yescharacteristic
72#88203—44no—
73C2×#6106—22no—
74C2×#6104—22no—
75C2×#6103—21no—
76C2×C4⋊Dic13D411yescharacteristic
77C2×#6099—21no—
78C2×#6099—21no—
79C22×#1055—21no—
80C2×C8×D13—22no—
81C2×C8×D13—22no—
82C2×C8×D13—44no—
83C4×D52—21no—
84C4×D52—21no—
85C4×D52D412yes—
86C4×D52C2312yes—
87C13×#106—21no—
88C13×#106—21no—
89C13×#103—21no—
90C13×#103—21no—
91C13×#102—21no—
92C13×#102—21no—
93C4×C8×C13D411yescharacteristic
94C4×C8×C13D411yescharacteristic
95C4×C8×C13C2311yescharacteristic
96C4×C8×C13C2311yescharacteristic
97#88075—22no—
98#88072—22no—
99#88060—22no—
100#88060—22no—
101#88049—22no—
102#88049—22no—
103#88048—22no—
104#88047—22no—
105C4⋊#1055—21no—
106C4⋊#1055—21no—
107C4⋊#1055—21no—
108C4⋊#1055C2312yes—
109C4⋊#1055C2312yes—
110C4⋊#1055C2312yes—
111#19254—131nomaximal, Sylow
112C23×D13—21no—
113C13×#41—42no—
114C13×#41—42no—
115C2×Q8×C13—21no—
116C2×Q8×C13—21no—
117C2×Q8×C13—21no—
118C2×D4×C13—21no—
119C2×D4×C13—21no—
120C2×D4×C13—21no—
121C2×D4×C13—41no—
122C22×C4×C13—41no—
123C22×C4×C13C2×D411yescharacteristic
124C22×C4×C13C2×D411yescharacteristic
125C22×C4×C13#4111yescharacteristic
126C22×Dic13—21no—
127C2×D52—21no—
128C2×D52—22no—
129C2×D52SD1614yescomplemented
130C2×D52C2×D412yes—
131C2×C4×D13—22no—
132C2×C4×D13—22no—
133C2×C4×D13—42no—
134C2×C4×D13—42no—
135C13×SD16—42no—
136C13×SD16—42no—
137C2×C8×C13—22no—
138C2×C8×C13—41no—
139C2×C8×C13C2×D411yescharacteristic
140C2×C8×C13C2×D411yescharacteristic
141C2×C8×C13C2×D411yescharacteristic
142C2×C8×C13C2×D411yescharacteristic
143C13×C4⋊C4—21no—
144C13×C4⋊C4—21no—
145C13×C4⋊C4—21no—
146C13×C4⋊C4—21no—
147C13×C4⋊C4—21no—
148C13×C4⋊C4—21no—
149C13×C4⋊C4—41no—
150C13×C4⋊C4—41no—
151C13×C22⋊C4—41no—
152C13×C22⋊C4—41no—
153C13×C22⋊C4—41no—
154C42×C13—41no—
155C42×C13C2×D411yescharacteristic
156C42×C13C2×D411yescharacteristic
157C42×C13#4111yescharacteristic
158C42×C13C2411yescharacteristic
159#6106—44no—
160#6104—44no—
161#6103—22no—
162#6103—22no—
163C4⋊Dic13—21no—
164C4⋊Dic13C2×D412yes—
165#6099—42no—
166#6099—42no—
167C2×#1055—22no—
168C2×#1055—22no—
169C2×#1055—22no—
170C2×#1055—42no—
171C8×D13—44no—
172C8×D13—44no—
173#3167D1311yescharacteristic, complemented
174#3077—131no—
175#3036—131no—
176C2×C8×D4—131no—
177#1679—132no—
178#1677—132no—
179#1671—132no—
180#1665—132no—
181#1566—131no—
182#1550—131no—
183#1547—131no—
184C23×C13—41no—
185C23×C13#12111yescharacteristic
186C22×D13—22no—
187C22×D13—24no—
188C22×D13—42no—
189C22×D13—42no—
190Q8×C13—41no—
191Q8×C13—41no—
192Q8×C13—41no—
193Q8×C13—41no—
194Q8×C13—81no—
195D4×C13—41no—
196D4×C13—41no—
197D4×C13—41no—
198D4×C13—41no—
199D4×C13—81no—
200D4×C13—82no—
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—82no—
208C2×C4×C13#12111yescharacteristic
209C2×C4×C13#12111yescharacteristic
210C2×C4×C13#12111yescharacteristic
211C2×C4×C13C2×SD1612yes—
212C2×C4×C13#13512yes—
213C2×C4×C13C22×D411yescharacteristic
214C2×C4×C13C22×D411yescharacteristic
215C2×C4×C13C2×#4111yescharacteristic
216C2×Dic13—22no—
217C2×Dic13—42no—
218D52—22no—
219D52—24no—
220D52C2×SD1614yes—
221C4×D13—44no—
222C4×D13—42no—
223C4×D13—42no—
224C8×C13—21no—
225C8×C13—21no—
226C8×C13—21no—
227C8×C13AGL(1,ℤ/8)12yescomplemented
228#1055—42no—
229#1055—42no—
230#1055—42no—
231C2×AGL(1,ℤ/8)—261no—
232C23×C8—261no—
233#533C2×D1311yescharacteristic
234#532C2×D1311yescharacteristic
235C2×C4×D4—131no—
236#498C2×D1311yescharacteristic, complemented
237#492C2×D1311yescharacteristic, complemented
238#487C2×D1311yescharacteristic, complemented
239#486C2×D1311yescharacteristic, complemented
240#478—262no—
241#476—132no—
242#474—132no—
243#465—21no—
244#465—21no—
245#463—262no—
246#461—132no—
247#459—132no—
248C4×SD16—21no—
249C4×SD16—21no—
250C8×D4—132no—
251C8×D4—132no—
252C8×D4—261no—
253C8×D4—261no—
254#428—261no—
255#423—131no—
256#423—131no—
257C2×C4⋊C8—131no—
258#418—261no—
259#417—261no—
260C2×C22⋊C8—261no—
261C2×C4×C8C2×D1311yescharacteristic
262#334—132no—
263#334—132no—
264#333—132no—
265#332—132no—
266#332—132no—
267#331—132no—
268C22×C13—21no—
269C22×C13—21no—
270C22×C13—81no—
271C22×C13—81no—
272C22×C13—81no—
273C22×C13#46512yes—
274C22×C13C2×#12111yescharacteristic
275C2×D13—24no—
276C2×D13—48no—
277C2×D13—42no—
278C2×D13—42no—
279C4×C13—21no—
280C4×C13—21no—
281C4×C13—21no—
282C4×C13—81no—
283C4×C13—81no—
284C4×C13—81no—
285C4×C13#46512yes—
286C4×C13C22×SD1611yescharacteristic
287C4×C13C2×#13511yescharacteristic
288C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
289C4×C13#57511yescharacteristic
290Dic13—42no—
291C2×#41—21no—
292C22×D4—131no—
293C23×C4—261no—
294AGL(1,ℤ/8)—524no—
295C2×SD16—21no—
296C2×SD16—21no—
297C2×SD16—262no—
298C2×D8—262no—
299C2×M16—261no—
300C2×M16—261no—
301C22×C8—261no—
302C22×C8—262no—
303C22×C8—262no—
304C22×C8—524no—
305C22×C8C22⋊D1311yescharacteristic
306C22×C8C22⋊D1311yescharacteristic
307C4⋊Q8C22×D1311yescharacteristic
308C4⋊Q8C22×D1311yescharacteristic
309C4⋊Q8C22×D1311yescharacteristic
310#127C22×D1311yescharacteristic
311#125—21no—
312#124—21no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315#121—21no—
316#121—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—132no—
322C4×D4—132no—
323C4×D4—261no—
324C4×D4—261no—
325#117—21no—
326C2×C4⋊C4—131no—
327C2×C22⋊C4—261no—
328C2×C42C22×D1311yescharacteristic
329#107—262no—
330#106—21no—
331#106—21no—
332#106—262no—
333C4⋊C8—132no—
334C4⋊C8—132no—
335C4⋊C8—132no—
336C4⋊C8—261no—
337C4⋊C8—261no—
338C4⋊C8—261no—
339#103—21no—
340#103—21no—
341#103—262no—
342#102—21no—
343#102—21no—
344#102—262no—
345#102—262no—
346#102—262no—
347C22⋊C8—262no—
348C22⋊C8—262no—
349C4×C8C22⋊D1311yescharacteristic
350C4×C8C22⋊D1311yescharacteristic
351C4×C8C22×D1311yescharacteristic
352C4×C8C22×D1311yescharacteristic
353C2×C13—22no—
354C2×C13—161no—
355C2×C13#303611yescharacteristic
356C2×C13C2×#46511yescharacteristic
357C2×C13#306711yescharacteristic
358D13—44no—
359C24—261no—
360#41—42no—
361#41—42no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—134no—
370C2×D4—132no—
371C2×D4—262no—
372C2×D4—261no—
373C22×C4—41no—
374C22×C4—261no—
375C22×C4—262no—
376C22×C4—262no—
377C22×C4—522no—
378C22×C4—522no—
379C22×C4D4×D1311yescharacteristic
380C22×C4#612911yescharacteristic
381C22×C4C2×C22⋊D1311yescharacteristic
382SD16—42no—
383SD16—42no—
384SD16—524no—
385D8—524no—
386M16—522no—
387M16—522no—
388C2×C8—22no—
389C2×C8—41no—
390C2×C8—262no—
391C2×C8—262no—
392C2×C8—262no—
393C2×C8—524no—
394C2×C8—522no—
395C2×C8—524no—
396C2×C8C2×C22⋊D1311yescharacteristic
397C2×C8C2×C22⋊D1311yescharacteristic
398C2×C8C2×C22⋊D1311yescharacteristic
399C2×C8C2×C22⋊D1311yescharacteristic
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—132no—
409C4⋊C4—261no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—262no—
414C22⋊C4—262no—
415C42—41no—
416C42D4×D1311yescharacteristic
417C42#612911yescharacteristic
418C42C2×C22⋊D1311yescharacteristic
419C42C23×D1311yescharacteristic
420C13#1925411yescharacteristic, complemented, Sylow
421C23—41no—
422C23—262no—
423C23—264no—
424C23—522no—
425C23—522no—
426C23#8819211yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—81no—
437D4—82no—
438D4—134no—
439D4—262no—
440D4—264no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—262no—
449C2×C4—524no—
450C2×C4—522no—
451C2×C4—522no—
452C2×C4—522no—
453C2×C4SD16×D1312yes—
454C2×C4#8817012yes—
455C2×C4#8819211yescharacteristic
456C2×C4#8819211yescharacteristic
457C2×C4#8819211yescharacteristic
458C2×C4C2×D4×D1311yescharacteristic
459C2×C4C2×#612911yescharacteristic
460C2×C4C22×C22⋊D1311yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—522no—
465C8—522no—
466C8—522no—
467C8#8820312yescomplemented
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—264no—
474C22—528no—
475C22—522no—
476C22—522no—
477C22#1171883212yes—
478C22C2×#8819211yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—522no—
486C4#1171883212yes—
487C4C2×SD16×D1311yescharacteristic
488C4C2×#8817011yescharacteristic
489C4C2×#8820311yescharacteristic
490C4#1171952711yescharacteristic
491C2—22no—
492C2—161no—
493C2—524no—
494C2#1312429611yescharacteristic
495C2C2×#1171883211yescharacteristic
496C2#1312455211yescharacteristic
subgroupG/H
centerC22C2×#88192
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3167C2
socleC22×C13C2×#121
prime radical#13124296C2
hypercenter#3167D13
hypocenterC13#19254
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #13124629C2semidirect
3. C2×C8×D52C2semidirect
4. #13114256C2semidirect
5. #13114242C2semidirect
6. #13114208C2semidirect
7. #13114174C2semidirect
8. #13113592C2semidirect
9. #13113526C2semidirect
10. #13113513C2semidirect
11. C13×#3167C2semidirect
17. C13×#498C22semidirect
18. C13×#492C22semidirect
19. C13×#487C22semidirect
20. C13×#486C22semidirect
43. #11718139C22semidirect
44. #11718139C22semidirect
45. #11718138C22semidirect
46. #11718136C22semidirect
47. #11718049C22semidirect
48. #11718049C22semidirect
129. C2×D52SD16semidirect
173. #3167D13semidirect
227. C8×C13AGL(1,ℤ/8)semidirect
236. #498C2×D13semidirect
237. #492C2×D13semidirect
238. #487C2×D13semidirect
239. #486C2×D13semidirect
420. C13#19254semidirect
467. C8#88203semidirect

Maximal subgroups

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

subgroupindex
#131246292
#131242962
C2×C8×D522
#131142562
#131142422
#131142082
#131141742
#131135922
#131135262
#131135132
C13×#31672
#1925413