Finite Group Databasebeta

← #15880097

Subgroups — #15880097

Order 3328.

1190 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88049C211yesmaximal, characteristic
2#13117713C211yesmaximal, characteristic, complemented
3#13117656C211yesmaximal, characteristic
4#13115520C212yesmaximal, complemented
5#13115461C212yesmaximal, complemented
6#13115098C212yesmaximal, complemented
7#13114859C211yesmaximal, characteristic, complemented
8#13114768C211yesmaximal, characteristic
9#13114650C212yesmaximal, complemented
10#13114610C211yesmaximal, characteristic, complemented
11C13×#1824C211yesmaximal, characteristic, complemented
12C23×D52C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C23×#1055C2211yescharacteristic
15C2×#88147C2212yes—
16C2×C22⋊C4×D13—21no—
17C2×C13×C22⋊C8C2211yescharacteristic
18C13×#380C2211yescharacteristic
19#11718873—21no—
20#11718873C2211yescharacteristic
21C4×C22⋊Dic13—21no—
22C4×C22⋊Dic13C2211yescharacteristic
23C2×#88061C2212yes—
24C2×#88061C2212yes—
25C2×C8×Dic13C2212yescomplemented
26#11718649C2212yes—
27#11718646C2212yescomplemented
28#11718642—22no—
29C4⋊C4×Dic13C2212yescomplemented
30C2×#88049C418yescomplemented
31C2×#88049C2211yescharacteristic
32C2×#88049C2211yescharacteristic
33C2×#88049C2211yescharacteristic
34C2×#88049C2211yescharacteristic
35C2×#88048C2212yescomplemented
36#11718616—21no—
37#11718326—22no—
38C13×#340C2212yescomplemented
39C13×#340C2212yescomplemented
40#11718142C2212yescomplemented
41#11718121C2212yescomplemented
42C24×D13—21no—
43C23×C4×C13—21no—
44C23×C4×C13C2311yescharacteristic
45C22×D52—24no—
46C22×D52—21no—
47C22×D52—22no—
48C22×D52—42no—
49C22×D52C2×C414yes—
50C22×D52C2311yescharacteristic
51C22×D52C2311yescharacteristic
52C22×C4×D13—22no—
53C22×C8×C13C2312yes—
54C2×C13×C4⋊C4—21no—
55C2×C13×C4⋊C4—22no—
56C2×C13×C4⋊C4C2×C414yescomplemented
57C2×C13×C4⋊C4C2312yes—
58C2×C13×C4⋊C4C2311yescharacteristic
59C2×C13×C4⋊C4C2311yescharacteristic
60C2×C13×C22⋊C4—21no—
61C2×C22⋊Dic13—21no—
62C2×C22⋊Dic13D411yescharacteristic
63C2×C22⋊Dic13D411yescharacteristic
64C2×#6103—22no—
65C2×#6103—22no—
66C2×#6103—22no—
67C2×C4⋊Dic13—22no—
68C2×C4⋊Dic13C2312yes—
69C2×C4×Dic13—22no—
70C2×C4×Dic13C2312yes—
71C22×#1055—42no—
72C22×#1055C2×C414yescomplemented
73C22×#1055C2311yescharacteristic
74C22×#1055C2311yescharacteristic
75#88147—24no—
76#88147—42no—
77C22⋊C4×D13—44no—
78C13×C22⋊C8—22no—
79C13×#95—21no—
80#88071—21no—
81#88071—22no—
82#88071—22no—
83#88071D412yes—
84#88061—24no—
85#88061—42no—
86C8×Dic13—42no—
87#88049C2×C414yes—
88#88049C2×C414yes—
89#88049C2×C414yescomplemented
90#88049C2×C414yescomplemented
91#88048—42no—
92#22574—131nomaximal, Sylow
93C24×C13C2×D411yescharacteristic
94C23×D13—21no—
95C23×D13—21no—
96C23×D13—24no—
97C23×D13—44no—
98C23×D13—42no—
99C23×D13—42no—
100C23×D13—82no—
101C22×C4×C13—22no—
102C22×C4×C13—22no—
103C22×C4×C13—21no—
104C22×C4×C13—21no—
105C22×C4×C13—24no—
106C22×C4×C13—22no—
107C22×C4×C13—82no—
108C22×C4×C13C22×C412yes—
109C22×C4×C13C2×D411yescharacteristic
110C22×C4×C13#4112yes—
111C22×C4×C13C2411yescharacteristic
112C22×Dic13—22no—
113C22×Dic13C2×D412yes—
114C22×Dic13C2×D412yes—
115C2×D52—21no—
116C2×D52—22no—
117C2×D52—22no—
118C2×D52—22no—
119C2×D52—28no—
120C2×D52—24no—
121C2×D52—21no—
122C2×D52—44no—
123C2×D52—48no—
124C2×D52—44no—
125C2×D52C22×C412yes—
126C2×D52C22×C412yes—
127C2×D52#4114yes—
128C2×C4×D13—44no—
129C2×C4×D13—44no—
130C2×C4×D13—82no—
131C2×C8×C13—22no—
132C2×C8×C13—22no—
133C2×C8×C13—22no—
134C13×C4⋊C4—24no—
135C13×C4⋊C4—21no—
136C13×C4⋊C4—21no—
137C13×C4⋊C4—22no—
138C13×C4⋊C4—22no—
139C13×C4⋊C4C22×C412yes—
140C13×C4⋊C4C22×C412yes—
141C13×C22⋊C4—42no—
142C22⋊Dic13—22no—
143C22⋊Dic13—42no—
144C22⋊Dic13SD1614yes—
145#6103—44no—
146#6103—44no—
147#6103—82no—
148C4⋊Dic13—22no—
149C4⋊Dic13—22no—
150C4⋊Dic13—82no—
151C4×Dic13—22no—
152C4×Dic13—22no—
153C4×Dic13—82no—
154C2×#1055—22no—
155C2×#1055—22no—
156C2×#1055—44no—
157C2×#1055C22×C412yes—
158C2×#1055C22×C412yes—
159C22×#102—131no—
160#2345—131no—
161D4×C22⋊C4—131no—
162#1934—132no—
163#1926—132no—
164#1864—132no—
165#1824D1311yescharacteristic, complemented
166#1807—131no—
167#1786—131no—
168C4×#102—132no—
169C8×C22⋊C4—131no—
170C23×C13—22no—
171C23×C13—22no—
172C23×C13C4×D412yes—
173C23×C13C2×SD1612yes—
174C23×C13#13512yes—
175C23×C13C22×D411yescharacteristic
176C22×D13—22no—
177C22×D13—24no—
178C22×D13—22no—
179C22×D13—48no—
180C22×D13—41no—
181C22×D13—41no—
182C22×D13—42no—
183C22×D13—42no—
184C22×D13—41no—
185C22×D13—41no—
186C22×D13—42no—
187C22×D13—48no—
188C22×D13—84no—
189C22×D13—88no—
190C22×D13—88no—
191C2×C4×C13—22no—
192C2×C4×C13—22no—
193C2×C4×C13—22no—
194C2×C4×C13—24no—
195C2×C4×C13—22no—
196C2×C4×C13—22no—
197C2×C4×C13—42no—
198C2×C4×C13—42no—
199C2×C4×C13—42no—
200C2×C4×C13—42no—
201C2×C4×C13—42no—
202C2×C4×C13—84no—
203C2×C4×C13C4×D412yes—
204C2×C4×C13C23×C411yescharacteristic
205C2×C4×C13C2×#4112yes—
206C2×C4×C13C2×#4111yescharacteristic
207C2×C4×C13H(5,𝔽2)12yes—
208C2×Dic13—22no—
209C2×Dic13—22no—
210C2×Dic13—22no—
211C2×Dic13—22no—
212C2×Dic13—22no—
213C2×Dic13—42no—
214C2×Dic13—42no—
215C2×Dic13—42no—
216C2×Dic13C2×SD1614yes—
217D52—22no—
218D52—22no—
219D52—22no—
220D52—24no—
221D52—48no—
222D52—42no—
223D52#11714yes—
224C4×D13—88no—
225C8×C13—42no—
226#1055—22no—
227#1055—22no—
228C23×D4—131no—
229C23×C8—131no—
230C2×C4×D4—132no—
231C22×C4⋊C4C2×D1311yescharacteristic
232C22×C22⋊C4—261no—
233C2×#106—132no—
234C2×#102—138no—
235C2×#102—131no—
236C2×#102—131no—
237C2×#102—131no—
238C2×#102—131no—
239C2×#102—132no—
240C2×#102—132no—
241C2×C22⋊C8C2×D1311yescharacteristic
242C2×C4×C8—132no—
243#390—132no—
244#389—132no—
245#387—262no—
246#386—262no—
247#380—131no—
248#380—261no—
249#380C2×D1311yescharacteristic
250#379—261no—
251C4×C4⋊C4—132no—
252C4×C22⋊C4—131no—
253C4×C22⋊C4—261no—
254#340—132no—
255#340C2×D1312yescomplemented
256#340C2×D1312yescomplemented
257#336—132no—
258C22×C13—22no—
259C22×C13—22no—
260C22×C13—24no—
261C22×C13—24no—
262C22×C13C4×SD1614yes—
263C22×C13C2×C4×D411yescharacteristic
264C22×C13#54612yes—
265C22×C13C22×SD1611yescharacteristic
266C22×C13C2×#13511yescharacteristic
267C22×C13#57612yes—
268C2×D13—24no—
269C2×D13—42no—
270C2×D13—42no—
271C2×D13—44no—
272C2×D13—44no—
273C2×D13—42no—
274C2×D13—42no—
275C2×D13—42no—
276C2×D13—816no—
277C2×D13—82no—
278C2×D13—82no—
279C4×C13—22no—
280C4×C13—42no—
281C4×C13—42no—
282C4×C13—82no—
283C4×C13C2×#11711yescharacteristic
284C4×C13#51811yescharacteristic
285C4×C13#55112yes—
286Dic13—24no—
287Dic13—42no—
288Dic13—82no—
289C25—261no—
290C22×D4—134no—
291C22×D4—131no—
292C22×D4—131no—
293C22×D4—264no—
294C22×D4—261no—
295C22×D4—262no—
296C22×D4—522no—
297C23×C4—21no—
298C23×C4—262no—
299C23×C4C22×D1311yescharacteristic
300C22×C8—131no—
301C22×C8—131no—
302C22×C8—134no—
303C22×C8—522no—
304C22×C8C22×D1312yes—
305C4×D4—264no—
306C4×D4—522no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—132no—
310C2×C4⋊C4—262no—
311C2×C4⋊C4C4×D1314yescomplemented
312C2×C4⋊C4C22×D1312yes—
313C2×C4⋊C4C22×D1311yescharacteristic
314C2×C4⋊C4C22×D1311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—131no—
317C2×C22⋊C4—131no—
318C2×C22⋊C4—262no—
319C2×C22⋊C4—262no—
320C2×C22⋊C4—261no—
321C2×C22⋊C4—262no—
322C2×C22⋊C4—524no—
323C2×C42—132no—
324C2×C42—262no—
325#106—522no—
326#102—134no—
327#102—134no—
328#102—134no—
329#102—134no—
330#102—264no—
331#102—522no—
332C22⋊C8—22no—
333C4×C8—522no—
334#95—21no—
335#95—132no—
336#95—262no—
337#95—262no—
338#95—261no—
339C2×C13—24no—
340C2×C13#234211yescharacteristic
341C2×C13C2×C4×SD1611yescharacteristic
342C2×C13#294811yescharacteristic
343C2×C13#329512yes—
344C2×C13#329712yes—
345D13—44no—
346D13—82no—
347C24—261no—
348C24—261no—
349C24—264no—
350C24—524no—
351C24—522no—
352C24—522no—
353C24—1042no—
354C24D4×D1311yescharacteristic
355C2×D4—132no—
356C2×D4—132no—
357C2×D4—134no—
358C2×D4—268no—
359C2×D4—264no—
360C2×D4—261no—
361C2×D4—261no—
362C2×D4—262no—
363C2×D4—262no—
364C2×D4—262no—
365C2×D4—524no—
366C2×D4—528no—
367C2×D4—524no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—82no—
375C22×C4—132no—
376C22×C4—132no—
377C22×C4—262no—
378C22×C4—524no—
379C22×C4—524no—
380C22×C4—1042no—
381C22×C4C2×C4×D1312yes—
382C22×C4D4×D1311yescharacteristic
383C22×C4#612912yes—
384C22×C4C23×D1311yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—132no—
389C2×C8—132no—
390C2×C8—262no—
391C2×C8—262no—
392C2×C8—524no—
393C4⋊C4—24no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—262no—
399C4⋊C4—262no—
400C4⋊C4—1042no—
401C4⋊C4C2×C4×D1312yes—
402C4⋊C4C2×C4×D1312yes—
403C22⋊C4—42no—
404C22⋊C4—134no—
405C22⋊C4—262no—
406C22⋊C4—524no—
407C22⋊C4—524no—
408C22⋊C4—522no—
409C22⋊C4—1042no—
410C42—262no—
411C42—262no—
412C42—1042no—
413C13#2257411yescharacteristic, complemented, Sylow
414C23—22no—
415C23—22no—
416C23—262no—
417C23—264no—
418C23—262no—
419C23—528no—
420C23—521no—
421C23—521no—
422C23—522no—
423C23—521no—
424C23—521no—
425C23—528no—
426C23—522no—
427C23—522no—
428C23—1044no—
429C23—1048no—
430C23—1048no—
431C23#8813512yes—
432C23SD16×D1312yes—
433C23#8817312yes—
434C23C2×D4×D1311yescharacteristic
435D4—134no—
436D4—262no—
437D4—262no—
438D4—262no—
439D4—264no—
440D4—528no—
441D4—522no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—84no—
454C2×C4—134no—
455C2×C4—262no—
456C2×C4—262no—
457C2×C4—262no—
458C2×C4—262no—
459C2×C4—262no—
460C2×C4—522no—
461C2×C4—522no—
462C2×C4—522no—
463C2×C4—1048no—
464C2×C4#8813512yes—
465C2×C4C22×C4×D1311yescharacteristic
466C2×C4C2×#612911yescharacteristic
467C2×C4#41×D1312yes—
468C2×C4#8825612yes—
469C8—42no—
470C8—262no—
471C8—262no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—264no—
477C22—524no—
478C22—522no—
479C22—524no—
480C22—522no—
481C22—522no—
482C22—522no—
483C22—522no—
484C22—10416no—
485C22—1042no—
486C22—1042no—
487C22#1171844914yes—
488C22C2×#8813511yescharacteristic
489C22#1171935912yes—
490C22C2×SD16×D1311yescharacteristic
491C22C2×#8817311yescharacteristic
492C22#1171947912yes—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—264no—
498C4—522no—
499C4—1042no—
500C4D13×#11711yescharacteristic
501C4#1171922611yescharacteristic
502C4#1171936312yes—
503C2—24no—
504C2—524no—
505C2—1042no—
506C2#1311767811yescharacteristic
507C2C2×#1171844911yescharacteristic
508C2#1312368411yescharacteristic
509C2#1312665812yes—
510C2#1312668412yes—
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1824C2
socleC23×C13C22×D4
prime radicalC23×D52C22
cosocleC22×C4×C13C24
hypercenter#1824D13
hypocenterC13#22574
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. #13117713C2semidirect
4. #13115520C2semidirect
5. #13115461C2semidirect
6. #13115098C2semidirect
7. #13114859C2semidirect
9. #13114650C2semidirect
10. #13114610C2semidirect
11. C13×#1824C2semidirect
25. C2×C8×Dic13C22semidirect
27. #11718646C22semidirect
29. C4⋊C4×Dic13C22semidirect
30. C2×#88049C4semidirect
35. C2×#88048C22semidirect
38. C13×#340C22semidirect
39. C13×#340C22semidirect
40. #11718142C22semidirect
41. #11718121C22semidirect
56. C2×C13×C4⋊C4C2×C4semidirect
72. C22×#1055C2×C4semidirect
89. #88049C2×C4semidirect
90. #88049C2×C4semidirect
165. #1824D13semidirect
255. #340C2×D13semidirect
256. #340C2×D13semidirect
311. C2×C4⋊C4C4×D13semidirect
413. C13#22574semidirect

Maximal subgroups

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

subgroupindex
C22×#880492
#131177132
#131176562
#131155202
#131154612
#131150982
#131148592
#131147682
#131146502
#131146102
C13×#18242
#2257413